Spectral Structures and Topological Methods in Mathematics 9783037191972

Table of contents :
Preface......Page 6
Introduction......Page 8
Contents......Page 12
Introduction......Page 16
Symmetric random matrices......Page 20
Non-symmetric random matrices......Page 23
Local spectral distributions......Page 27
Connections between probability theory and number theory......Page 31
Analogies between classical and free probability......Page 35
References......Page 38
Fokker–Planck–Kolmogorov equations......Page 44
Three selected results on SPDEs......Page 55
References......Page 67
Analysis on manifolds......Page 70
Analysis on metric measure spaces......Page 78
Homology theory on graphs......Page 83
References......Page 86
Introduction......Page 90
Complex systems......Page 92
Markov evolutions......Page 96
Birth-and-death evolutions......Page 99
Vlasov-type scalings......Page 103
Kinetic equation......Page 107
References......Page 118
Large deviations......Page 122
Kramers' Law......Page 128
Parabolic SPDEs......Page 132
Unstable periodic orbits......Page 136
Mixed-mode oscillations......Page 140
References......Page 141
Equivariant evolution equations......Page 144
The freezing method......Page 148
Applications......Page 152
Relative Equilibria......Page 159
Nonlinear eigenvalue problems......Page 167
References......Page 171
Introduction......Page 174
Nonlinear Schrödinger equations on compact manifolds......Page 180
Nonlinear systems on Euclidean space......Page 188
References......Page 194
Introduction......Page 198
Variational solutions to the Dirichlet problem......Page 200
Ellipticity and coercivity of nonlocal operators......Page 204
(Weak) Harnack inequalities, and Hölder regularity......Page 206
References......Page 209
Introduction......Page 212
Weak model sets......Page 213
A decorated quasiperiodic tiling with mixed spectrum......Page 218
Random inflations......Page 225
Enumeration of lattices......Page 228
References......Page 232
Introduction......Page 236
Special Kähler geometry in local coordinates......Page 238
Some global aspects of special Kähler geometry on P 1......Page 245
References......Page 247
The poset of non-crossing partitions......Page 250
Non-crossing partitions in free probability......Page 255
Braid groups......Page 262
Non-crossing partitions in Coxeter groups......Page 274
The Hurwitz action......Page 280
Non-crossing partitions arising in representation theory......Page 282
Generalised Cartan lattices......Page 284
Braid group actions on exceptional sequences......Page 285
References......Page 287
Introduction......Page 290
Preliminaries......Page 291
Types of localisation......Page 294
Cohomological localisations for the projective line......Page 300
Exotic localisations......Page 307
References......Page 311
Introduction: From group theory to topology......Page 314
Brown's criterion......Page 316
Combinatorial Morse theory......Page 318
Matching complexes for graphs and surfaces......Page 320
Higher generation in symmetric groups and braid groups......Page 323
The braided Thompson group Vbr......Page 325
A cube complex for Vbr......Page 327
The Morse function and its descending links......Page 329
Connectivity of descending links......Page 331
References......Page 335
Introduction......Page 338
Zeta integrals......Page 339
The Trace formula......Page 349
Unipotent terms in the trace formula......Page 353
References......Page 357
Zeta functions associated to groups and rings......Page 360
Submodule zeta functions......Page 362
Representation zeta functions for unipotent group schemes......Page 370
References......Page 376
Conjectures of Brumer, Gross and Stark......Page 380
Preliminaries......Page 381
The abelian case......Page 383
The general case......Page 385
Relations to further conjectures and results......Page 394
References......Page 401
Introduction......Page 404
Frames......Page 405
Displays......Page 408
Classification of p-divisible groups......Page 412
The nilpotency condition......Page 413
Isogenies......Page 417
References......Page 422
List of contributors......Page 424
Index......Page 426

Citation preview

Michael Baake, Friedrich Götze and Werner Hoffmann, Editors

This book is a collection of survey articles about spectral structures and the application of topological methods bridging different mathematical disciplines, from pure to applied. The topics are based on work done in the Collaborative Research Centre (SFB) 701. Notable examples are non-crossing partitions, which connect representation theory, braid groups, non-commutative probability as well as spectral distributions of random matrices. The local distributions of such spectra are universal, also representing the local distribution of zeros of L-functions in number theory. An overarching method is the use of zeta functions in the asymptotic counting of sublattices, group representations etc. Further examples connecting probability, analysis, dynamical systems and geometry are generating operators of deterministic or stochastic processes, stochastic differential equations, and fractals, relating them to the local geometry of such spaces and the convergence to stable and semi-stable states.

ISBN 978-3-03719-197-2

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SCR Baake et al. | Egyptienne F | Pantone 116, 287 | RB 30 mm

Spectral Structures and Topological Methods in Mathematics

Spectral Structures and Topological Methods in Mathematics

Michael Baake, Friedrich Götze and Werner Hoffmann, Editors

Series of Congress Reports

Series of Congress Reports

Spectral Structures and Topological Methods in Mathematics Michael Baake Friedrich Götze Werner Hoffmann Editors

EMS Series of Congress Reports

EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowron´ski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.) Representations of Algebras and Related Topics, Andrzej Skowron´ski and Kunio Yamagata (eds.) Contributions to Algebraic Geometry. Impanga Lecture Notes, Piotr Pragacz (ed.) Geometry and Arithmetic, Carel Faber, Gavril Farkas and Robin de Jong (eds.) Derived Categories in Algebraic Geometry. Toyko 2011, Yujiro Kawamata (ed.) Advances in Representation Theory of Algebras, David J. Benson, Henning Krause and Andrzej Skowron´ski (eds.) Valuation Theory in Interaction, Antonio Campillo, Franz-Viktor Kuhlmann and Bernard Teissier (eds.) Representation Theory – Current Trends and Perspectives, Henning Krause, Peter Littelmann, Gunter Malle, Karl-Hermann Neeb and Christoph Schweigert (eds.) Functional Analysis and Operator Theory for Quantum Physics. The Pavel Exner Anniversary Volume, Jaroslav Dittrich, Hynek Kovarˇ ík and Ari Laptev (eds.) Schubert Varieties, Equivariant Cohomology and Characteristic Classes, Jarosław Buczyn´ski, Mateusz Michałek and Elisa Postinghel (eds.) Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, Fritz Gesztesy, Harald Hanche-Olsen, Espen R. Jakobsen, Yurii Lyubarskii, Nils Henrik Risebro and Kristian Seip, Editors

Spectral Structures and Topological Methods in Mathematics Michael Baake Friedrich Götze Werner Hoffmann Editors

Editors: Michael Baake Friedrich Götze Werner Hoffmann Fakultät für Mathematik Universität Bielefeld Postfach 100131 33501 Bielefeld Germany

[email protected] [email protected] [email protected]

2010 Mathematics Subject Classification: Primary 58J65, 52C23, 20F65, 11M41; secondary 46L54, 60J45, 35C07, 35Q55, 43A25, 20F36, 11R42, 14L05. Key words: Universal distributions, free probability, Markov processes, Schrödinger operators, heat kernel, spatial ecology, metastability, numerical analysis, critical regularity, aperiodic order, dynamical systems, special Kähler structure, non-crossing partitions, localising subcategory, braided groups, zeta functions, subgroup growth, representation growth, Brumer–Stark conjecture, p-divisible groups.

ISBN 978-3-03719-197-2 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2019

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Typeset using the authors’ TeX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface This book is a collection of survey articles on several fields of mathematics in which spectral structures appear and topological methods are applied. Those were the overarching themes under which a large group of researchers joined their efforts in the Collaborative Research Centre (SFB) 701 over three funding periods from 2005 until 2017. The topics span diverse mathematical disciplines from stochastics and dynamical systems via global analysis and representation theory to arithmetic geometry. Each article exposes recent results obtained by members and guests of the SFB 701 and embeds them into the general state of the art in the pertinent field. The interrelations between seemingly disparate areas are demonstrated by the introduction, by several joint papers and by various cross-references between the individual articles. For example, universal probability distributions (Chapter 1) have conjectural connections to the zero distribution of zeta functions in number theory, and the special values of those zeta functions carry arithmetic information (Chapter 16). Zeta functions are also used to study the growth of the number of representations of a group depending on the degree (Chapter 15). Noncrossing partitions appear in representation theory (Chapter 11), asymptotic distributions (Chapters 1, 11) and geometric group theory (Chapters 11, 13). Another example are generating operators of stochastic processes, which are studied in the framework of stochastic differential equations (Chapter 2), of Markov processes in continuum (Chapter 4) and of processes on more general spaces such as fractals (Chapter 3). Thus, the current volume gives insight into recent developments, and highlights the unity of mathematics. We hope that the joint index helps to enhance the usefulness of this publication. The editors would like to use this occasion to express their gratitude to the numerous people who made the SFB run smoothly, including Nadja Epp and Stephan Merkes, who managed the general administration including the visitor programme, the student assistants, who ran the IT infrastructure, and the secretaries of the members, in particular Anita Lydia Cole, who supported the speaker. Also, we express our gratitude to Britta Heitbreder for her expert LATEX work in preparing this volume. Last not least, we thank the German Research Foundation (DFG) for having invested in this project and for efficient and supportive procedures.

Michael Baake

Friedrich Götze

Werner Hoffmann

Introduction “Spectral structures and topological methods in Mathematics” Collaborative Research Centre 701 (2005–2017) Over the 12 years of funding, the CRC covered a broad spectrum of research. Its participants were driven by the vision of reinforcing and building bridges between various branches of theoretical and applied mathematics. Many significant developments in mathematics are related to spectral structures and use topological methods. Frequently, they have their origins in applied fields, for example in new concepts of mathematical physics and fluid dynamics, crystallography and materials science. The research pursued in the framework of the CRC may be attributed to one or several of the following broad mathematical topics:  asymptotics and universality  lattices

 representation theory

 harmonic and geometric methods

 deterministic and stochastic dynamics  moduli spaces

 p-adic L-functions

 p-adic cohomology

In this volume, we would like to highlight some illustrating examples of this research, to embed it into a wider mathematical context, and to emphasise connections within and between these topic areas. Let us now give a synopsis of the various chapters. Asymptotic approximations are a major tool for the analysis of distributions in different areas of mathematics. In Chapter 1 (Götze and Kösters), they are used to investigate the accuracy of universal statistical laws for local and global distribution of spectral values of random matrices and sums with independent parts. Here, universality for large times or for large complexity means that these distributions show an emergent collective behaviour in the limit which is independent of special properties of the model, such as the starting distribution, the details of the dynamics or the details of the distribution of the constituent parts. The asymptotic growth of numbers of geometric or algebraic objects are a common theme of Chapter 9 (Baake, Gähler, Huck and Zeiner) as well as Chapter 15 (Voll). The first one considers the enumeration of particular lattices in Euclidean space, the second one centres around the enumeration of subrings and representations of unipotent group schemes. In both cases, these numbers are studied via an analytic encoding in zeta functions, which generalise those of Hecke and Tate.

viii Other generalisations of these zeta functions with applications to Arthur–Selberg trace formulas in the framework of the Langlands program are reviewed in Chapter 14 by Hoffmann. Their analytic continuation and functional equations are obtained by tools of harmonic analysis like Poisson summation. Apart from enumeration, Chapter 9 (Baake, Gähler, Huck and Zeiner) also focuses on harmonic properties of lattices and quasiperiodic sets as well as on spectral implications of quasiperiodic tilings, their generation and connection to dynamical systems. Similarly, spectral properties of nonlinear dispersive equations of Schrödinger type, studied in Chapter 7 (Herr), are closely tied to spectral sets of lattice points which are investigated by methods of harmonic analysis parallel to those used in analytic number theory and the geometry of numbers studied in Chapter 1 (Götze and Kösters). Stochastic dynamical systems and their spectral properties represent another main topic of the CRC 701. In Chapter 5, Gentz studies metastability in parabolic SPDEs and other noise-induced phenomena in coupled dynamics by means of harmonic and stochastic analysis, large-deviation methods and random Poincaré maps. Equivariant dynamical systems are investigated in Chapter 6 by Beyn and Otten, where the surprising stability of equivariant evolution equations and their relative equilibra is studied under numerical discretisation. The stability of waves can be analysed using holomorphic nonlinear eigenvalue problems. In Chapter 8, Kassmann applies methods from partial differential equations and nonlocal operators in Euclidean spaces to study variational solutions of fractional Dirichlet problems and related Harnack inequalities. In Chapter 4, Kondratiev, Kutovyi and Tkachov study Markov birth and death processes in spatial ecologies by means of evolutions of configuration sets and semigroup methods, passing from microscopic stochastic configuration processes to mesoscopic kinematic model equations. In Chapter 2, Röckner discusses open problems and new approaches in solving Fokker–Planck–Kolmogorov equations in finite and in particular in infinite-dimensional spaces. He also reviews important results on the corresponding stochastic differential equations in this general infinite-dimensional setup and discusses applications to stochastic differential equations such as the stochastic porous media equation. A panorama of views showing the interplay of different fields of mathematics is developed for the notion of a poset (or lattice) of non-crossing partitions in Chapter 11 (Baumeister, Bux, Götze, Kielak and Krause). Non-crossing partitions are fundamental in moment computations for universal laws of non-commutative convolutions in free probability, and the Kreweras complement in this poset allows to analyse multiplicative non-commutative convolutions. Similarly, it can be used to determine the classifying spaces of braid groups and to describe the Hurwitz action in finite Coxeter systems. Last but not least, non-crossing partitions lie at the heart of bijections between subcategories of thick and coreflective subcategories related to crystallographic Coxeter systems in representation theory. These fruitful connections have been the topic of several conferences hosted by the CRC 701 in the last years on

ix free probability, quantum groups, algebraic combinatorics, buildings and representation theory. Connections between the algebraic geometry of the projective line and its cohomological localisations in representation theory are reviewed in Chapter 12 by Krause and Stevenson. The example of the projective line shows that the classification of thick subcategories via non-crossing partitions that arises in representation theory is nicely complemented by the classification of thick tensor ideals arising in algebraic geometry. Harmonic analysis and stochastic dynamics on Riemanian manifolds together with associate heat kernel bounds and escape rates are studied in Chapter 3 (Grigor’yan). Some of these results can be transferred to (ultra-)metric measure spaces by means of Dirichlet forms. For discrete spaces like graphs, the notions of Hochschild homology and other fundamental homology constructions like Künneth’s formula can be partially extendend. In a similar vein, complexes for graphs and surfaces are studied in Chapter 13 (Bux) together with Morse functions on cell complexes in order to access higher finiteness properties of braided version of certain groups. Chapter 10 (Callies, Haydys) is devoted to the interplay of local and global geometry and harmonic analysis for special affine Kähler structures. Finally, p-adic analysis, number theory and geometry are in the focus of Chapter 16 (Nickel), reporting evidence for conjectures by Gross and Stark on vanishing orders and leading terms of p-adic L-functions and complex L-functions at zero. The theory of displays and the classification of p-divisible groups and with its important recent applications are studied in Chapter 17 by Zink. The research within the CRC 701 established viable connections at the interface between theoretical and applied mathematics: Algebraic geometry and dynamical systems, representation theory and probability theory, stochastic analysis and numerics, harmonic analysis connecting nonlinear partial differential equations, stochastics and analytic number theory. The special added value of the CRC 701 has been to realise the full potential of the mathematical theories around these interfaces, which motivated the newly recruited researchers to engage themselves into a coherent, stimulating research environment. Establishing such a framework of interactions between different fields of mathematics might be viewed as the most important legacy of the CRC 701. Friedrich Götze

Michael Röckner

Thomas Zink

Contents

1 Convergence and asymptotic approximations to universal distributions in probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by F. Götze, H. Kösters 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Symmetric random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Non-symmetric random matrices . . . . . . . . . . . . . . . . . . . . . . . 1.4 Local spectral distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Connections between probability theory and number theory . . . . . 1.6 Analogies between classical and free probability . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Kolmogorov operators and SPDEs . . . . . . . by M. Röckner 2.1 Fokker–Planck–Kolmogorov equations 2.2 Three selected results on SPDEs . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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3 Analysis and stochastic processes on metric measure spaces . by A. Grigor’yan 3.1 Analysis on manifolds . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analysis on metric measure spaces . . . . . . . . . . . . . . . 3.3 Homology theory on graphs . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Markov evolutions in spatial ecology: From microscopic dynamics to kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by Yu. Kondratiev, O. Kutovyi, P. Tkachov 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Complex systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Markov evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Birth-and-death evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Vlasov-type scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Metastability in randomly perturbed dynamical systems: Beyond largedeviation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by B. Gentz 5.1 Large deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Kramers’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Parabolic SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Unstable periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Mixed-mode oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computation and stability of waves in equivariant evolution equations by W.-J. Beyn, D. Otten 6.1 Equivariant evolution equations . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The freezing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Relative Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Nonlinear eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Initial value problems for nonlinear dispersive equations at critical regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . by S. Herr 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Nonlinear Schrödinger equations on compact manifolds 7.3 Nonlinear systems on Euclidean space . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational solutions to nonlocal problems . . . . . . . . . by M. Kaßmann 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Variational solutions to the Dirichlet problem . . . . 8.3 Ellipticity and coercivity of nonlocal operators . . . 8.4 (Weak) Harnack inequalities, and Hölder regularity References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Spectral and arithmetic structures in aperiodic order . . . by M. Baake, F. Gähler, C. Huck, P. Zeiner 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Weak model sets . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 A decorated quasiperiodic tiling with mixed spectrum 9.4 Random inflations . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Enumeration of lattices . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Affine special Kähler structures in real dimension two . . . . by M. Callies, A. Haydys 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Special Kähler geometry in local coordinates . . . . . . . 10.3 Some global aspects of special Kähler geometry on P 1 . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Non-crossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . by B. Baumeister, K.-U. Bux, F. Götze, D. Kielak, H. Krause 11.1 The poset of non-crossing partitions . . . . . . . . . . . . . 11.2 Non-crossing partitions in free probability . . . . . . . . . 11.3 Braid groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Non-crossing partitions in Coxeter groups . . . . . . . . . 11.5 The Hurwitz action . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Non-crossing partitions arising in representation theory 11.7 Generalised Cartan lattices . . . . . . . . . . . . . . . . . . . . 11.8 Braid group actions on exceptional sequences . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The derived category of the projective line . . . . . . . . . by H. Krause, G. Stevenson 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Types of localisation . . . . . . . . . . . . . . . . . . . . . 12.4 Cohomological localisations for the projective line 12.5 Exotic localisations . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 Higher finiteness properties of braided groups . . . . . . . . . . by K.-U. Bux 13.1 Introduction: From group theory to topology . . . . . . . . 13.2 Brown’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Combinatorial Morse theory . . . . . . . . . . . . . . . . . . . . 13.4 Matching complexes for graphs and surfaces . . . . . . . . 13.5 Higher generation in symmetric groups and braid groups 13.6 The braided Thompson group V br . . . . . . . . . . . . . . . . 13.7 A cube complex for V br . . . . . . . . . . . . . . . . . . . . . . . 13.8 The Morse function and its descending links . . . . . . . . . 13.9 Connectivity of descending links . . . . . . . . . . . . . . . . . 13.10 Finiteness properties of V br : Proof of Theorem 13.6.1 . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14 Zeta functions and the trace formula . . . by W. Hoffmann 14.1 Introduction . . . . . . . . . . . . . . . . . 14.2 Zeta integrals . . . . . . . . . . . . . . . . 14.3 The Trace formula . . . . . . . . . . . . 14.4 Unipotent terms in the trace formula References . . . . . . . . . . . . . . . . . . . . . .

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15 Zeta functions of groups and rings—functional equations and analytic uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by C. Voll 15.1 Zeta functions associated to groups and rings . . . . . . . . . . . . . . . 15.2 Submodule zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Representation zeta functions for unipotent group schemes . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Conjectures of Brumer, Gross and Stark . . . . by A. Nickel 16.1 Preliminaries . . . . . . . . . . . . . . . . . . . . 16.2 The abelian case . . . . . . . . . . . . . . . . . . 16.3 The general case . . . . . . . . . . . . . . . . . . 16.4 Relations to further conjectures and results References . . . . . . . . . . . . . . . . . . . . . . . . . .

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409

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411

17 Displays and p-divisible groups . . . . . . by T. Zink 17.1 Introduction . . . . . . . . . . . . . . . . 17.2 Frames . . . . . . . . . . . . . . . . . . . 17.3 Displays . . . . . . . . . . . . . . . . . . 17.4 Classification of p-divisible groups 17.5 The nilpotency condition . . . . . . . 17.6 Isogenies . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Convergence and asymptotic approximations to universal distributions in probability F. Götze and H. Kösters The limiting distributions of functionals depending on a large number of independent random variables of comparable size are often universal, leading to a vast number of convergence and approximation results. We discuss some general principles that have emerged in recent years. Examples include classical and entropic central limit theorems in classical and free probability, distributions of zeros of random polynomials of high degree and related distributions of algebraic numbers, as well as global and local universality results for spectral distributions of random matrices.1

1.1 Introduction In random matrix theory, the distributions of the eigenvalues of various kinds of random matrices are investigated, see e.g. [3]. More precisely, we are interested in the asymptotic behaviour of the eigenvalues as the matrix size tends to infinity. Let us begin with one of the most famous results in random matrix theory, the semi-circle law. For each n 2 N, let X n D .n 1=2 Xj k /16j;k6n be a symmetric Wigner matrix, i.e. a symmetric n  n matrix such that .Xj k /16j 6k6n is a family of independent and identically distributed (i.i.d.) real random variables satisfying the moment conditions and EXj2k D 1

(1.1.1)

sup EjXj k j2Cı < 1 :

(1.1.2)

EXj k D 0 and, for some ı > 0, sup

n2N 16j;k6n

The random variables Xj k may depend on n, but this dependence is suppressed in the notation. Let us mention that all the results described in this section continue to hold for symmetric random matrices with non-identically distributed entries, and also for Hermitian Wigner matrices. P Let 1 ; : : : ; n denote the eigenvalues of X n , and write X n WD n1 njD1 ıj for the (empirical) spectral distribution of X n . The investigation of the limiting spectral 1 Projects

A4, B1

2

F. Götze, H. Kösters

distribution (after appropriate rescaling) has a long history and dates back to work by Wigner in the 1950s. The famous semi-circle law states that q 1 lim EX n D sc .dx/ WD 2 .4 x 2 /C dx (1.1.3) n!1

in the sense of weak convergence. Similar results hold without the expectation, with weak convergence in probability or almost surely. It is worth emphasising that the limiting spectral distribution is always given by the semi-circle distribution sc , irrespective of the distribution of the random variables Xj k . In this respect, the semicircle distribution is universal. There are two main approaches to prove the semi-circle law, the method of moments and the method of Stieltjes transforms. If, for each m 2 N, the moments EjXj k jm are finite and uniformly bounded in n, one can use the method of moments. Here one shows using combinatorial arguments that Z Z   lim x m .EX n /.dx/ D lim E n1 tr X m D x m sc .dx/ (1.1.4) n n!1

n!1

for any m 2 N0 . For m D 2l even, the limit is the lth Catalan number, or the number of non-crossing pair partitions of 2l elements. Since convergence in moments implies convergence in distribution when the limiting distribution is determined by its moments, this proves (1.1.3). For any probability measure  on the real line, the Stieltjes transform is the analytic function on the upper half-plane CC WD fu C iv 2 C W u; v 2 R; v > 0g defined by Z 1 m .z/ WD .dx/ ; z 2 CC : (1.1.5) x z We shall write mX n instead of m and msc instead of msc . It is well known Xn that the pointwise convergence of Stieltjes transforms (to a limit which is itself the Stieltjes transform of a probability measure) is equivalent to the weak convergence of the underlying probability measures. In the context of Wigner matrices, the usefulness of the Stieltjes transform comes from the observation that, for each z 2 CC , mX n .z/ D

1 n

tr R n D D

1 n

n X

j D1 1 n

n X

j D1

n

1=2 X

jj

z

1 z

1 n

tr R n

n

1 P

1

C En D

k;l¤j

z

.j / Xj k Xj l Rkl

(1.1.6)

1 C En : mX .z/ n

.j / / / Here, R n WD .X n zI n / 1 D .Rkl / and R .j WD .X .j zI n 1 / 1 D .Rkl / n n .j / .j / denote the resolvents of the matrices X n and X n , where X n is obtained from X n

3

Convergence and Asymptotic Approximations

by deleting the j th row and the j th column, and En is an error term (see below). If this error term were zero, (1.1.6) would yield the self-consistency equation m2 .z/ C zm.z/ C 1 D 0 ;

(1.1.7)

whose solution is given by the Stieltjes transform of the semi-circle distribution. Hence, to prove the semi-circle law, one must show that En ! 0 as n ! 1 (in the appropriate sense). More precisely, it is not hard to see that the error term in (1.1.6) is given by .z C mX n .z//En D

1 n

n X

j D1 1 n

 Rjj n

X

k¤j

.Xj2k

1=2

X

1 n

Xjj

.j /

Xj k Xj l Rkl

k;l¤j Wk¤l .j / 1/Rkk

C

1 n

tr.R n

/ R .j n /



(1.1.8) ;

so it remains to show that the sums on the right-hand side in (1.1.8) tend to zero as n ! 1. To this end, results from classical probability theory concerning the moments and distributions of (possibly random) linear and quadratic forms of independent random variables are useful. The problem of optimal approximations of distributions of linear and nonlinear functions of independent random variables is a classical problem of probability and harmonic analysis, which originated from classical analytic number theory. For linear functionals, the arithmetic structure of the summands strongly influences the distribution of the sums, as in the Littlewood–Offord problem [96] and in the central limit theorem (CLT). For nonlinear functionals like quadratic forms, this influence is already reduced and studied intensively in the value distribution on lattices by Landau (1924) and the multivariate CLT for balls by Esséen (1945). Hence, one expects that distributions of generic highly non-linear functionals of vectors and matrices of independent variables (e.g. eigenvalues of matrices) exhibit a smooth distributional behaviour irrespective of a possible arithmetic/lattice structure in the distribution of these variables. This expected behaviour applies as well to the distribution of roots of high-degree polynomials with independent random coefficients, which is related to the distribution of algebraic numbers of growing height. Random matrix theory is also related to free probability [3, 102]. To explain this, let us start from the observation that, by generalisation of Eq. (1.1.4), the calculation of the expected traces of more general products of random matrices is also of interest. If .X n / and .Y n / are independent sequences of Wigner matrices as in (1.1.4), it turns out that   k   j j k lim E n1 tr X n1 . lim 1 E tr X 1 /I n Y n1 . lim 1 E tr Y  1 /I n


n!1 !1 !1    km jm k jm 1 1 . lim  E tr Y  m /I n D 0 (1.1.9) . lim  E tr X  /I n Y n Xn !1

!1

for all m 2 N and all j1 ; : : : ; jm ; k1 ; : : : ; km 2 N. By linearity and by induction, this allows us to reduce the expected traces of the products X jn1 Y kn1


X jnm Y knm to those

4

F. Götze, H. Kösters

of the powers X jn and Y kn . The relation (1.1.9) is called the asymptotic freeness of the sequences .X n / and .Y n /. The relation (1.1.9) is the asymptotic counterpart of the notion of freeness in free probability. Here, one considers a unital algebra A endowed with a tracial unital linear functional ' W A ! C and studies the moments '.an / of elements a 2 A. Two elements a; b 2 A are called free (with respect to each other) if
' .aj1 '.aj1 //.b k1 '.b k1 //


.ajm '.ajm //.b km '.b km // D 0

(1.1.10)

for all m 2 N and all j1 ; : : : ; jm ; k1 ; : : : ; km 2 N. Just as above, this allows us to reduce the mixed moments '.aj1 b k1


ajm b km / to the moments '.aj / and '.b k /. The notion of freeness may be regarded as a (non-commutative) analogue of the notion of independence in classical probability theory. Moreover, it allows for the development of a free probability with many parallels to classical probability. For instance, if a and b are free, the moments of the sum a C b and the product ab depend only on the moments of a and b. Thus, provided that all the moment sequences correspond to probability measures a , b etc. on the real line, one may define the free additive and multiplicative convolution of a and b by setting a
b WD aCb

and a  b WD ab :

For the multiplicative convolution, we shall assume without further notice that the distributions a and b are supported on the positive half-line. One may then investigate similar questions (limit theorems, infinite divisibility, asymptotic approximations and expansions, . . . ) as for the classical convolutions. For instance, the free CLT shows p D  weakly, which is the that if '.a/ D 0 and '.a2 / D 1, then limn!1 
n sc a= n direct analogue of the classical CLT. Thus, the semi-circle distribution plays a similar role in free probability as the normal distribution in classical probability. When the probability measures a and b have compact support, one may also take an analytic approach based on suitable transforms, namely Voiculescu’s R- and S -transforms [3, 102], instead of the combinatorial approach outlined above. These transforms are analytic on certain domains in the complex plane and satisfy Ra
 D Ra C R b

b

and Sa  D Sa
S ; b

b

and hence may be viewed as analogues of the logarithmic Fourier transform and the Mellin transform in classical probability theory. Incidentally, the R- and S -transforms are also closely related to the Stieltjes transform in (1.1.5). For instance, for the R-transform, we have Ra .z/ D . ma .z//h 1i z 1 , where .
/h 1i denotes the inverse function. In terms of random matrices, the free convolutions may be interpreted as follows: Suppose that .X n / and .Y n / are sequences of self-adjoint random matrices of increasing dimension whose mean spectral distributions converge to X and Y in moments and which are asymptotically free. (Roughly speaking, this means that their

Convergence and Asymptotic Approximations

5

eigenspaces are in general position to one another.) Then, the limiting mean spectral distributions of X n C Y n and X n Y n are given by X
Y and X  Y , respectively. Thus, the limiting spectral distributions of certain composite random matrices may be investigated using tools from free probability. As already mentioned, the semi-circle distribution may be regarded as the counterpart of the normal distribution in free probability. It seems natural to study functionals under which these distributions have certain extremal properties. For instance, it is well known that the normal distribution maximises Shannon entropy among all distributions with mean 0 and variance 1. Similarly, the semi-circle distribution maximises Voiculescu’s free entropy in the same class. Thus, it seems natural to investigate limit theorems with respect to the divergence measures associated with these entropies. This brings us to the field of entropic limit theorems; see Section 1.6.2. In the semi-circle law, in the limit as n ! 1, the eigenvalues are confined to a bounded interval. This is the global (or macroscopic) level, where one studies the weak convergence of the empirical eigenvalue distribution to some limiting distribution (typically with compact support). One may also consider the local (or microscopic) level, where the eigenvalues are rescaled in such a way that the mean spacing between neighbouring eigenvalues is of the order 1. One then studies the asymptotic correlations between a small number of eigenvalues which are close to one another. Interestingly, here the limiting distributions are often universal too, which means that the same limits arise for different kinds of random matrices (from the same symmetry class). See Section 1.4 below for a sample of results. Moreover, these limits also appear in a variety of other contexts. For instance, the asymptotic local spectral distributions for Hermitian Wigner matrices also show up in representation theory (asymptotics of Young diagrams), probability theory (non-colliding stochastic processes, repulsive particle systems), number theory (zeros of L-functions) and physics (quantisations of chaotic dynamical systems).

1.2 Symmetric random matrices In this section we discuss several extensions of the semi-circle law that have been obtained in the last few years. We continue with the notation and the assumptions for symmetric Wigner matrices from the Introduction. 1.2.1 Rate of convergence in the semi-circle law It is natural to ask for the rate of convergence in the semi-circle law. Given two probability measures  and  on the real line with distribution functions F and G, we write k

k1 WD sup jF .x/

G.x/j

x2R

for the Kolmogorov distance between  and . Then, one may consider the distance either for the mean spectral distribution, n WD kEX n

sc k1 ;

6

F. Götze, H. Kösters

or for the spectral distribution, n WD kX n

sc k1 :

The problem to establish upper bounds on n has a long history. In 1993, Bai [7] derived the rate O .n 1=4 / under a 4th moment condition. The rate O .n 1=2 / was obtained independently by Girko (1998, 2002), Bai, Miao and Tsay (2002), and Götze and Tikhomirov (2003) under various moment conditions; see [69] for references. Next, the optimal rate O .n 1 / was obtained for the special case of random matrices with Gaussian entries, first in the Hermitian case [65] and then in the symmetric case [100]. Using concentration of measure techniques, Bobkov, Götze and Tikhomirov [22] obtained the rate O .n 2=3 / under the assumption that the matrix entries satisfy a Poincaré inequality. The optimal rate of convergence O .n 1 / under weak moment conditions was finally established by Götze and Tikhomirov [68, 69], initially under an 8th moment condition [68] and later under a .4 C ı/th moment condition [69]. More precisely, it was shown in [69] that if M4Cı WD sup

sup

n2N 16j;k6n

EjXj k j4Cı < 1

(1.2.1)

for some ı > 0, then there exists a constant C D C.ı; M4Cı / such that n 6 C n

1

(1.2.2)

for all n 2 N. The proof of this result required three major ingredients: 1. A suitable variant of the smoothing inequality to bound the Kolmogorov distance in terms of the difference of the corresponding Stieltjes transforms, see Proposition 2.1 in [68]. This inequality uses a special contour which stays away from the end-points ˙2 of the support of sc , where the Stieltjes transforms are more difficult to control. 2. A recursive argument to derive good bounds on the diagonal entries of the resolvent close to the real axis, see Section 5 in [68]. Roughly speaking, this argument shows that a bound on EjRjj j2p at distance v from the real axis entails a bound on EjRjj jp at distance v=s0 from the real axis. It was inspired by similar results, albeit under stronger moment conditions, by Cacciapuoti, Maltsev and Schlein [26]. The proof under weak moment conditions is more involved, and uses recursive expansions for the resolvent entries similar to (1.1.6), as well as Burkholder’s inequality for martingale difference sequences to estimate the resulting quadratic forms. 3. A recursive inequality for EjmX n .z/ mSC .z/j2 , see Lemma 7.24 in [68]. This inequality is based on a Stein-type expansion adapted to the self-consistency equation (1.1.7), which facilitates the recursion argument considerably. Using moment matching techniques, a simplified proof of (1.2.2) could be given in [61].

Convergence and Asymptotic Approximations

7

Let us now turn to upper bounds on n . In 1997, Bai, Miao and Tsay [9] obtained the rate OP .n 1=4 / under a 4th moment condition. The rate OP .n 1=2 / was established in [64] under a 12th moment condition and in [10] under a 6th moment condition. More recent results by Erd˝os, Yau and Yin [42] imply that n D OP .n 1.log n/C log log n /, see e.g. Section 1 in [68]. In [60], complemented by additional material in [58, 59], Götze, Naumov and Tikhomirov proved that, under the condition (1.2.1), n D OP .n

1

log n/ ;

(1.2.3)

with some explicit constant  D .ı/. In view of a result by Gustavsson [72] for GUE matrices (i.e. Hermitian Wigner matrices with Gaussian entries), it seems clear that the optimal rate cannot be better than OP .n 1 log1=2 n/. Thus, a result of the form (1.2.3) is optimal up to logarithmic factors. Götze, Naumov, Tikhomirov and Timushev [61] improved the result (1.2.3) by showing that it is possible to take  D 2. More generally, they showed that, under the condition (1.2.1), there exist positive constants C D C.ı; M4Cı / and c D c.ı; M4Cı / such that for 1 6 p 6 c log n, one has P.n > K/ 6

C p log2p n K p np

for all K > 0 :

These results for n were obtained by refining the methods developed in [68]. For instance, in [60], the previous estimates from [68] were extended to the off-diagonal entries of the resolvent, and Stein-type expansions were employed systematically in order to bound the pth moment of the error term En in (1.1.8). Moreover, in [61], the authors used moment matching techniques (motivated by results in [37, 83]) to compare a general Wigner matrix to a suitable Wigner matrix with sub-Gaussian entries. Finally, an essential ingredient in all these works was an appropriate version of the local semi-circle law, which will be described in the next subsection. 1.2.2 Local semi-circle law As mentioned below (1.1.5), the proof of the semicircle law amounts to showing that jmX n .u C iv/ msc .u C iv/j ! 0 as n ! 1, for any fixed u 2 R, v > 0. In the last few years, similar results have been obtained for the situation where v may tend to zero as n tends to infinity, but not too fast [40, 38, 60, 61]. More precisely, under suitable moment conditions, jmX n .u C iv/

msc .u C iv/j 6

.log n/C log log n nv

(1.2.4)

with high probability, uniformly in u 2 R and 1 > v > log n=n, say. A result of the form (1.2.4) is called a local semi-circle law in the literature, since it may be used as a starting point for the investigation of the local distribution of the eigenvalues. In fact, the local semi-circle law was originally developed by Erd˝os, Schlein and Yau [40] on their way to proving the universality of the local spectral distribution of

8

F. Götze, H. Kösters

general Wigner matrices, see also Section 1.4. The first version of the local semicircle law was derived under the assumption that the underlying matrix entries have finite exponential moments (as well as further regularity properties). This assumption was successively relaxed to .4 C ı/th moments in a series of papers by Erd˝os, Knowles, Schlein, Yau and Yin; compare [38] and the references therein. The paper [60] provided an alternative self-contained proof of the local semicircle law, also under a .4 C ı/th moment condition, by building upon the techniques developed for the investigation of the rate of convergence [68]. One of the main advantages of this approach is that the exponent  of log n in (1.2.4) is reduced from C log log n to a constant (at least in a certain region for the arguments u and v), with a precise dependence on ı. These results were further improved by Götze, Naumov, Tikhomirov and Timushev [61], who showed that for any ı > 0, there exist constants C0 ; C1 ; C2 depending only on ı and M4Cı (see (1.2.1)) such that   C0 p p EjmX n .u C iv/ msc .u C iv/jp 6 nv for all 1 6 p 6 C1 log n, 1 > v > C2 log n=n and juj 6 2 C v. By taking p of the order log n and using Markov’s inequality, one re-obtains (1.2.4) for 1 > v > C2 log n=n and juj 6 2 C v, but with the constant exponent  D 1 instead of  D C log log n for the logarithmic factor. As already mentioned, this version of the local semi-circle law plays a major role in recent advances on the rate of convergence for n . Further applications include the delocalisation of eigenvectors and the rigidity of eigenvalues; see [42, 60, 61] and the references given there. In view of the results by Gustavsson [72], all these results seem to be optimal up to logarithmic factors.

1.3 Non-symmetric random matrices The spectral distributions of non-symmetric random matrices have also been investigated. 1.3.1 Circular law For each n > 1, let X n D .n 1=2 Xj k /16j;k6n be a real Girko– Ginibre matrix, i.e. an n  n matrix such that .Xj k /16j;k6n is a family of i.i.d. real random variables satisfying the moment conditions EXj k D 0; EXj2k D 1 and, for some ı > 0, sup

sup

n2N 16j;k6n

.1 6 j; k 6 n/

EjXj k j2Cı < 1 :

Again, the random variables Xj k are allowed to depend on n. Also, all the results described in this section continue to hold for non-symmetric random matrices with

Convergence and Asymptotic Approximations

9

non-identically distributed matrix entries, as well as for complex Girko–Ginibre matrices. Similarly as above, let 1 ; : : : ; n denote the eigenvalues of X n , and write X n for the (empirical) spectral distribution of X n . Of course, this is in general a probability measure on the complex plane now. The famous circular law states that lim EX n D circ .dz/ WD

n!1

1 

1 fjzj 2, n ; n > .n C 1/=3 holds, which follows from a counting result for pairs of real algebraic conjugate numbers in certain intervals of given large height. This is in turn a consequence of results in metric number theory which generalises results of Baker, Schmidt, Bernik, Kleinbock and Margulis. For an overview of techniques and results, see [14]. 1.5.3 Effective bounds in Diophantine inequalities A classical problem of effective bounds means to determine the size of the smallest vector m 2 Z d n f0g such that the Diophantine inequality jQŒmj < 1 holds. Here, QŒx denotes an (nondegenerate) indefinite quadratic form with real coefficients on Rd . For forms with integer coefficients it is known by a result of Meyer (1884) that there exists an m 2 Z d n f0g with QŒm D 0 provided that d > 5. As for the size of this solution vector, classical results of Birch and Davenport (1958) using the geometry of numbers show that for indefinite integral quadratic forms 0 < QC Œm < cd j det Qj holds, where QC D .QT Q/1=2 is positive definite. For diagonal indefinite quadratic forms Q, Birch and Davenport (1958) have shown that there exists a nontrivial solution m ¤ 0 to jQŒmj <  of size kmk ı  2Cı in dimension d D 5. In higher dimensions solutions may be generated via embedding. Using theta functions, methods from the geometry of numbers and the asymptotic orbit behaviour of unipotent subgroups of SL2 .R/, Götze and Margulis proved in [54] for general indefinite QŒx a bound of order 0 < kmk ı  kd Cı ; kd D 12; 8:5; 7 for d D 5; 6 and d > 7. 1.5.4 Lattice point counting problems The existence of m ¤ 0 with jQŒmj <  is a consequence of precise results for counting lattice pointsin elliptic as well as hyperbolic shells defined via a quadratic form QŒx in Rd and Ea;r WD fQŒx 2 Œa; rg for

18

F. Götze, H. Kösters

Q positive definite and Hr .a; b/ WD fQŒx 2 Œa; b; x 2 rC g (C suitable compact convex body) in the case of Q indefinite as r in Hr .a; b/ and Ea;r tends to infinity. The counting error is measured relative to volume, say jE0;r j, resp. jHr .a; b/j of these compact rescaled regions, say Br , via ˇ ˇ ˇ card.Z d \ B / jB j ˇ = jB j D O .ı r 2˛ /; r ! 1; (1.5.1) r r r r

where ˛ 6 1. In case that the optimal exponent ˛ D 1 can be shown, the factor ır may be bounded below or tend to zero depending on the Diophantine properties of the coefficients of Q. Such error bounds have been studied for a long time, starting with Landau (1915, 1924), Jarnik (1928) for special positive definite forms QŒx being diagonal or with integer coefficients, where ˛ D 1 could be shown. Optimal exponents have been shown for d > 9 and definite as well as indefinite general forms QŒx first by Bentkus and Götze (1997). For dimension d > 5, the optimal exponent was shown in [45] for E0;r and in [54] for Hr .a; b/ such that b a tends to infinity. As for the dependence of Diophantine properties of Q, Davenport and Lewis (1972) conjectured that for positive definite irrational forms the distances of the ordered elements of QŒZ d  for d > 5 converge to 0. This is related to the famous conjecture by Oppenheim (1929) that QŒZ d  is dense in R for indefinite irrational forms for d > 5, which was proved by Margulis (1986) even for d > 3. Quantitative versions of these conjectures for irrational forms were shown for d > 9 in Bentkus and Götze (1997) as well as for d > 5 and Q positive-definite in [45]. In [54], the problem for both cases has been solved for d > 5 by means of a unified approach. Lower bounds for the error for E0;r in dimensions d D 2; 3; 4 show that the optimal exponents satisfy ˛ > 1=4; 1=2; 1 respectively, multiplied by logarithmic factors like .log r/ˇ ; ˇ > 0; d D 2; 3 resp. .log log r/ ; > 0; d D 4, whereas lower error bounds for dimensions 5 and higher are just given by .r 2 /. Crucial for the proof of ˛ D 1 and ır ! 0 in (1.5.1) is the investigation of theta sums on the generalised Siegel upper half-plane, that is on matrices iQ C A, where A; Q are real symmetric d  d matrices with Q positive definite, given by X .iQ C A/ D expŒ mT Q m C imT A m: (1.5.2) m2Z d

The number of lattice points m such that Q.m/ 6 r may be expressed as an integral along a theta function  .sQ/ expŒit r=.2s/ on a line s D t C ir 2 ; t 2 R (that is a degenerate horocycle), with  given by (1.5.2). The following inequality between theta functions turned out to be an essential step: X j.sQ/j2 6 .t/ WD c.Q/r d expŒ Ht .m; n/; (1.5.3) m;n

where Ht denotes the t-dependent quadratic form Ht .m; n/ WD r 2 km

t Q nk2 C r

2

knk2

(1.5.4)

Convergence and Asymptotic Approximations

19

of .m; n/ 2 Z 2d . Using lattice density bounds for .t/, via the first d successive Minkowski minima Mt;1 6 Mt;2 6 : : : 6 Mt;d of Ht , that is via .t/ 6 c.Q/

rd ; Mt;1 : : : Mt;d

(1.5.5)

the problem of estimating the lattice point remainder is transferred to the estimation of an integral along .sQ/ together with questions in metric number theory and the geometry of numbers. This approach is closely connected to the study of ergodic properties of unipotent and quasi-geodesic flows in the papers of Eskin, Margulis and Mozes (1998). Indeed, we may write the quadratic form Ht .m; n/ as Ht .m; n/ D k r ut .m; n/k2 ;

r WD diag.r 2 Id; r

2

Id/;

ut .m; n/ WD .m

tQn; n/; (1.5.6)

where r 2 SL.2d; R/; r > 0 denotes a quasi-geodesic flow and ut 2 SL.2d; R/; t 2 R a unipotent flow in SL.2d; R/. 1.5.5 Central limit theorems Let B denote a domain in Rd with 0 2 B and smooth boundary, which is symmetric to 0. Let X1 ; : : : ; Xn denote independent and identically distributed Rd -valued random vectors with EX1 D 0; EkX1 k4 < 1 and identity covariance. A classical problem in probability theory is the question of optimal error bounds in the central limit theorem for the distribution of sums Sn D X1 C


CXn of random vectors on the system of sets Br D r B; r > 0, i.e. the determination of the exponent ˛ in ˇ ˚ ˇ ˇ ˇ sup ˇP Sn 2 n1=2 Br PfS 2 Br gˇ D O .n ˛ /; (1.5.7) r>0

where S denotes a random vector Rd with a standard Gaussian distribution. For non-degenerated ellipsoids B, Esséen (1945) has shown ˛ D d=.d C 1/, which has been extended to uniformly convex bodies with smooth boundary. In the case of random vectors with independent coordinates and coefficient matrices which are diagonal with respect to a lattice basis, the exponent ˛ D 1 has been shown for d > 5 by Bentkus and Götze (1996). Identifying n with r 2 , there exists an obvious correspondence of error bounds in the CLT, that is in (1.5.7), for ellipsoids symmetric to 0 with bounds for the relative lattice point remainder in (1.5.1). The optimal error order which holds uniformly in the distribution of the random vectors subject to centering, moment and covariance conditions only is given by ˛ D 1 in (1.5.7) for d > 9 for quadrics (Bentkus–Götze 1997) and ˛ D k=2 for regions defined by special k-th order polynomials, k > 3 and large d (Götze 1989). The optimal rate for quadrics and d > 9 required new techniques from analytic number theory (Bentkus–Götze 1999). This result has been extended to general U -statistics and quadratic forms of n dimensional vectors with independent components (Götze–Tikhomirov 2002) provided that their rank d is at least 12 or larger. For a detailed review of the connections between

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lattice point problems and the CLT and the extensive literature in this field, see the reviews in [44] and [71]. The approximation results for Hr .a; b/ described above were essential as well for proving the above rate of convergence ˛ D 1 for quadrics in the CLT down to dimension 5 in [71]. Here, Götze and Zaitsev proved, for example, an explicit error bound for the elliptic or hyperbolic regions of type B WD fQŒx 6 1g in dimensions d > 5, sup j PfQŒSn  6 nr 2 g r>0

PfQŒS  6 r 2 g j 6 cd n

1

j det Qj

1=2

EkQ

1=2

X k4 :

This result concludes a long series of investigations starting with the seminal results by Esséen (1945) mentioned above. It relies on the fact that the characteristic function of quadratic forms Q.Sn / for sums of independent identically distributed vector summands may be estimated via the average over random matrices A in the characteristic functions of bilinear forms hAT1 ; T2 i of two independent sums T1 ; T2 of Rademacher vectors taking values in the lattice Z d . These characteristic functions in turn may be estimated again using local limit theorems via theta sums as outlined in (1.5.3). Hence, the decisive step of controlling the local fluctuations of the distribution of such indefinite quadratic forms could be reduced to the estimates outlined above in order to obtain a full correspondence in terms of dimensions and rates between both areas. In order to investigate distribution functions of quadratic forms, a crucial technical obstacle had been to extend the averaging (in t) of the characteristic described in (1.5.5) from the measure dt (sufficient for narrow shells Hr .a; b/ with bounded intervals Œa; b) to the harmonic measure dt=t on the real line. For lower dimensions, for example d D 3, one cannot expect these optimal rates in view of the correspondence to the lattice point error problem in these dimensions.

1.6 Analogies between classical and free probability In non-commutative probability theory the addition and multiplication of ‘free random variables’ corresponds to the ‘free’ additive and multiplicative convolution of the corresponding spectral probability measures which are represented by Hermitian operators in certain von Neumann algebras A with a finite normalised faithful trace  , see e.g. the survey [102]. A guiding principle for the investigation of free convolutions of spectral measures is the analogy to the classical theory of convolutions of probability measures. Within the framework of comparison studies between classical and free probability, Chistyakov and Götze investigated the free additive convolution with respect to a classification of indecomposable elements and infinite divisibility. The final results now appeared in [29]. They represent an analog of the classical results of infinite divisibility via the so-called Bercovici–Pata bijection, which yields a correspondence of classical Lévy measures to free Lévy measures. The latter appear in free Khintchine type decompositions via integral representations of reciprocal Stieltjes transforms,

Convergence and Asymptotic Approximations

21

which are analytic functions from CC to C and hence closely related to the class of Nevanlinna functions. A remarkable difference described for free measure decompositions in [29] as compared to classical ones is that the infinitely divisible measures without indecomposable factors are trivial Dirac measures only. This observation holds not only for the free additive convolution, but also for the free multiplicative convolution on the positive half-line as well as on the unit circle [29]. As for analogs of results of classical parametric statistics, the independence of sample mean and sample covariance for Gaussian random variables has a counterpart in free probability for free semi-circular random variables in von Neumann algebras; see Chistyakov, Götze and Lehner [32]. 1.6.1 Asymptotic approximations in free probability In [28], Chistyakov and Götze investigated the rate of convergence of the distribution function of an n-fold normalised free additive convolution of a spectral measure  with itself to that of Wigner’s semi-circle law. Since their result showed a complete analogy to the Berry– Esséen theorem in classical probability, i.e. a rate of order O.n 1=2 / assuming the existence of a third moment of , they investigated higher order approximations for this n-fold free convolution of non-trivial measures. Unlike the classical case there are no arithmetic obstructions to the smoothness of such convolutions: for n larger than a finite threshold (depending on ), the resulting measure admits a density with respect to Lebesgue measure. Thus one would hope that, similarly to Edgeworth expansions in classical probability, moment conditions of order k C 2 suffice to define asymptotic approximations up to an error of order o.n k=2 / involving free cumulants. This is indeed the case for k D 1, and the expansion term involves derivatives of the semi-circle density. Differences to the classical case appear for the expansion term of order k D 2, which cannot be written in terms of derivatives of the semi-circle density anymore: one needs signed measures and a slightly shifted support when the third cumulant does not vanish. These problems are essentially caused by the singularity of the semi-circle density at the boundary of the support. Technically the proofs for the expansions are analytically much more involved then in the classical case. For k D 1; 2 one has to determine a subordinating function as a particular solution of a 3rd resp. 5th order equation, which degenerates into the quadratic equation (1.1.7) for the Stieltjes transform of Wigner’s semi-circle law as n ! 1, and study its explicit dependence on n. An alternative way to derive expansions of n 1=2 .X1 C : : : C Xn / in the free CLT for free identically distributed random variables Xj and related approximation problems for random matrices starts from a universal scheme of expansions for sequences of symmetric functions, see [62]. This scheme is an umbrella limit theorem for all Gaussian limits and related Edgeworth-type expansions in permutation symmetric functions of many variables by rotational invariant functions and corrections via polynomials in power sums of degree at least two. Applications to the highly non-linear free convolution were done in [63]. It involved derivatives with respect to j at zero of distribution functions like W C 1 X1 C 2 X2 , where W has semicircle distribution. In the interior of the limiting spectral support the validity of these ex-

22

F. Götze, H. Kösters

pansions is due to the smoothness of the distribution functions of 1 X1 C : : : C n Xn of .1 ; : : : ; n / 2 Rn in certain domains. 1.6.2 Entropic limit theorems Let us return to the classical central limit theorem. Let X1 ; X2 ; X3 ; : : : be i.i.d. real random variables with mean 0 and variance 1, and let Sp 1 ; S2 ; S3 ; : : : denote the associated partial sums. Let fn denote the density of Sn = n (if existent), and let ' denote the density of the standard normal distribution. The classical local limit theorem states that kfn

'k1 ! 0

if and only if there exists some n0 2 N such that fn0 is bounded. Since the existence of bounded densities is still a strong condition, it seems natural to look for similar characterisations under weaker conditions. One such result is the entropic central limit theorem by Barron (1985), which states that D.fn j '/ ! 0 if and only if there exists some n0 2 N such that D.fn0 j '/ < 1. Here, Z D.f j '/ WD L.f .x/='.x// '.x/dx denotes the relative entropy of f with respect to ', and L.x/ WD x log x. A nice interpretation of this result arises from the observation that D.fn j '/ D H.'/ H.fn /, where H denotes Shannon entropy, and that the standard normal distribution maximises Shannon entropy among all random variables of mean 0 and variance 1. Thus, the system converges to the state of maximal entropy. In this respect, let us also mention the result by Artstein, Ball, Barthe and Naor [5] that the entropy tends to that of the standard normal distribution monotonously, in line with the second law of thermodynamics. Moreover, in [6], these authors proved that the rate of convergence is O .n 1 / if the random variables Xn satisfy the Poincaré inequality. However, for more general probability densities, the question of the rate of convergence remained open. This question was answered completely in a series of papers by Bobkov, Chistyakov and Götze [17, 20] from the last few years. Their analysis combined tools from information theory (such as the entropy convolution inequality) with more classical results on characteristic functions and their asymptotic approximation. In [17], the authors derived Edgeworth-type expansions for the entropic central limit theorem. In particular, if the underlying random variables Xj have finite fourth moments, the rate of convergence is still O .n 1/, and hence much better than in the classical central limit theorem. This is related to the fact that the term of order O .n 1=2/ in the classical asymptotic expansion of the density fn is an odd function, and hence vanishes when taking the entropy integral. Moreover, optimal rates for the case where the random variables Xj have finite fractional moments of order 2 < s < 4 have also been obtained [17]. These results rest on technically involved approximations in the local limit theorem for sums of i.i.d. random variables in the case of fractional moments. Furthermore, Berry–Esséen bounds in the entropic central limit theorem have also been established [20].

Convergence and Asymptotic Approximations

23

Similar results were also derived for related notions from information theory, e.g. Fisher information [18], as well as for other limit laws, e.g. stable distributions [19]. However, in the latter situation some complications arise. For instance, since higher moments do not exist, asymptotic expansions are not available anymore. Moreover, the full analogue of the entropic limit theorem can only be obtained for the nonextremal stable laws. For the extremal stable laws, additional technical conditions are needed [19]. This might be related to the fact that (non-Gaussian) stable distributions do not maximise the entropy functional anymore. Chistyakov and Götze [31] studied related questions for the entropic free central limit theorem, i.e. they studied the convergence of Voiculescu’s free entropy to its maximum value (under fixed variance) assumed for the semi-circle distribution. Assuming a finite moment of order four, they showed that the rate of convergence in the entropic free CLT is also of the order O.n 1 /. Furthermore, they obtained an expansion up to an error of order o.n 1/. These results were based on previous results about expansions for densities in the free CLT [30]. Besides sums of i.i.d. random variables, maxima of sums of i.i.d. random variables have also been considered (Bobkov–Chistyakov–Kösters [21]). Here, the limiting distribution is given by the one-sided normal distribution, which also maximises a suitably defined entropy functional. Furthermore, one also obtains a characterisation result here: The maxima of the sums converge to the one-sided normal distribution in relative entropy if and only if the original random variables Xj have finite relative entropy with respect to the one-sided normal distribution. The proof is also based on a combination of the entropy convolution inequality with more classical results for characteristic functions, including Spitzer’s formula.

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[91] M. Rudelson, Invertibility of random matrices: norm of the inverse, Ann. of Math. 168 (2008), 575–600. [92] M. Rudelson and R. Vershynin, The Littlewood–Offord problem and invertibility of random matrices, J. Adv. Math. 218 (2008), 600–633. [93] K. Schubert and M. Venker, Empirical spacings of unfolded eigenvalues, Electron. J. Probab. 20 (2015), no. 120, 1–37. [94] T. Shcherbina, On the correlation function of the characteristic polynomials of the Hermitian Wigner ensemble, Commun. Math. Phys. 308 (2011), 1–21. [95] E. Strahov and Y.V. Fyodorov, Universal results for correlations of characteristic polynomials: Riemann–Hilbert approach, Commun. Math. Phys. 241 (2003), 343–382. [96] T. Tao and V. Vu, Inverse Littlewood–Offord theorems and the condition number of random discrete matrices. Ann. of Math. 169 (2009), 595–632. [97] T. Tao and V. Vu, Random matrices: universality of ESDs and the circular law. With an appendix by M. Krishnapur, Ann. Probab. 38 (2010), 2023–2065. [98] T. Tao and V. Vu, Random matrices: universality of local eigenvalue statistics, Acta Math. 206 (2011), 127–204. [99] D.A. Timushev and A.N. Tikhomirov, Limit theorems for spectra of sums of random matrices, Proc. Komi Science Center of Ural Division of RAS 187 (2014), 24–32. [100] D.A. Timushev, A.N. Tikhomirov and A.A. Kholopov, On the accuracy of the approximation of the GOE spectrum by the semi-circular law, Theory Probab. Appl. 52 (2008), 171–177. [101] M. Venker, Particle systems with repulsion exponent ˇ and random matrices, Electron. Commun. Probab. 18 (2013), no. 83 (12 pp). [102] D. Voiculescu, Lectures on free probability theory. In Lectures on Probability Theory and Statistics (P. Bernard, ed.), LNM 1738, Springer, Berlin, 2000, 279–349.

Chapter 2

Kolmogorov operators and SPDEs M. Röckner The purpose of this paper is to survey a number of selected results on Kolmogorov operators and SPDEs. It consists of two parts: One part is related to Kolmogorov operators and is devoted to the corresponding linear Fokker–Planck–Kolmogorov equations (see [11]), the other part is about three key results about SPDEs obtained resp. published during the last funding period, namely an existence and uniqueness result for L2 -initial data for the stochastic total variation flow, a new approach to SPDEs and a pathwise uniqueness result of SDEs on Hilbert spaces with a merely bounded drift part. Both parts are survey type papers coauthored by regular visitors of the CRC 701. The reader who is interested in a more comprehensive and less selective overview of the results of Project B4 and also Project A9 is referred to the monographs [1, 5, 11, 19] and the references therein.1

2.1 On recent progress and open problems in the study of linear Fokker–Planck–Kolmogorov equations V.I. Bogachev, M. Röckner and S.V. Shaposhnikov This part reports on some recent progress and remaining challenging open problems in the study of linear elliptic and parabolic Fokker–Planck–Kolmogorov equations obtained in Project B4. 2.1.1 Introduction Let us define (linear) Fokker–Planck–Kolmogorov equations in the elliptic and parabolic cases and formulate several problems related to these equations. Consider the Kolmogorov operator Lu.x/ D

d X

i;j D1

aij .x/@x @x u.x/ C i

j

d X

b i .x/@x u.x/; i

i D1

where aij and b i are Borel functions on Rd such that A D .aij /16i;j 6d is a nonnegative symmetric matrix.

1 Projects

A9, B4

30

M. Röckner

We say that a bounded Borel measure  on Rd (possibly signed) satisfies the Fokker–Planck–Kolmogorov equation L  D 0

(2.1.1)

if aij ; b i 2 L1loc .jj/, where jj denotes the variation of , and for every function u 2 C01 .Rd / Z Lu d D 0: Rd

If  is given by a density % with respect to the Lebesgue measure, Eq. (2.1.1) can be written as an equation for the density, d X

d X

@x @x .aij %/ i

j

i;j D1

i D1

@x .b i %/ D 0:

(2.1.2)

i

A parabolic Fokker–Planck–Kolmogorov equation is introduced similarly. Consider T > 0, and let d X

Lu.x; t/ D

i;j D1

ij

a .x; t/@x @x u.x; t/ C i

j

d X

b i .x; t/@x u.x; t/; i

i D1

where aij and b i are Borel functions on Rd  .0; T / such that A D .aij /16i;j 6d is a non-negative symmetric matrix. We say that a bounded (i.e. of bounded variation) Borel measure  on Rd  .0; T / (possibly signed) is defined by a family of Borel locally bounded measures .t /0 1 such that, for A D Id (the unit matrix), there are several different (hence infinitely many) probability solutions to the equation L  D 0. Similarly, in the parabolic case, we have constructed examples of infinitely differentiable b.x/ on Rd with d > 3 such that for A D Id the Cauchy problem with some initial probability measure  has infinitely many probability solutions. Therefore, certain additional conditions rather than smoothness are needed. We have found conditions of this sort expressed in terms of integrability of the coefficients with respect to solutions and (an alternative set of assumptions) in terms of so-called Lyapunov functions. We have also found sufficient conditions for the uniqueness of integrable solutions. In particular, we have shown that the uniqueness conditions for the class I of integrable solutions essentially differ from those for the class P . For example, the following results have been obtained (see [11]). Theorem 2.1.1. The elliptic equation L  D 0 has at most one probability solution in either of the following cases, assuming in these cases (for simplicity) that A is locally Lipschitz and b is locally bounded. (i) There is a non-negative function V 2 C 2 .Rd / such that lim V .x/ D C1 jxj!1

and LV .x/ 6 C V .x/ for some C > 0. (ii)

bi aij ; 2 L1 ./ for some probability solution . 1 C jxj2 1 C jxj

Kolmogorov operators and SPDEs

33

Note that (i) does not assume (and does not ensure) the existence of a probability solution, while (ii) being satisfied by some solution, guarantees that this solution is the only one. However, we do not know whether it can happen that the elliptic equation L  D 0 with A D Id and smooth b has no probability solutions, but has a nonzero signed solution in the class of bounded measures (this is impossible when d D 1). It follows from the previous theorem that, in case of the unit diffusion matrix (or a nondegenerate Lipschitz diffusion matrix) and bounded b, a probability solution is unique (if it exists). However, the case of irregular A has not been studied. Even the case of bounded continuous nondegenerate A (and bounded b) has not been investigated. The problem of existence of solutions has been better studied. Here are parabolic analogues. Theorem 2.1.2. Let A.x/ be locally Lipschitz and nondegenerate and let b.x; t/ be locally bounded. Suppose that there is a positive function V 2 C 2 .Rd / such that lim V .x/ D C1 and

jxj!1

LV .x; t/ 6 C C C V .x/ for some C > 0. Then, for any probability measure  on Rd , there is at most one probability solution to the Cauchy problem, with initial condition . Theorem 2.1.3. Let A D Id, and let b be a locally bounded vector field. Then, for the uniqueness of a probability solution to the Cauchy problem, it suffices to have a function V 2 C 2 .Rd / with lim V .x/ D C1 and jrV .x/j 6 C1 such that jxj!1

LV .x; t/ 6 C2 ; while for the uniqueness of an integrable solution the inequality LV .x; t/ >

C2

is sufficient. In the case of a radial function V , such conditions actually mean that, for the uniqueness of a probability solution, the quantity .b.x; t/; x/ should not tend to C1 too quickly, and for the uniqueness of an integrable solution, .b.x; t/; x/ should not tend to 1 too quickly. Such a function V is called a Lyapunov function. A challenging open problem in this area concerns the cases d D 1 and d D 2: it is still unknown whether in these cases, for A D Id and smooth b, the Cauchy problem with a probability initial data has at most one probability solution (as noted above, there are counterexamples for all d > 3). We now proceed to less regular diffusion matrices. Let U.x; r/ denote the ball of radius r centred at x.

34

M. Röckner

Let g be a bounded function on Rd C1 . We set O.g; R/ D

sup

sup r

.x;t /2Rd C1 r6R

2

jU.x; r/j

2

tZ Cr 2 t

“

jg.y; s/ g.z; s/j dy dz ds:

y;z2U.x;r/

If lim O.g; R/ D 0, the function g is said to belong to the class VMOx .Rd C1 /. If R!0

g 2 VMOx .Rd C1 /, then one can always assume that O.g; R/ 6 w.R/ for all R > 0, where w is a continuous function on Œ0; C1/ and w.0/ D 0. The following condition on A is used in our uniqueness result in case of A of low regularity. (H1) for every ball U  Rd , there exist numbers D .U / > 0 and M D M.U / > 0 such that .A.x; t/y; y/ > jyj2 ; kA.x; t/k 6 M for all .x; t/ 2 U  Œ0; T  and y 2 Rd .

Theorem 2.1.4. Suppose that aij 2 VMOx;loc .Rd  Œ0; T / and that the matrix A D .aij / satisfies condition (H1). Then, the set

M D f 2 P W aij ; b i 2 L1 .; Rd  Œ0; T /g consists of at most one element. We emphasise that this theorem gives uniqueness not in the whole class of probability solutions but only in its subclass specified by the integrability of the drift b (for uniformly bounded b, this condition holds automatically). Let us say a few words about the infinite-dimensional case which attracts many researchers due to applications in stochastic partial differential equations, infinitedimensional diffusions, and mathematical physics. For simplicity, we consider equations on the space of real sequences R1 (the countable power of the real line). This space is a complete separable metric space with the distance d.x; y/ D

1 X

nD1

2

n

min.1; jxn

yn j/:

Due to the special structure of this space, the main concepts (but not the results) connected with Fokker–Planck–Kolmogorov equations on it are very similar to the case of Rd . Namely, given Borel functions aij and b i on R1 such that the finitedimensional submatrices .aij /i;j 6n are symmetric non-negative-definite, we say that a Borel probability measure  on R1 satisfies the equation L  D 0

35

Kolmogorov operators and SPDEs

with the operator L D

1 X

i;j D1

ij

a @x @x C i

j

1 X

b i @x

i

i D1

if the functions aij , b i are -integrable and for every function ' in finitely many variables belonging to the corresponding class Cb1 one has Z L' d D 0: R1

Obviously, the function L' is -integrable since the series becomes a finite sum of -integrable functions. Set b WD .b i /1 i D1 . Similarly, one defines a parabolic equation and the corresponding Cauchy problem. As in the finite-dimensional case, interesting problems arise already for aij D ıij (which in Rd would mean the unit diffusion matrix). For example, if b.x/ D x, the standard Gaussian measure on R1 (the countable power of the standard Gaussian measure on the real line) is a solution to the corresponding equation. This measure is the only probability solution. Indeed, for a general b, the projection of any solution  to Rn , denoted by n , satisfies the equation on Rn whose diffusion coefficients anij and drift coefficients bni are the conditional expectations of the functions aij and b i with i; j 6 n with respect to the measure  and the  -algebra Bn generated by the first n coordinates. This is obvious from the definitions. Therefore, in case of constant aij , we have anij D aij and, in case of b i with i 6 n depending on x1 ; : : : ; xn , we have bni D b i . In particular, for b i .x/ D xi , we have bni .x/ D xi whenever i 6 n. Therefore, in the situation under consideration, the projection of any probability solution to Rn satisfies the same equation as the standard Gaussian measure on Rn , but this equation admits only one probability solution. However, the situation may change for other b. Actually, examples of linear b are known such that the corresponding equation (with aij D ıij ) has several different probability solutions that are Gaussian measures (hence there exist also non-Gaussian solutions, their convex combinations). A relatively simple case arises if we take b.x/ D

x C v.x/;

where the perturbation v takes values in the Hilbert space H D l 2 and is bounded in the usual l 2 -norm. In this case, it is known that every probability solution  to the elliptic equation L  D 0 is absolutely continuous with respect to and is unique (it is also known that a solution exists under the stated assumptions). Certainly, the assumption that v is a bounded l 2 -valued perturbation is very restrictive. As a typical result on uniqueness in infinite dimensions, we mention the following theorem from [9]. Let us consider the following Cauchy problem. Let B D .B i / be a sequence of Borel functions on R1  .0; T0 /, where T0 > 0 is fixed, and let aij be Borel functions on R1  .0; T0 /. Let us consider the Cauchy

36

M. Röckner

problem



@t  D L ; jt D0 D ;

(2.1.5)

where L is the formal adjoint operator for the differential operator L defined by L'.x; t/ D

1 X

aij .x; t/@x @x '.x; t/ C i

i;j D1

j

1 X

B i .x; t/@x '.x; t/; i

i D1

for every smooth function ' depending on finitely many coordinates of x. We consider the following condition. (A) aij D aj i , each function aij depends only on the variables t; x1 ; x2 ; : : : ; xmaxfi;j g and is continuous and, for every natural number N , the matrix AN D .aij /16i;j 6N satisfies the following condition: there exist positive numbers N , N and ˇN 2 .0; 1 such that for all x; y 2 RN and t 2 Œ0; T0  one has

N jyj2 6 hAN .x; t/y; yi 6 N 1 jyj2 ; kAN .x; t/

AN .y; t/k 6 N jx

yjˇN ;

where k : k is the operator norm and j : j is the standard Euclidean norm. Let  be a Borel probability measure on R1 and let P be some convex set of probability solutions  D t .dx/ dt to (2.1.5), i.e., t > 0 and t .R1 / D 1 for every t 2 .0; T0 /, such that jB k j 2 L2 ./ for each k 2 N and the following condition holds: (B) for every " > 0 and every natural number d , there exist a natural number N D N."; d / > d and a Cb2;1 -mapping .b"k /N W RN  Œ0; T0  ! RN such kD1 that Z T0 Z jAN .x; t/ 1=2 .BN .x; t/ b".x1 ; : : : ; xN ; t//j2 t .dx/ dt < "; 0

R1

where BN D .B 1 ; : : : ; B N /. We do not indicate dependence on d where it is meant. Theorem 2.1.5. Assume that conditions (A) and (B) hold. Then, the set P contains at most one element. Let us illustrate condition (B) by several examples. We shall use the following notation: given a sequence  D .n /n>1 of positive numbers, the weighted Hilbert space 1 n o X 2 2 l D x D .xn /W kxkl 2 D n xn2 < 1 

nD1

will be equipped with the inner product hx; yi D

P1

nD1

n x n y n .

37

Kolmogorov operators and SPDEs

Example 2.1.6. (i) Let B k depend only on the variables t; x1 ; x2 ; : : : ; xk . Then, in order to ensure our condition (B), we need only the inclusion jB k j 2 L2 ./ for all k > 1. Indeed, we set N D d and approximate each function B k separately. (ii) Let ˛ D .˛k /k>1 , ˛k > 0 for each k 2 N and 1=˛ WD .˛k 1 /k>1 . Suppose that aij satisfy condition (A) and that there exists a positive number C independent of N such that jAN .x; t/ 1=2yj 6 C kykl 2 1=˛

for all x, t and y D .y1 ; y2 ; : : : ; yN ; 0; 0; : : :/. For example, this is true if aij D 0 for i ¤ j and ai i D ˛i . 2 Let .B k .x; t// 2 l1=˛ for -almost every .x; t/ and let kBkl 2 2 L2 ./. For 1=˛ every " > 0 and every natural number d , we pick a number M > d such that Z T0 Z 1 X ˛k 1 jB k j2 dt dt < "=2: R1

0

kDM C1

Then, for every B k , we find a smooth function b"k depending on the first nk variables such that Z T Z 0 ˛k 1 jB k b"k j2 dt dt < ".2M / 1 ; k D 1; : : : ; M: 0

R1

Set N D maxfM; n1 ; n2 ; : : : ; nM g and b"k  0 for k > N . Then, N Z X

kD1

0

T0

Z

R1

D

˛k 1 jB k

M Z X

kD1

C

T0

0

b"k j2 dt dt

Z

R1

Z N X

kDM C1

˛k 1 jB k

0

T0

Z

R1

b"k j2 dt dt ˛k 1 jB k j2 dt dt < ":

(iii) Finally, for aij as in (ii), we can combine both examples. Let B D G C F , where G k ; F k 2 L2 ./; and

2 G k .x; t/ D G k .x1 ; x2 ; : : : ; xk ; t/; F .x; t/ 2 l1=˛

kF k1=˛ 2 L2 ./:

Obviously, for given B k of this type, the set of all probability solutions  D t .dx/ dt to (2.1.3) satisfying the previous integrability conditions is convex.

38

M. Röckner

An important and interesting problem concerns the study of conditions for uniqueness of solutions to nonlinear FPK equations. The accomplished results and our analysis of the linear case can give some insight and serve as auxiliary tools. Another important direction pursued by many researchers is related to the study of uniqueness for martingale problems associated with second order operators for which we consider FPK equations. Certainly, when dealing with uniqueness for martingale problems and connections between various kinds of uniqueness (the Cauchy problem for FPK equations, martingale problems, uniqueness of semigroups, etc.), it is quite natural to consider also pseudo-differential operators. 2.1.3 Bounds for solution densities Here, we mention some typical results on upper bounds for solution densities. Theorem 2.1.7. Suppose that  is a probability measure on Rd such that L  D 0, where A and A 1 are uniformly bounded, A is Lipschitz on every ball of radius 1 with a constant independent of the centre of the ball, and b satisfies the following condition with some p > d : either supx2Rd kbkLp .U.x;1// < 1

or supx2Rd kbkLp .U.x;1/;/ < 1.

Then, the continuous version % of the density of  is uniformly bounded. If jbj 2 Lp ./, then % 2 W p;1 .Rd /. In the next theorem, the conditions ensure bounds of the form %.x/ 6 C=ˆ.x/. For simplicity, we formulate this result for the unit diffusion matrix, but a Sobolev differentiable matrix leads only to slightly longer assumptions. Theorem 2.1.8. Suppose that  D % dx is a probability measure on Rd satisfying the elliptic equation L  D 0 with A D Id and b such that either jbj 2 L˛loc .Rd / and

or jbj 2 L˛loc .jj/, where ˛ > d ,

jbj 2 Lˇ .jj/;

where ˇ > 1:

Let ˆ 2 C 1 .Rd / be a positive function such that, for some  > d , one has ˆ 2 L1 ./; jrˆj 2 L ./: Then, %.x/ 6

C ˆ.x/

with some constant C . For example, if jbj is locally bounded and belongs to L2 ./, the only real restriction in this theorem is the integrability of jrˆj % over the whole space. In turn, this

39

Kolmogorov operators and SPDEs

integrability in many cases can be checked by means of suitable Lyapunov functions. For example, for obtaining a polynomial decay of %, it suffices to use ˆ.x/ D jxjm , which requires the integrability of jxjN %.x/ for a sufficiently large N . To ensure this, it is enough that .x; b.x// be estimated by a negative constant outside of a ball. If we want to get a bound %.x/ 6 Ce

kjxj

;

it suffices that b is locally bounded and ek jxj %.x/ is integrable for some  > d . For a Gaussian decay 2 %.x/ 6 Ce kjxj ; we need (in addition to the local boundedness of b or integrability to some power 2 larger than d ) the integrability of ek jxj %.x/ with some  > d . The latter can be checked in terms of Lyapunov functions. For example, it suffices to have a bound .b.x/; x/ 6 c1 c2 jxj2 . An interesting feature of such sufficient conditions is that the behaviour of the component of b.x/ orthogonal to x does not matter. Similar results have been obtained in the parabolic case. Again for simplicity, we consider the unit diffusion matrix. Theorem 2.1.9. Let ˆ 2 C 2 .Rd / be a positive function such that Z ˆ.x/%.x; t/ dx < 1; sup t 2.0;T / Rd

jˆj ˆ; .jbj jrˆj/ ˆ

1

; jbj2 ˆ; jrˆj2 ˆ2

1

2 L1 ./;

where > .d C 2/=2. Then, there exists a number C > 0 such that %.x; t/ 6 C

1 1 t d=2 ˆ.x/

for all .x; t/ 2 Rd  .0; T /.

There are also lower bounds on solution densities in the elliptic and parabolic cases, which is connected with Harnack’s inequality. A detailed discussion of such results is given in [11]. An interesting direction of research in this area concerns the infinite-dimensional case. There, many natural questions of a very basic character remain open. For example, there are no broad sufficient conditions ensuring that probability solutions to stationary and parabolic FPK equations are strictly positive when applied to balls in Hilbert spaces (a property which in the finite-dimensional case is much weaker than the positivity of densities). Returning to the framework described at the end of the previous section, one can consider the elliptic equation on R1 with aij D ıij and continuous bounded b i . Suppose that  is a probability solution on R1 . Let E be a Hilbert space continuously embedded into R1 (say, a weighted Hilbert space of sequences) with .E/ D 1. Is it true that  is positive on balls in E? Of course, this is stronger than being positive on non-empty open sets in R1 , since such open sets contain open cylinders with finite-dimensional bases, so that the positivity on them follows from the finite-dimensional case. However, the finite-dimensional case does not help when we consider balls in E.

40

M. Röckner

Remark 2.1.10. There is an obvious connection between the results in this subsection and the general theory of heat kernel estimates, which are also intensively studied in Chapter 3. For a particularly singular case see [37].

2.2 Three selected results on SPDEs V. Barbu and M. Röckner 2.2.1 The stochastic total variation flow The main reference for this section is [6]. Consider the nonlinear diffusion equation h i dX.t/ D div sign .rX.t// dt C X.t/ dW .t/ on .0; T /  O ; X D 0

on .0; T /  @O ;

(2.2.1)

2

X.0/ D x 2 L .O /;

where T > 0 is arbitrary and O is a bounded, convex, open set in RN , @O smooth; W .t; / WD

1 X

k ek ./ˇk .t/;

kD1

.t; / 2 .0; 1/  O

with ˇk , k 2 N, independent Brownian motions on .; F ; .Ft /; P/, k 2 R and ek , k 2 N, the eigenbasis of the Dirichlet Laplacian D on O . Furthermore, the N mapping signW RN ! 2R (multi-valued!) is defined by 8 0,  2 O .

Kolmogorov operators and SPDEs

41

Remark 2.2.1. (i) In nonlinear diffusion theory, (2.2.1) is derived from the continuity equation perturbed by a Gaussian process proportional to the density X.t/ of the material, that is, dX.t/ D div J.rX.t// dt C X.t/ dW .t/;

where J D sign is the flux of the diffusing material. (See [15, 16, 17].) (ii) (2.2.1) is also relevant as a mathematical model for faceted crystal growth under a stochastic perturbation as well as in materials science (see [18] for the deterministic model and complete references on the subject). The resulting equations are differential gradient systems corresponding to a convex and nondifferentiable potential (energy). (iii) Other recent applications refer to the PDE approach to image recovery (see, e.g., [12] and also [4, 13]). In fact, if x 2 L2 .O / is the blurred image, one might find the restored image via the total variation flow X D X.t/ generated by (2.2.1). In its deterministic form, this is the so-called total variation based image restoration model and its stochastic version (2.2.1) arises naturally in this context as a perturbation of the total variation flow by a Gaussian (Wiener) noise.

In [6], we prove the existence and uniqueness of variational solutions to (2.2.1) in all dimensions N > 1 and for all initial conditions x 2 L2 .O /. We would like to stress that one main difficulty occurs when x 2 L2 .O / n H01 .O /, while the case x 2 H01 .O / is more standard. Furthermore, we prove the finite-time extinction of solutions with positive probability, if N 6 3, generalising corresponding results from [2] and [3] obtained in the deterministic case. Let us explain and formulate both results in more detail. Below, we use the following standard notation. Lp .O / WD standard Lp -spaces with norm j
jp ; p 2 Œ1; 1

1;p W.0/ O WD standard (Dirichlet)Sobolev spaces in Lp .O/; p 2 Œ1; 1/;

with norm kuk1;p WD

Z

O

p

jruj d

1=p

.d D Lebesgue measure on O /

H01 .O / WD W01;2 .O /: BV .O / WD space of functions u W O ! R with bounded variation Z  1 N kDuk WD sup u div ' d W ' 2 C0 .O I R /; j'j1 6 1 O   Z jruj d; if u 2 W 1;1 .O / : D O

0

BV .O / WD all u 2 BV .O / vanishing on @O :

42

M. Röckner

Define .u/ WD

(

kDuk C C1;

R

@O

j 0 .u/j dHN

1

;

if u 2 BV .O / \ L2 .O /; if u 2 L2 .O / n BV .O /;

where 0 .u/ is the trace of u on the boundary and dHN Then, for the subdifferential @ of , one can show that

1

is the Hausdorff measure.

@.u/ D div sign.ru/ (as multivalued maps). Hence, we can rewrite (2.2.1) as dX.t/ C @.X.t// dt 3 X.t/ dW .t/; t 2 Œ0; T ;

X.0/ D x 2 L2 .O /:

(2.2.2)

However, since the multi-valued mapping @ W L2 .O / ! L2 .O / is highly singular, for arbitrary initial conditions x 2 L2 .O /, no general existence result for stochastic infinite-dimensional equations of subgradient type is applicable to the present situation. Our approach is to rewrite (2.2.2) as a stochastic variational inequality (SVI), which is equivalent to (2.2.2) if  is regular enough. Definition 2.2.2. Let 0 < T < 1 and let x 2 L2 .O /. A stochastic process X W Œ0; T  ! L2 .O / is said to be an SVI-solution to (2.2.1) if the following conditions hold. (i) X is .Ft /-adapted, has P-a.s. continuous sample paths in L2 .O / and X.0/ D x. (ii) X 2 L2 .Œ0; T   I L2 .O //, .X / 2 L1 .Œ0; T   /. (iii) For each .Ft /-progressively measurable process G 2 L2 .Œ0; T   I L2 .O // and each .Ft /-adapted L2 .O /-valued process Z with P-a.s. continuous sample paths such that Z 2 L2 .Œ0; T   I H01 .O // which solve the equation Z t Z t Z.t/ Z.0/ C G.s/ ds D Z.s/ dW .s/; t 2 Œ0; T ; 0

0

we have

1 EjX.t/ 2

Z.t/j22

Z

t

1 Ejx Z.0/j22 2 0 Z tZ Z t 1 .X. / Z. //2 d d CE .Z. // d C E 2 0 O 0 Z t CE hX. / Z. /; G. /i d; t 2 Œ0; T ; CE

.X. // d 6

0

where h
;
i denotes the inner product in L2 .O /.

43

Kolmogorov operators and SPDEs

Now, we can state the two main results in this section. Theorem 2.2.3. Let O be a bounded and convex open subset of RN , N 6 3, with smooth boundary and T > 0. For each x 2 L2 .O /, there exists an SVI-solution X to (2.2.1), and X is the unique solution in the class of all solutions X such that, for some ı > 0, X 2 L2Cı .I L2 .Œ0; T I L2 .O ///: Furthermore, X has the following properties: (i) X 2 L2 .I C.Œ0; T I L2 .O ///:  2 p .p (ii) sup EŒjX.t/jp2  6 exp C1 2 t 2Œ0;T 

 1/ kxkp2 ; for all p 2 Œ2; 1/:

(iii) Let x; y 2 LN .O / and X x ; X y be the corresponding variational solutions with initial conditions x; y, respectively. Then, for some positive constant C D 2 C.N; C1 /, # " E

sup jX x . /

 2Œ0;T 

X y . /jN N

6 2jx

CT yjN : Ne

(iv) If x > 0, then X.t/ > 0 8 t 2 Œ0; T .

(v) If x 2 H01 .O /, then, for some C > 0 (independent of x), # " E

sup kX.t/k21;2

t 2Œ0;T 

6 C kxk21;2 ;

hence X 2 L2 .I L1 .Œ0; T I H01 .O ///. In the proof of Theorem 2.2.3, we approximate (2.2.1) by dX D div e .aX / dt C X dW X D 0 on .0; T /  @O ;

in .0; T /  O ;

(2.2.3)

X .0/ D x 2 L2 .O /;

where  2 .0; 1, e .u/ D  .u/ C u; 8u 2 RN . Here, mation of the function .u/ D sign u, that is, 8 1 if juj 6 ; <  u;  .u/ D : u ; if juj > : juj



Then we prove convergence when  ! 0 (see [6] for details).

is the Yosida approxi-

44

M. Röckner

Theorem 2.2.4 (Finite time extinction). Let N D 2 or 3. Let X be as in Theorem 2.2.3, with initial condition x 2 LN .O /, and let  WD infft > 0 W jX.t/jN D 0g. Then, we have Z t  1  PŒ 6 t > 1  1 e C s ds jxjN ; 8t > 0: 0

Here,  WD inffjyjW 1;1 .O/ =jyj N W y 2 W01;1 .O /g and C  WD N 1 0 particular, if jxjN < =C  , then PŒ < 1 > 0:

2 C1 2

.N

1/. In

2.2.2 A new approach to SPDEs The main reference for this section is [7]. Consider the following stochastic differential equation with linear multiplicative noise on a separable Hilbert space .H; h ; i/ dX.t/ C A.t; X.t// dt D X.t/ dW .t/; X.0/ D x 2 H:

t 2 Œ0; T ;

(2.2.4)

Example 2.2.5 (Stochastic porous media equation). dX.t/

. .X.t// dt D X.t/ dW .t/

1

on H

.O /; O  Rd ; O open.

 W R ! R continuous, increasing  r .r/ > c1 r p c2 , c1 ; c2 2 .0; 1/, p 2 .1; 1/  j .r/j 6 c3 jrjp 1 C c4 , c3 ; c4 2 .0; 1/ The state space H consists of functions or Schwartz distributions on an open O  Rd ; @O smooth; e.g. H D L2 .O/ or H D Sobolev space H0k .O/ or H k .O/, k 2 N. Assume there exists a reflexive Banach space V  H , continuously and densely, hence we obtain the Gelfand triple 0

0

0

V  H . H /  V :

We may assume that V , V are strictly convex. (Otherwise, we change to an appropriate equivalent norm on V ; according to Asplund’s theorem.) A typical example is given by H01 .O /  L2 .O /  H

1

.O /:

The noise in (2.2.4) is of type W D W .t/; t > 0; Wiener process on H of type W .t; / WD

1 X

j D1

j ej ./ˇj .t/;  2 O ; t > 0;

45

Kolmogorov operators and SPDEs

where: (a) ˇj ; j 2 N, are independent (real) Brownian motions on a stochastic basis .; F ; .Ft /; P/, (b) ej 2 C 2 .ON / \ H; j 2 N, is an ONB of H , and there exist j 2 .0; 1/ such that 8j 2 N jy ej jH 6 j jej j1 jyjH ; y 2 H I (c) j 2 R; j 2 N, such that  WD

1 X

j D1

2j .1 C 2j / jej j21 < 1:

Our assumptions on the drift A (where we write A.t; u/ instead of A.t; u; !/) are: 0 AW Œ0; T   V   ! V is progressively measurable and there exist p 2 .1; 1/, ı 2 Œ0; 1/, c1 2 .0; 1/, ci 2 R, i 2 f2; : : : ; 6g such that on , 8t 2 Œ0; T , 0  (“demicontinuous”) A.t;
/ W V ! V is strongly-weakly continuous,  (“quasi-monotone”) V

0

hA.t; u/

 (“coercive”) V

0

A.t; v/; u

ı ju

viV >

vj2H

8u; v 2 V;

hA.t; u/; uiV > c1 jujpV C c2 juj2H C c3

 (“boundedness”)

jA.t; u/jV 0 6 c4 jujVp

1

C c5 jujH C c6

We fix p 2 .1; 1/ as above from now on and set p0 WD

8u 2 V; 8u 2 V:

p . p 1

Definition 2.2.6. An H -valued, .Ft /-adapted process X.t/, t 2 Œ0; T , on .; F ; P/, with P-a.s. continuous sample paths is called solution to (2.2.4), if

X 2 L2 .0; T /  I H \ Lp .0; T /  I V and

X.t/ C

Z

0

t

A.s; X.s// ds D x C

Z

0

t

X.s/ dW .s/;

t 2 Œ0; T :

Theorem 2.2.7. Under the above hypotheses on the noise and drift, (2.2.4) has a unique solution for every x 2 H . Moreover, t 7! e W .t / X.t/ is V 0 -absolutely continuous on Œ0; T  P-a.s and Z T ˇ ˇp0 ˇ W .t / d ˇ .e W .t / X.t//ˇ 0 dt < 1: E ˇe V dt 0

46

M. Röckner

Let us perform the following rescaling transformation. Set y.t/ WD e

W .t /

X.t/; t 2 Œ0; T 

Then, by Itô’s formula, we obtain (proof non-trivial!) the equivalent random ODE dy C e dt y.0/ D x;

W .t /

A.t; eW .t / y.t// C y.t/ D 0

for dt-a.e. t 2 .0; T /

(2.2.5)

and vice versa, where ./ WD

1 1X 2 2 j ej ./;  2 O : 2 j D1

Definition 2.2.8. An H -valued, .Ft /-adapted process y.t/, t 2 Œ0; T , on .; F ; P/ with P-a.s. continuous sample paths is called solution to (2.2.5) if

eW y 2 L2 .0; T /  I H \ Lp .0; T /  I V ;

and P-a.s., t 7! y.t/ is V 0 -absolutely continuous on Œ0; T  satisfying (2.2.5), and ˇ 0 ZT ˇ ˇ W .t / dy ˇp ˇ dt < 1: ˇ E ˇe dt ˇ 0 V

0

Theorem 2.2.9. Under the above hypotheses on the noise and drift, (2.2.5) has a unique solution for every x 2 H . Remark 2.2.10. Theorem 2.2.9 ” Theorem 2.2.7. By the transformation y 7! e tone”, i.e. V

0

hA.t; u/

.Cı/t

A.t; v/; u

y, we may assume that A is “strongly mono-

viV > ju

vj2H

8u; v 2 V:

One main problem is the following: y 7! e

W .t /

A.t; eW .t / y/ is no longer monotone from V to V 0 for fixed t 2 Œ0; T ; ! 2 :

So, we change to another Gelfand triple

V  H  V 0; with spaces V , H, V 0 consisting of stochastic processes with norms scaled by eW and study the resulting equations there. This is the main idea of the new approach to SPDEs, announced in the title of this section.

47

Kolmogorov operators and SPDEs

a) For p > 2:

V WD all V -valued, .Ft /-adapted processes y.t/; t 2 Œ0; T ; such that ! p1 Z Tˇ ˇp ˇ W .t / ˇ jyjV WD E < 1; y.t/ˇ dt ˇe 0

V

H WD all H -valued, .Ft /-adapted processes y.t/; t 2 Œ0; T ; such that ! 21 Z Tˇ ˇ2 ˇ ˇ W .t / y.t/ˇ dt jyjH WD E < 1; ˇe 0

H

V 0 WD all V 0 -valued, .Ft /-adapted processes y.t/; t 2 Œ0; T ; such that ! 10 Z Tˇ p ˇp0 ˇ W .t / ˇ < 1: jyjV 0 WD E y.t/ˇ 0 dt ˇe 0

V

b) For p 2 .1; 2/: Replace V above by V \ H and V 0 by V 0 C H. Now, consider (2.2.5) as an equation on this new Gelfand triple

V  H  V 0; namely as an operatorial equation:

B y C Ay D 0;

(2.2.6)

where A W V ! V 0 and B W D.B /  V ! V 0 are defined by .Ay/.t/ D e W .t / A.t/.eW .t / y.t// y.t/; a.e. t 2 .0; T /; y 2 V ; dy .t/ C . C /y.t/; a.e. t 2 .0; T /; y 2 D.B /; .B y/.t/ D dt n o dy D.B / D y 2 V W y 2 AC.Œ0; T I V 0 / \ C.Œ0; T I H /; P-a.s.; 2 V 0 ; y.0/ D x : dt

Here, AC.Œ0; T I V 0 / is the space of all absolutely continuous V 0 -valued functions on Œ0; T . Remark 2.2.11. (2.2.4) ” (2.2.5) ” (2.2.6). Idea of proof of Theorem 2.2.9. By Remark 2.2.11, it suffices to solve (2.2.6). Claim: A and B are maximal monotone from V to V 0 . To prove that A is maximal monotone is straightforward. Let us prove that B is monotone.

48

M. Röckner

For all y 2 D.B /, by Itô’s formula (non-trivial !), and t 2 Œ0; T , e

W .t /

 Zt  Zt W .s/ dy W .s/ e y.t/ D y.0/ C C e y.s/ ds C eW .s/ y.s/ dW .s/; ds 0

0

and thus by the known Itô formula for j
Zt ˇ2 1 1 ˇˇ W .t / ˇ 2 y.t/ˇ D jy.0/jH C ˇe H 2 2 0

C

j2H :

  E D W .s/ W .s/ dy C y.s/ ; e y.s/ e ds 0 ds V V

Zt D

eW .s/ y.s/; eW .s/ y.s/ dW .s/

0

E

H

Zt 1 ˇ2 1 X 2 ˇˇ W .s/ ˇ j ˇe y.s/ej ˇ ds; t 2 Œ0; T : C H 2 j D1

0

Then, take y1 ; y2 2 D.B /, and set y WD y1 y2 to obtain   Z TD E W .t / W .t / dy C y.t/ ; e y.t/ dt e 0 hB .y/; yiV D E V dt 0 Z TD E eW .t / y.t/; eW .t / y.t/ dt CE 0

ˇ2 1 ˇˇ ˇ > E ˇeW .T / y.T /ˇ > 0; H 2 where the penultimate inequality follows by the definition of  and the above Itô formula for j
j2H . To prove maximal monotonicity is hard and technical. We refer to [7] for details. Using the claim, we obtain that A is max. monotone from V to V 0 and D.A/ D V and that B is max. monotone from V to V 0 . This implies A C B is max. monotone from V to V 0 . Therefore, by the coercivity assumption on A, it is standard to prove that A C B is bijective from D.B / to V 0 . Hence, in particular, there exists a solution to (2.2.6).  Applications of the new approach are numerous. One obtains, e.g., new spatial regularity results for solutions of SPDEs. For details, we refer to [7]. For further analysis of special properties of solution to SPDEs as for example long time behaviour and, in particular, random attractors we refer e.g. to [8]. We conclude this section with the “subgradient case”: Assume that A is the subdifferential of a continuous convex function 'W V ! R, i.e., for t 2 Œ0; T , ! 2 , u 2 V; A.t; u; !/ D @'.t; u; !/ D f 2 V 0 W V 0 h; u

ziV > '.t; u; !/

'.t; z; !/ 8z 2 V g:

49

Kolmogorov operators and SPDEs

Then, A W V ! V 0 defined as above is itself the subdifferential @ˆ W V ! V 0 of the convex lower-semicontinuous function ˆ W V ! R defined by ˆ.y/ D E

Z

T

0



'.t; eW .t / y.t//

 jeW .t / y.t/j2H dt;

8y 2 V :

Let ˆ W V 0 ! R be the conjugate of ˆ, that is, ˆ .v/ WD supfV 0 hv; uiV

ˆ.u/ W u 2 V g:

Then, the solution y to (2.2.6) (equivalently, (2.2.5) or (2.2.4)) is the solution to a minimisation problem. More precisely, y D argmin fˆ.y/Cˆ . B y/ C V 0 hB y; yiV W y 2 D.B /g : 2.2.3 Pathwise uniqueness for SDEs on Hilbert spaces with a merely bounded measurable drift part The main reference for this section is [14]. Consider on a separable (real) Hilbert space .H; h ; iH / (norm j:jH ) for a cylindrical Brownian motion Wt , t > 0, on .; F ; .Ft /; P/, and BW H ! H bounded, Borel, dXt D .AXt rV .Xt / C B.Xt // dt C dWt ; X0 D z 2 H:

t 2 Œ0; T 

(2.2.7)

Example 2.2.12 (Stochastic reaction diffusion equation with bounded measurable perturbation). H WD L2 ..0; 1/; d/;

dXt D .Xt

2

Xt jXt j

p 2 Œ1; 1/

p 1

/ dt C B.Xt / dt C dWt ;

X0 D z 2 L ..0; 1/; d/:

Assumptions: .H1/ AW D.A/  H ! H self-adjoint and A 6 of trace class. .H 2/

t 2 Œ0; T ;

!I for some ! > 0, with A

1

V W H ! . 1; C1 convex, lower-semicontinuous, lower bounded function and let DV be the set of all x 2 fV < 1g such that its Gâteaux derivative rV exists at x.

.H 3c/0 There exists a separable Banach space E  H; continuously and densely embedded, such that E  DV , .E/ D 1 and on E the function V is twice Gâteaux-differentiable such that for all x 2 E its second Gâteaux derivative VE00 .x/ 2 L.E; E 0 / (with E 0 being the dual of E) extends by continuity to an element in L.H; E 0 / such that kVE00 .x/kL.H;E 0 / 6 ‰.jxjE /

50

M. Röckner

for some convex function ‰ W Œ0; 1/ ! Œ0; 1/. Furthermore, for a.e. initial condition z 2 E, there exists a (probabilistically) weak solution X V D X V .t/; t 2 Œ0; T , to (2.2.8) so that E

Z

T

0

‰.jX V .s/jE / ds < 1:

Here, dXt D .AXt X0 D z:

rV .Xt // dt C dWt

(2.2.8)

and WD N.0; Q/, i.e. the centred Gaussian measure in H with covariance Q D 1 A 1 and 2 Z 1 V .x/ .dx/ WD e

.dx/; Z WD e V .x/ .dx/: Z H

Definition 2.2.13. A solution of (2.2.7) in H is a filtered probability space .; F ; .Ft /t >0 ; P/, an H -cylindrical .Ft /-Brownian motion .Wt /t >0 on this space, and a continuous .Ft /-adapted process .Xt /t >0 on this space such that: RT i) Xs .!/ 2 DV for dt ˝ P-a.e. .s; !/ and 0 jhrV .Xs / ; hiH j ds < 1 P-a.s., for every T > 0 and h 2 D .A/;

ii) for every h 2 D .A/ and t > 0, one has P-a.s. Z t
˛

˝ rV Xs ; h H dsChWt ; hiH : hXs ; AhiH C B Xs hXt ; hiH D hz; hiH C 0

W

If X is F -adapted, where F W D FtW W , we say that X is a strong solution.




t >0

is the normal filtration generated by

The main result of this section is the following: Theorem 2.2.14. There is a Borel set SV  H such that .H n SV / D 0, having the following property: If z 2 SV and X , Y are two solutions to (2.2.7) with initial condition z, defined on the same filtered probability space .; F ; .Ft /t >0 ; P/ and w.r.t. the same cylindrical Brownian motion W , then X and Y are indistinguishable processes. Hence by the Yamada–Watanabe theorem they are (probabilistically) strong solutions and have the same law. Remark 2.2.15. Theorem 2.2.14 generalises the seminal work [20] from Rd to infinitedimensional (Hilbert) spaces.

51

Kolmogorov operators and SPDEs

Idea of proof of Theorem 2.2.14. The idea of proof is to rewrite (2.2.7) in “elliptic coordinates” as follows: Let i 2 .0; 1/; i 2 N, be the eigenvalues of A. Let  > 4kjBjH k21 and consider the following PDE’s for i 2 N: . C i /ui

L ui

hB; Dui iH D B i ;

where B i .x/ WD hB.x/; ei iH ; x 2 H . Then, one can prove that each ui is Lipschitz, and 1 X U.x/ WD ui .x/ei ; x 2 H; i D1

defines a bounded, Lipschitz-continuous map U W H ! H , so that, for  large enough, kU kLip 6 21 . Then the new “elliptic” coordinates are given by ' i W H ! R, where ' i .x/ WD h'.x/; ei iH ; i 2 N, with '.x/ WD x C U.x/; x 2 H: Since kU kLip 6 21 , we have 8x; y 2 H 1 jx 2

yjH 6 j'.x/

'.y/jH 6

3 jx 2

yjH :

Rewritten in these new coordinates, Eq. (2.2.7) becomes




i Xti Di V Xt dt C  C i ui Xt dt d' i Xt D ˛
˝ C Dui Xt ; dWt H C dWti ;

(2.2.9)

where i 2 N, Xti WD hXt ; ei i and Wti WD hWt ; ei i (so B does not appear anymore). Fix z 2 SV and let X; Y be two solutions of (2.2.7) on .; F ; .Ft /; P/. Then, by (2.2.9), we have for all i 2 N dŒ' i .Xt /

' i .Yt / D

Œi .Xti

Yti / C Di V .Xt / Di V .Yt / dt C . C i /Œui .Xt /

ui .Yt / dt C hDui .Xt / Dui .Yt /; dWt iH :

Hence, by Itô’s formula, we get for At WD 2

Z

0

t

jrV .Xs / j'.Xs /


2 Z tX 1 ui .Xs / ui .Ys / rV .Ys /jH ds C 2 i ds '.Ys /jH j'.Xs / '.Ys /j2H 0 i D1 ˇ2 ˇ i Z tX 1 ˇ Du .Xs / Dui .Ys /ˇH ds; t > 0; C j'.Xs / '.Ys /j2H 0 i D1

52

M. Röckner

that h E e

At

'.Yt /j2H

j'.Xt / 6 2E

Z

CE CE E

1 62 jXs Ys jH 6j'.Xs / '.Ys /jH

t

e

As

0

CE

i

Z

j'.Xs /

t

e

As

j'.Xs /

e

As

j'.Xs /

e

As

j'.Xs /

e

As

j'.Xs /

0

Z

t

0

Z

t

0

Z

t

0

'.Ys /jH

‚ …„ ƒ jU.Xs / U.Ys /jH

ds

2 jrV .Xs / rV .Ys /jH ds j'.Xs / '.Ys /jH 1 X .ui .Xs / ui .Ys //2 '.Ys /j2H 2 i ds j'.Xs / '.Ys /j2H i D1 ˇ2 ˇ i 1 ˇ X Du .Xs / Dui .Ys /ˇH 2 '.Ys /jH ds j'.Xs / '.Ys /j2H i D1 '.Ys /j2H

'.Ys /j2H dAs :

So, we can apply Gronwall to get P-a.s. 1 jX 4 t

Yt j2H 6 j'.Xt /

'.Yt /j2H D 0;

t > 0:

It remains to show that At < 1

P-a.s.

For the proof of this, which is quite hard and technical, we refer to [14].



The case where in (2.2.7) the nonlinear part of the drift is not gradient type is much more difficult and is at present under further study.

References [1] S. Albeverio, Y.G. Kondratiev, Y. Kozitsky and M. Röckner, The Statistical Mechanics of Quantum Lattice Systems: A Path Integral Approach, EMS Tracts in Mathematics 8, European Mathematical Society, Zürich, 2009. [2] F. Andreu, V. Caselles, J. Díaz and J. Mazón, Some qualitative properties for the total variation flow, J. Funct. Anal. 188(2) (2002), 516–547. [3] F. Andreu-Vaillo, V. Caselles and J.M. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progress in Mathematics 223, Birkhäuser, Basel, 2004. [4] T. Barbu, V. Barbu, V. Biga and D. Coca, A PDE variational approach to image denoising and restoration, Nonlinear Anal. Real World Appl., 10(3) (2009), 1351–1361.

Kolmogorov operators and SPDEs

53

[5] V. Barbu, G. Da Prato and M. Röckner, Stochastic Porous Media Equations, LNM 2163, Springer, Cham, 2016. [6] V. Barbu and M. Röckner, Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise, Arch. Ration. Mech. Anal. 209(3) (2013), 797–834. [7] V. Barbu and M. Röckner, An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise, J. Eur. Math. Soc. (JEMS) 17(7) (2015), 1789–1815. [8] W.-J. Beyn, B. Gess, P. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations 36(3) (2011), 446– 469. [9] V.I. Bogachev, G. Da Prato, M. Röckner and S.V. Shaposhnikov, An analytic approach to infinite-dimensional continuity and Fokker–Planck–Kolmogorov equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 14(3) (2015), 983–1023. [10] V.I. Bogachev, N.V. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Commun. Partial Diff. Equs. 26(11–12) (2001), 2037–2080. [11] V.I. Bogachev, N.V. Krylov, M. Röckner and S.V. Shaposhnikov, Fokker–Planck–Kolmogorov Equations, Mathematical Surveys and Monographs 207, American Mathematical Society, Providence, RI, 2015. [12] A. Chamballe and P.L. Lions, Image recovery via total variation minimization, Numer. Math. 76(2) (1997), 17–31. [13] T. Chan, S. Esedogly, F. Park and A. Yip, Total variation image restoration. Overview and recent developments, In Handbook of Mathematical Models in Computer Vision, Springer, New York (2006), 17–31. [14] G. Da Prato, F. Flandoli, M. Röckner and A. Veretennikov, Strong uniqueness for SDEs in Hilbert spaces with nonregular drift, Ann. Probab. 44(3) (2016), 1985–2023. [15] Y. Giga and R. Kobayashi, On constrained equations with singular diffusivity, Methods Appl. Anal. 10(2) (2003), 253–278. [16] Y. Giga and R.V. Kohn, Scale invariant extinction time estimates for some singular diffusion equations, Discr. Cont. Dynam. Syst. A 30(2) (2011), 509–535. [17] M.H. Giga and Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal. 159(4) (2001), 295–333. [18] R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Stat. Phys. 95(5–6) (1999), 1187–1220. [19] W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Springer, Cham, 2015. [20] A.Y. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sbornik 39 (1981), 387–403.

Chapter 3

Analysis and stochastic processes on metric measure spaces A. Grigor’yan This contribution deals with the properties of certain differential and nonlocal operators on various spaces, with the emphasis on the relationship between the analytic properties of the operators in question and the geometric properties of the underlying space. In most situations, these operators are Markov generators. In such cases, we are also concerned with probabilistic aspects, such as the path properties of the corresponding Markov process.1

3.1 Analysis on manifolds The main object of interest in this part of the project is the Laplace–Beltrami operator  on a Riemannian manifold M . In most cases M can be assumed to be geodesically complete and non-compact. Denote by B.x; r/ the geodesic ball on M of radius r centered at x 2 M , and by V .x; r/ the Riemannian volume of B.x; r/. A manifold M is called parabolic if any positive superharmonic function on M is constant. It is known that the following properties are equivalent:  M is parabolic;

 there is no positive Green function of  on M ;  Brownian motion on M is recurrent;

(see [16]). 3.1.1 Elliptic operators 3.1.1.1 Semi-linear elliptic inequalities Consider on M the differential inequality u C u 6 0

(3.1.1)

where  > 1 is a constant, and ask if it has a positive solution u on M . This question was initially motivated by certain problems in differential geometry, but after many years of research of many authors it has become a popular question in PDEs. 1 Project

A6

56

A. Grigor’yan

A classical result of Gidas and Spruck [15] says that the equation u C u D 0

in Rn

has no positive solution, whereas for any  > nC2 this with n > 2 and  < nC2 n 2 n 2 equation has a positive solution. The case of an inequality (3.1.1) has a different answer: If  6 n n 2 then (3.1.1) has no positive solution, whereas for  > n n 2 there are positive solutions. The existing methods of handling the differential inequality (3.1.1) and various generalisations use quite strongly specific properties of PDEs in Rn (see, for example, [47]). Here, we are interested in understanding minimal geometric assumptions needed for the non-existence of a positive solution of (3.1.1). Assume that M is geodesically complete. A famous theorem of Cheng and Yau [10] says that if, for some x and all r  1, V .x; r/ 6 C r 2 ; (3.1.2) then M is parabolic. Since a solution of (3.1.1) is superharmonic, we see that under (3.1.2) the inequality (3.1.1) has no positive solution. The following is a combined result of [29] and [43]. Theorem 3.1.1. Let M be a geodesically complete, non-compact manifold. If, for some x 2 M and all r  1, V .x; r/ 6 C r p logq r;

(3.1.3)

where

2 1 and q D ;  1  1 then the inequality (3.1.1) has no positive solution. p D

(3.1.4)

Note that p > 2 so that the assumption (3.1.3) is weaker than (3.1.2). The conditions (3.1.3)–(3.1.4) are sharp in the following sense: if p D

2 

1

and q >

1 

1

;

then there is an example of M satisfying (3.1.3) and having a positive solution of (3.1.1). 3.1.1.2 Negative eigenvalues of Schrödinger operators Let ˆ be a non-negative function on Rn . Denote by Neg.ˆ; Rn / the number of negative eigenvalues of the Schrödinger operator H D  ˆ.x/ on Rn , assuming that ˆ is such that the operator H with domain C01 .Rn / is essentially self-adjoint in L2 .R/. In the case n > 3, it is known that Z n ˆn=2 dx; (3.1.5) Neg.ˆ; R / 6 Cn Rn

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which is the content of a celebrated theorem of Cwikel–Lieb–Rozenblum (see [11, 46, 48]). In the case n D 2, this estimate is not true, and an equally good upper bound for Neg.ˆ/ is still unknown. However, Neg.ˆ; R2 / admits a lower bound: Z ˆdx; Neg.ˆ; R2 / > c R2

where c > 0 is an absolute constant, which was proved in [36]. Obtaining good enough upper bounds for Neg.ˆ; R2 / is unexpectedly difficult. A major contribution to this area was done by M. Solomyak [50], which was then improved by E. Shargorodsky [49]. In [35] we obtained a new type of upper bounds. Fix some p > 1 and define, for any non-negative integer n, the following quantities: Z ˇ
ˇ ˆ.x/ 1 C ˇ ln jxjˇ dx; An .ˆ/ D n 1

fe2

Bn .ˆ/ D

Z

n

0, we have   R M g.x; z/ g.z; y/ ˆ.z/dz : (3.1.7) gˆ .x; y/ > g.x; y/ exp g.x; y/ A striking feature of this result is that it does not require any restriction on M . Moreover, the same result holds in a higher generality of abstract harmonic spaces. 3.1.2 The heat equation A central object in the analysis on manifolds is the heat kernel pt .x; y/, that is, the fundamental solution of the heat equation @t u D u; where t > 0 denotes time and x; y are points of M . For example, if M D Rn , then the heat kernel is given by the classical Gauss–Weierstrass formula ! 1 jx yj2 pt .x; y/ D : exp 4t .4 t/n=2 The problem of obtaining heat kernel estimates under certain geometric assumptions on the underlying manifold M has been extensively studied for several decades (see [8], [12, 51]). For example, if the manifold M is geodesically complete and has non-negative Ricci curvature, then, by a theorem of Li and Yau [45],   d 2 .x; y/ C ; (3.1.8) pt .x; y/  p exp ct V .x; t/ where d is the geodesic distance on M , and C; c are positive constant. The sign  means that both 6 and > are true, but with different values of C; c. 3.1.2.1 Heat kernels on connected sums Here, we consider heat kernel estimates on the connected sum M1 #M2 of two manifolds M1 ; M2 of equal dimensions. By definition, M1 #M2 denotes any manifold that is obtained by connecting exterior domains in M1 and M2 via a compact connected manifold. For example, even estimating the heat kernel on Rn #Rn is a highly non-trivial task. Although the first approach to the latter problem was initiated in [6] in 1996, the full answer was only obtained in [41] in 2009.

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Theorem 3.1.4. If x; y are two points lying on different sheets of M D Rn #Rn with n > 3, then, for large enough t; jxj ; jyj,   d 2 .x;y/ C 1 1 ct pt .x; y/  n=2 : e C t jxjn 2 jyjn 2 More generally, consider a connected sum M D M1 # : : : #Mk , where we assume that, for each manifold Mi , the heat kernel satisfies the two-sided Li–Yau estimate (3.1.8). The question of estimating the heat kernel on such a manifold M was largely solved in a series of papers of A. Grigor’yan and L. Saloff-Coste culminating in [41]. A remarkable observation of [41] is that one has to distinguish parabolic and non-parabolic ends Mi . The results of [41] are exhaustive when the manifold M is non-parabolic, that is, when at least one end Mi is non-parabolic. Assume also that, for some oi 2 Mi and all large enough r, V .oi ; r/ ' r ˛i ; where ˛i > 0. Denote jxj D d.x; oi /. Theorem 3.1.5. Assume that ˛i ¤ 2 for all 1 6 i 6 k. Set  ˛i ; if ˛i < 2;  ˛i D 4 ˛i ; if ˛i > 2; and

˛ D min f˛i W 1 6 i 6 kg :

Then, for all t  1, x 2 Mi and y 2 Mj with i ¤ j and large enough jxj ; jyj, ! 1 1 1 C  C ˛ =2 pt .x; y/  C     t ˛=2 jxj˛i 2 jyj˛j 2 t j jxj˛i 2 t ˛i =2 jyj˛j 2   d 2 .x; y/ 2 ˛j / C .2 ˛i /C . exp  jxj : jyj ct In the case when k D 2 and ˛1 D ˛2 D n > 2, we get a D ˛i D n, and we obtain the estimate of Theorem 3.1.4. Consider an example of a mixed case M D M1 #M2 with M1 D R1C  S2

and M2 D R3 :

In this case, the manifold M1 is parabolic with the volume growth exponent ˛1 D 1, and M2 is non-parabolic with ˛2 D 3. It follows that ˛1 D 4

˛1 D 3;

˛2 D ˛2 D 3;

and ˛ D min.˛1 ; ˛2 / D 3:

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Hence, if x 2 M1 and y 2 M2 , Theorem 3.1.5 implies   d 2 .x;y/ jxj C ct e : pt .x; y/ D 3=2 1 C jyj t The parabolic case was treated in [41] only in a special case, while the general parabolic case still remains open. In particular, the following estimate was proved for M D R2 #R2 (equivalently, for a catenoid) in [41]. Theorem 3.1.6. Let x and y p be two points that lie on different sheets of M D R2 #R2 . Then, for jxj ; jyj > t  1,   d 2 .x;y/ C 1 1 ct pt .x; y/  e C ; t log jxj log jyj p while for jxj ; jyj 6 t we have p p
C 2 : pt .x; y/  log p t C log t log jxj log jyj t log2 t The proofs in [41] are based on [39, 38, 40, 42]. 3.1.2.2 Heat kernels of Schrödinger operators Consider in Rn the Schrödinger operator H D Cˆ

where ˆ > 0 is a smooth function, and let ptˆ .x; y/ be the heat kernel of H . Here, we describe some results about the estimates of ptˆ obtained in [17] using the method of h-transform from [40]. It is well known that, if n > 2 and, for some " > 0, ˆ.x/ 6 C jxj

.2C"/

; for all jxj > 1;

then ptˆ .x; y/ 

C

jx yj2 ct

: (3.1.10) t This estimate reflects the fact that potentials with the upper bound (3.1.9) are small perturbations of the Laplace operator (so-called short range potentials), so that the estimate (3.1.10) is obtained by a perturbation argument. The case n D 2 is quite different as stated below. Set n=2

e

(3.1.9)

hxi WD 2 C jxj : Theorem 3.1.7. Let ˆ be a non-zero function with compact support in R2 . Then, the heat kernel of H satisfies ptˆ .x; y/ 

C loghxi loghyi p p e t log.hxi C t / log.hyi C t/

jx yj2 ct

:

(3.1.11)

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Analysis and stochastic processes on metric measure spaces

In particular, in the range t > hxi2 C hyi2 , we have loghxi loghyi : t log2 t

ptˆ .x; y/ '

Now, let us consider the most interesting potential ˆ.x/ D b jxj

2

; for all jxj > 1;

(3.1.12)

which is on the borderline between the short and long range potentials. Theorem 3.1.8. Let ˆ be a potential (3.1.12) in Rn with n > 2. Then, the heat kernel of H , for all t > 0 and x; y 2 Rn , satisfies the estimate: ptˆ .x; y/



C t n=2Cˇ



1 1 p C hxi t



where ˇ D

n C1C 2

ˇ



r

1 1 p C hyi t n 2

1

2



ˇ

e

jx yj2 ct

;

(3.1.13)

C b:

In particular, in the most interesting range t > hxi2 C hyi2 ; the estimate (3.1.13) becomes hxiˇ hyiˇ ptˆ .x; y/ ' n=2Cˇ : t Note that the value of the coefficient b in (3.1.12) determines the exponent n2 C ˇ of the power decay of the heat kernel as t ! 1. Since b takes values in .0; 1/, the exponent of t ranges in . n2 ; 1/. For comparison, let us mention that, for a long range potential ˆ.x/ D b jxj

.2 ˛/

; for all jxj > 1;

with ˛ 2 .0; 2/, the long time decay of the heat kernel is already superpolynomial as follows:  2 ˛ ptˆ .0; 0/  C exp ct 2C˛ : 3.1.2.3 Heat kernels of operators with singular drift Consider in Rn n f0g the operator Lu D u r
ru with a singular potential .x/ D jxj

˛

;

where ˛ > 0. We have proved in [37] the following estimates of the heat kernel of L.

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Theorem 3.1.9. For all 0 < t < 1, we have  sup pt .x; y/ 6 exp C t

˛ ˛C2

x;y

and

 sup pt .x; x/ > exp ct x

for some positive constants C; c.

˛ ˛C2



 ;

The singularity of the drift term at the origin causes a higher rate of blow up of the heat  kernel  at t ! 0, and the fact that the latter should be given by the term ˛ exp t ˛C2 is not obvious at all and was not predicted by any “physical” argument. By a suitable transformation, we reduce the problem to estimating the heat kernel of a weighted Laplace operator, and the latter amounts to proving a certain isoperimetric inequality on a weighted manifold .Rn ; /, where the measure  is given by   1 dx: d D exp jxj˛ Due to specific properties of this measure , the previously known methods for obtaining isoperimetric inequalities on warped products did not work, and we had to develop [37] a new machinery for that. 3.1.3 Escape rate of Brownian motion A manifold M is called stochastically complete if Brownian motion on M has lifetime 1, which is equivalent to the condition Z pt .x; y/dy D 1 M

for all x 2 M and t > 0. It is known that a geodesically complete Riemannian manifolds is stochastically complete provided Z 1 rdr D 1 (3.1.14) log V .x; r/ for some x 2 M . It was proved in [19] that, under the condition (3.1.14), one can also obtain quantitative estimate on how fast Brownian motion escapes to 1. The following result was proved in [19]. Theorem 3.1.10. Let M be a Cartan–Hadamard manifold satisfying (3.1.14). Fix a point x 2 M and define a function '.t/ for large t by the identity Z '.t / r dr t D : log V .x; r/ 1 Then, Brownian motion on M at time t stays in the ball B.x; '.C t// for large enough t with probability 1, where C > 0 is an absolute constant (for example, C D 130).

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63

In other words, the function R.t/ D '.C t/ is an upper rate function of Brownian motion. Examples of spherically symmetric manifolds show that this estimate of the escape rate in terms of V is essentially sharp. For example, if V .x; r/ ' r ˛ ; we obtain an upper rate function

R.t/ D const

p

t log t:

(3.1.15)

Note for comparison that, by Khinchine’s law of the iterated logarithm, an optimal upper rate function in Rn is p R.t/ D .4 C "/t log log t:

The function (3.1.15) is therefore not optimal in Rn because of the distinction between log log t and log t, but it is sharp in the class of all manifolds with polynomial volume growth (see [28]). Historically, the upper rate function (3.1.15) was obtained by Hardy and Littlewood in 1914 for sums of independent Bernoulli random variables, which, however, was superseded within ten years by Khinchine’s law. From the modern point of view, the Hardy–Littlewood function (3.1.15) still make sense as an optimal upper rate function for Brownian motion on manifolds with polynomial volume growth.

3.2 Analysis on metric measure spaces 3.2.1 Heat kernels on fractal-like spaces Let .M; d; / be a metric measure space, that is, .M; d / is a metric space and  is a Radon measure on M with full support. We denote by B.x; r/ the metric balls in M and assume that all metric balls are precompact. Set V .x; r/ D  .B.x; r//. Let .E ; F / be a regular Dirichlet form in L2 .M; / (see [14]). We investigate the properties of the Hunt process associated to the Dirichlet form, and its heat kernel pt .x; y/ that is defined as the integral kernel (provided it exists) of the corresponding heat semigroup. We distinguish two main cases: when the Dirichlet form .E ; F / is local, that is, the associated process is a diffusion, and when the Dirichlet form .E ; F / is of jump type, that is, it is given by Z Z .f .x/ f .y//.g.x/ g.y//J.x; y/d.x/d.y/; (3.2.1) E .f; g/ D M

M

where J is a symmetric jump kernel.

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Fix two positive parameters ˛; ˇ. We look for conditions on the measure and energy that would ensure the following heat kernel bounds:  sub-Gaussian bound in the local case: 0 C pt .x; y/  ˛=ˇ exp @ c t

ˇ

d .x; y/ t

! ˇ1 1 1

AI

(3.2.2)

 stable-like bound in the jump case: pt .x; y/ 

C t ˛=ˇ

  d.x; y/ 1 C 1=ˇ t

.˛Cˇ /

D

Ct t 1=ˇ C d.x; y/


˛Cˇ : (3.2.3)

It was proved in [23] that, in both cases, ˛ is the Hausdorff dimension of .M; d / and, moreover, V .x; r/ ' r ˛ : (3.2.4) In the case of (3.2.2), the parameter ˇ is called the walk dimension, which is an invariant of .M; d / as well. By [23], ˇ > 2 in this case (in fact, ˇ > 2 for the most interesting fractals). In the case of (3.2.3), the parameter ˇ is called the index of the associated jump process. There are many reasons for considering these two types of estimates. Firstly, both are known to hold on various families of fractals, in particular, on the Sierpinski gasket and carpet (cf. [2]). Secondly, the following dichotomy was proved in [30]: if pt .x; y/ satisfies the estimate   d.x; y/ ˛=ˇ pt .x; y/  C t ˆ t 1=ˇ with some function ˆ, then this has to be either (3.2.2) or (3.2.3). It was proved in [3] that the sub-Gaussian estimate (3.2.2) is equivalent to the parabolic Harnack inequality . An important problem is to find some practical conditions on .M; d; / and .E ; F / that should be equivalent to (3.2.2) resp. (3.2.3). Some results about existence of the heat kernel and its upper bounds were obtained in [20, 22, 27]. In order to state the results about equivalent conditions for the estimates (3.2.2), let us first define the following notions. Definition 3.2.1. A function u 2 F is called harmonic in an open set   M if

E .u; '/ D 0 for all ' 2 F \ C0 ./.

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65

Definition 3.2.2. We say that the uniform elliptic Harnack inequality is satisfied if there is a constant C such that, for any function u 2 F that is harmonic and nonnegative in a ball B.x; r/  M , ess sup u 6 C ess inf u: B.x;r=2/

B.x;r=2/

Definition 3.2.3. For any compact set K  M and open set   K, define the capacity of the capacitor .K; / by ˚ cap.K; / D inf E .'; '/ W ' 2 F \ C0 ./; 'jK  1 : The series of works [22, 44, 21] leads to the following result.

Theorem 3.2.4. Under a certain connectivity property of .M; d /, the sub-Gaussian estimate (3.2.2) is equivalent to the conjunction of the following three conditions:  the volume regularity (3.2.4);  the uniform elliptic Harnack inequality;  the capacity condition: for all balls B D B.x; r/ and 2B D B.x; 2r/, cap.B; 2B/ ' r ˛

ˇ

:

(3.2.5)

Of course, the elliptic Harnack inequality is quite difficult to verify in general, so the search for better conditions goes on. If M is a complete Riemannian manifold with the canonical Dirichlet form, then the Gaussian heat kernel bound (that is, the case ˇ D 2 in (3.2.2)) is known to be equivalent to the conjunction of the following two conditions:  the volume regularity (3.2.4);  the Poincaré inequality Z Z c .f jrf j2 d > 2 r B.x;r/ B.x;2r/ R 1 where f D .B.x;r// B.x;r/ f d.

f /2 d;

(3.2.6)

In the most interesting case ˇ > 2 that typically occurs in fractals, one replaces the Poincaré inequality (3.2.6) by the ˇ-Poincaré inequality Z Z c d€hf; f i > ˇ .f f /2 d; (3.2.7) r B.x;r/ B.x;r/ where €hf; f i is the energy measure of f . Then, both (3.2.4) and (3.2.7) are also necessary for (3.2.2), but not sufficient. In order to state the next result, we need the notion of a generalised capacity.

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Definition 3.2.5. Let u 2 F \ L1 .M /. For any compact set K  M and open set   K, define the generalised capacity of the capacitor .K; / by ˚ capu .K; / D inf E .u2 '; '/ W ' 2 F \ C0 ./; 'jK  1 : The following theorem is a slightly reformulated result of [25].

Theorem 3.2.6. The estimate (3.2.2) is equivalent to the conjunction of three properties:  the volume regularity (3.2.4);  the ˇ-Poincaré inequality (3.2.7);  the generalised capacity estimate: for any function u 2 F \ L1 and for any two concentric balls B1 WD B.x; R/ and B2 WD B.x; R C r/; Z C u2 d: (3.2.8) capu .B1 ; B2 / 6 ˇ r B However, the latter condition is still difficult to check. Our conjecture is that it can be replaced by a simpler capacity condition (3.2.5). Note that (3.2.8) with u D 1 and R D r is equivalent to (3.2.5). A similar question is in place for the stable-like estimate (3.2.3). Some approach to upper bounds was developed in [24]. The equivalent conditions for the two-sided estimates (3.2.3) in the case ˇ < 2 were obtained by Z.-Q. Chen and T. Kumagai [9], who proved that (3.2.3) is equivalent to the volume regularity (3.2.4) and the following estimate of the jump kernel J : J.x; y/ '

1 d.x; y/˛Cˇ

(3.2.9)

(jump kernels of this type in Rn are considered also in Chapter 8). The condition (3.2.9) replaces the Poincaré inequality in this case. The case ˇ > 2 is still open. There is one specific setting though where obtaining heat kernel bounds for the jump kernel J.x; y/ D d.x; y/ .˛Cˇ / is relatively easy for any ˇ > 0: this is the case when .M; d / an ultra-metric space. The theory of Markov processes on ultrametric spaces was developed in [5], using specific properties of an ultra-metric. In particular, this theory applies when M D Q p is the space of p-adic numbers with the p-adic distance, and yields the estimate (3.2.3) with ˛ D 1 (see the estimate (3.2.12) in Section 3.2.3 below). 3.2.2 Stochastic completeness of jump processes In [26], we investigated the stochastic completeness of the jump process associated with the Dirichlet form (3.2.1). We say that the distance function d.x; y/ and the jump kernel J.x; y/ are adapted to each other, if there exists a constant C such that Z .1 ^ d.x; y/2/J.x; y/d.y/ 6 C for all x 2 M: (3.2.10) M

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67

For example, the jump kernel in Rn , J.x; y/ D

const jx

yjnC˛

is adapted to the Euclidean distance provided ˛ 2 .0; 2/. Moreover, by the Lévy– Khinchine theorem, the Lévy measure W .dy/ of any Lévy process in Rn satisfies the condition Z   1 ^ jyj2 W .dy/ < 1: Rn nf0g

Since W .dy/ corresponds in our notation to J.x; y/d.y/, we see that the Euclidean distance in Rn is adapted to any Lévy process. The main result of [26] is the following theorem. Theorem 3.2.7. If J and d are adapted and if, for some x 2 M and c > 0, V .x; r/ 6 exp.cr log r/ for all large enough r ; then the jump process with the jump kernel J is stochastically complete. 3.2.3 Jump processes on ultra-metric spaces An ultra-metric space is a metric space .M; d / where the distance function satisfies the ultra-metric inequality d.x; y/ 6 max.d.x; z/; d.z; y//; that is obviously stronger than the usual triangle inequality. The ultra-metric inequality implies that any two metric balls B.x; r/, B.y; r/ of the same radius are either disjoint or identical. This in turn implies that, for any non-negative real r, the family of all distinct balls of radius r form a partition of M . Let .M; d / be a locally compact ultra-metric space. A model example is the field Q p of p-adic numbers with the p-adic distance or its straightforward generalisation Q np . Fix a Radon measure  on M with full support, a probability distribution function  .r/ on Œ0; C1/ and define the following operator P on functions on M :  Z Z 1 1 f d d .r/ (3.2.11) Pf .x/ D .B.x; r// B.x;r/ 0 (cf. [4, 5]). This operator is clearly a Markov operator. As it follows from the aforementioned property of ultra-metric balls, P is a bounded non-negative definite, self-adjoint operator in the Hilbert space L2 .M; /. The latter allows us to define the heat semigroup fPt gt >0 simply by Pt D P t and, hence, the associated continuous time random walk fXt gt >0 on M (note that typically Markov operators are not positive definite, so that the operator P t cannot be defined in general). The spectral decomposition for Pt follows easily from the representation given in (3.2.11), which leads to explicit expression for the heat kernel pt .x; y/ of Pt and then also to simple estimates of pt .x; y/ (see [5]).

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For example, consider M D Q p with the p-adic distance d.x; y/ D kx and the Haar measure . Then, .B.x; r// ' r. Choose

ykp

 .r/ D exp. .c=r/˛ /; where ˛; c > 0: Theorem 3.2.8. In Q p , the heat kernel of the heat semigroup fPt g with the above probability distribution function  .r/ satisfies the estimate pt .x; y/ '

t .t 1=˛

C kx

ykp /1C˛

;

(3.2.12)

for all t > 0 and x; y 2 Q p . Consequently, the Green function g.x; y/ of fPt g is finite if and only of ˛ < 1, and in this case g.x; y/ ' kx

yk.˛ p

1/

:

As a locally compact abelian group, Q p has the dual group that is again Q p . Hence, the Fourier transform is defined as a unitary operator in L2 .Q p ; /. Using the Fourier transform, Vladimirov and Volovich [52, 53] introduced a class D˛ of fractional derivatives on functions on Q p . This operator acts as follows: Z p˛ 1 f .x/ f .y/ ˛ D f .x/ D d.y/: ˛ 1 1 p kx yk1C˛ p Qp

The following theorem was proved in [5]. Theorem 3.2.9. The operator D˛ coincides with the generator of the semigroup fPt g with the probability distribution function  .r/ D exp. .p=r/˛ /: Consequently, the heat kernel of D˛ satisfies (3.2.12). It does not seem possible to obtain this estimate of the heat kernel of D˛ by using Fourier analysis approach.

3.3 Homology theory on graphs In a series of papers [31, 32, 33, 34], we introduced the notion of a differential form on a digraph (= directed graph) with the exterior derivative d , as well as the dual object— a @-invariant path with the boundary operator @, which leads to the dual notions of cohomology and homology of graphs.

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Let V be a finite set. An elementary p-path on V is any sequence i0 ; : : : ; ip of .p C 1/ vertices of V , which is also denoted by ei0 :::ip . The formal linear combinations of all ei0 :::ip with coefficients from a field K form a linear space ƒp . Define a linear boundary operator @ W ƒp ! ƒp 1 by @ei

0 :::ip

p X

D

. 1/k e

i0 ::: b ik :::ip

kD0

;

where ibk means omission of ik . Let G D .V; E/ be a digraph, where E is the set of directed edges (=arrows) on V . A p-path ei0 :::ip is called allowed if all the pairs ik ikC1 are arrows. Denote by Ap the subspace of ƒp generated by all allowed p-paths. In general, if v 2 Ap , @v does not have to be in Ap 1 . For example, on the digraph 0

 ! 1 ! 2

the 2-path e012 is allowed and, hence, lies in A2 while its boundary @e012 D e12

e02 C e01

(3.3.1)

is not in A1 because e02 is not allowed. This observation motivates the following definition. Definition 3.3.1. Define the subspace p of Ap by ˚ p D p .G/ D v 2 Ap W @v 2 Ap The elements of p are called @-invariant p-paths.

1



:

For example, if G contains the following ‘triangle’ 1

& 0  ! 2 %

then the 2-path e012 is @-invariant by (3.3.1). If G contains the following ‘square’ 1

! 3 " ! 2

 " 0 

then the 2-path v D e013 @v D e13

e03 C e01




e023 is @-invariant, because v 2 A2 and
e23 e03 C e02 D e13 C e01 e23

It is easy to see that @ maps p to p chain complex  .G/ :::

@

p

@ 1

p

1

@

e02 2 A1 :

and that @2 D 0. Hence, we obtain a pC1

:::

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where p > 0 and  1 D f0g. The homology groups Hp D Hp .G/ of this chain complex are called the path homologies of G. There are also dual notions of d -invariant p-forms, cochain complex  .G/ and path cohomologies H  .G/ of G that we do not address here. If G is an (undirected) graph, G can always be considered as a digraph, by turning each edge of G into a double arrow. There has been a number of attempts to define the notion of .co/homology for graphs. For example, one can consider a graph as an one-dimensional simplicial complex, or take into account all its cliques (= complete subgraphs) as simplexes of the corresponding dimensions. However, such homologies do not ususally have the necessary functorial properties. Another approach to homologies of digraphs can be realised via Hochschild homologies, using a natural path algebra of a graph. However, it is known that, in this case, the Hochschild homologies of order > 2 are trivial, which makes this approach useless. In singular homology theories of graphs, certain "small" graphs are predefined as basic cells. However, simple examples show that the singular homology groups do depend essentially on the choice of the basic cells. Our notion of path homologies of digraphs has the following advantages.  The path homologies of all dimensions can be non-trivial; even for planar graphs, they can be non-trivial in dimension 2.  The path homologies can be easily computed using any software package containing operations with matrices.  The path homology theory is compatible with the homotopy theories of graphs [1] and digraph [31].  The path homologies have good functorial properties with respect to graphtheoretical operations; for example, the homologies of the Cartesian product of digraphs (as well as of the join) satisfy the Künneth formula.  The path homology theory is dual to the cohomology theory of digraphs. The latter was introduced independently by A. Dimakis and F. Müller-Hoissen [13], using the classification of Bourbaki [7] of exterior derivations on algebras. One of the most essential and technically difficult results of our work is the Künneth formula for products. For two digraphs X and Y , denote by X  Y their Cartesian product, that is, the product based on the pattern . Theorem 3.3.2. For any two finite digraphs X and Y , we have  .X  Y / Š  .X / ˝  .Y /; that is, for any integer r > 0, r .X  Y / Š

M

fp;q>0WpCqDrg


p .X / ˝ q .Y / :

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Consequently, by the abstract theorem of Künneth, the same isomorphism holds for homologies: H .X  Y / Š H .X / ˝ H .Y /: The fact that the Künneth formula holds at the level of chain complexes is very surprising. This is in contrast with classical algebraic topology, where the Künneth formula holds only in homologies. This result provides indirect evidence that our notion of the chain complex  .G/ for digraphs is meaningful by itself. Define the join X  Y of digraphs X; Y as a digraph whose set of vertices is the disjoint union of the sets of vertices of X and Y , and the set of arrows of X  Y consists of all the arrows of X; Y as well as of new arrows from any vertex of X to e. any vertex of Y . In the next result, we use the the augmented chain complex  Theorem 3.3.3. For any two finite digraphs X , Y and for any integer r > have M
e r .X  Y / Š e p .X / ˝  e q .Y / :  

1, we

fp;q> 1WpCqDr 1g

It follows that, for any r > 0, e r .X  Y / Š H

M

fp;q>0WpCqDr 1g


e p .X / ˝ H e q .Y / : H

References [1] E. Babson, H. Barcelo, M. de Longueville and R. Laubenbacher, Homotopy theory of graphs, J. Algebr. Comb. 24 (2006), 31–44. [2] M.T. Barlow, Diffusions on Fractals, Springer, Berlin, 1998. [3] M.T. Barlow, A. Grigor’yan and T. Kumagai, On the equivalence of parabolic Harnack inequalities and heat kernel estimates, J. Math. Soc. Japan 64 (2012), 1091–1146. [4] A. Bendikov, A. Grigor’yan and Ch. Pittet, On a class of Markov semigroups on discrete ultra-metric spaces, Potential Analysis 37 (2012), 125–169. [5] A. Bendikov, A. Grigor’yan, Ch. Pittet and W. Woess, Isotropic Markov semigroups on ultrametric spaces, Russian Math. Surveys 69 (2014), 589–680. [6] I. Benjamini, I. Chavel and E.A. Feldman, Heat kernel lower bounds on Riemannian manifolds using the old ideas of Nash, Proc. London Math. Soc. 72 (1996), 215–240. [7] N. Bourbaki, Elements of Mathematics. Algebra I. Chapters 1-3, Herman, Paris, 1989. [8] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984. [9] Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d -sets, Stochastic Process. Appl. 108 (2003), 27–62. [10] S.Y. Cheng and S.-T. Yau, Differential equations on riemannian manifolds and their geometric applications, Commun. Pure Appl. Math. 28 (1975), 333–354. [11] W. Cwikel, Weak type estimates for singuar values and the number of bound states of Schrödinger operators, Ann. of Math. 106 (1977), 93–100.

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[12] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. [13] A. Dimakis and F. Müller-Hoissen, Discrete differential calculus: graphs, topologies, and gauge theory, J. Math. Phys. 35 (1994), 6703–6735. [14] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter, Berlin, 1994. [15] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math. 34 (1981), 525–598. [16] A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135–249. [17] A. Grigor’yan, Heat kernels on weighted manifolds and applications, Contemp. Math. 398 (2006), 93–191. [18] A. Grigor’yan and W. Hansen, Lower estimates for a perturbed Green function, J. d’Analyse Math. 104 (2008), 25–58. [19] A. Grigor’yan and E.P. Hsu, Volume growth and escape rate of Brownian motion on a CartanHadamard manifold, Int. Math. Ser. (N. Y.) 9 (2009), 209–225. [20] A. Grigor’yan and J. Hu, Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces, Invent. Math. 174 (2008), 81–126. [21] A. Grigor’yan and J. Hu, Heat kernels and Green functions on metric measure spaces, Canad. J. Math 66 (2014), 641–699. [22] A. Grigor’yan and J. Hu, Upper bounds of heat kernels on doubling spaces, Moscow Math. J. 14 (2014), 505–563. [23] A. Grigor’yan, J. Hu and K.-S. Lau, Heat kernels on metric-measure spaces and an application to semi-linear elliptic equations, Trans. Amer. Math. Soc. 355 (2003), 2065–2095. [24] A. Grigor’yan, J. Hu and K.-S. Lau, Estimates of heat kernels for non-local regular Dirichlet forms, Trans. Amer. Math. Soc. 366 (2014), 6397–6441. [25] A. Grigor’yan, J. Hu and K.-S. Lau, Generalized capacity, Harnack inequality and heat kernels on metric spaces, J. Math. Soc. Japan 67 (2015), 1485–1549. [26] A. Grigor’yan, X.-P. Huang and J. Masamune, On stochastic completeness of jump processes, Math. Z. 271 (2012), 1211–1239. [27] A. Grigor’yan and N. Kajino, Localized heat kernel upper bounds for diffusions via a multiple Dynkin–Hunt formula, Trans. Amer. Math. Soc. 369 (2017), 1025–1060. [28] A. Grigor’yan and M. Kelbert, On Hardy-Littlewood inequality for Brownian motion on Riemannian manifolds J. London Math. Soc. 62(2) (2000), 625–639. [29] A. Grigor’yan and V.A. Kondratiev, On the existence of positive solutions of semi-linear elliptic inequalities on Riemannian manifolds, Around the research of Vladimir Maz’ya II, A. Laptev (ed.), Int. Math. Series 12 (2010), 203–218. [30] A. Grigor’yan and T. Kumagai, On the dichotomy in the heat kernel two sided estimates, Proc. Symposia in Pure Mathematics 77 (2008), 199–210. [31] A. Grigor’yan, Y. Lin, Yu. Muranov and S.-T. Yau, Homotopy theory for digraphs, Pure Appl. Math. Quaterly 10 (2014), 619–674. [32] A. Grigor’yan, Yu. Muranov and S.-T. Yau, Graphs associated with simplicial complexes, Homology, Homotopy Appl. 16(1) (2014), 295–311.

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[33] A. Grigor’yan, Yu. Muranov and S.-T. Yau, Cohomology of digraphs and (undirected) graphs, Asian J. Math. 19 (2015), 887–932. [34] A. Grigor’yan, Yu. Muranov and S.-T. Yau, On a cohomology of digraphs and Hochschild cohomology, Homotopy Rel. Structures 11 (2016), 209–230. [35] A. Grigor’yan and N. Nadirashvili, Negative eigenvalues of two-dimensional Schrödinger equations, Archive Rat. Mech. Anal. 217 (2015), 975–1028. [36] A. Grigor’yan, Yu. Netrusov and S.-T. Yau, Eigenvalues of elliptic operators and geometric applications, Surveys Diff. Geom. IX (2004), 147–218. [37] A. Grigor’yan, S.-X. Ouyang and M. Röckner, Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities, J. Reine Angew. Math. (Crelle) 736 (2018), 1–31. [38] A. Grigor’yan and L. Saloff-Coste, Dirichlet heat kernel in the exterior of a compact set, Commun. Pure Appl. Math. 55 (2002), 93–133. [39] A. Grigor’yan and L. Saloff-Coste, Hitting probabilities for brownian motion on riemannian manifolds, J. Math. Pures et Appl. 81 (2002), 115–142. [40] A. Grigor’yan and L. Saloff-Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier, Grenoble 55 (2005), 825–890. [41] A. Grigor’yan and L. Saloff-Coste, Heat kernel on manifolds with ends, Ann. Inst. Fourier, Grenoble 59 (2009), 1917–1997. [42] A. Grigor’yan and L. Saloff-Coste, Surgery of the Faber-Krahn inequality and applications to heat kernel bounds, Nonlinear Analysis 131 (2016), 243–272. [43] A. Grigor’yan and Y. Sun, On non-negative solutions of the inequality u C u 6 0 on Riemannian manifolds, Commun. Pure Appl. Math. 67 (2014), 1336–1352. [44] A. Grigor’yan and A. Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Prob. 40 (2012), 1212–1284. [45] P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153–201. [46] E.H. Lieb, Bounds on the eigenvalues of the Laplace and Schrödinger operators, Bull. Amer. Math. Soc. 82 (1976), 751–753. [47] E. Mitidieri and S.I. Pokhozhaev, Apriori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234 (2001), 1–362. [48] G.V. Rozenblum, The distribution of the discrete spectrum for singular differential operators, Dokl. Akad. Nauk SSSR 202 (1972), 1012–1015. [49] E. Shargorodsky, On negative eigenvalues of two-dimensional Schrödinger operators, Proc. London Math. Soc. 108 (2014), 441–483. [50] M. Solomyak, Piecewise-polynomial approximation of functions from H ` ..0; 1/d /; 2` D d , and applications to the spectral theory of the Schrödinger operator, Israel J. Math. 86 (1994), 253–275. [51] N.Th. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992. [52] V.S. Vladimirov, Generalized functions over the field of p-adic numbers, Uspekhi Mat. Nauk 43 (1988), 17–53, 239. [53] V.S. Vladimirov and I.V. Volovich, p-adic Schrödinger-type equation, Lett. Math. Phys. 18 (1989), 43–53.

Chapter 4

Markov evolutions in spatial ecology: From microscopic dynamics to kinetics Yu. Kondratiev, O. Kutovyi and P. Tkachov In this summary, we construct Markov statistical dynamics for a class of birth-anddeath ecological models in the continuum. Mesoscopic scaling limits for these dynamics lead to the kinetic equations for the density of a population. The resulting evolution equations are non-local and non-linear ones. We study properties of solutions to kinetic equations which strongly depend on characteristics of the models considered.1

4.1 Introduction Dynamics of interacting particle systems appear in several areas of complex systems theory. In particular, we observe a growing activity in the study of Markov dynamics for continuous systems. The latter fact is motivated, in particular, by modern problems of mathematical physics, ecology, mathematical biology, and genetics; compare [19, 15, 16, 36] and the literature cited therein. Moreover, Markov dynamics are used for the construction of social, economic and demographic models. Notice that Markov processes for continuous systems are considered in the stochastic analysis as dynamical point processes [31, 28, 27], and they even appear in the representation theory of big groups [6, 7]. A mathematical formalisation of the problem may be described as follows. As a phase space of the system, we use the space €.Rd / of locally finite configurations in the Euclidean space Rd . A heuristic Markov generator which describes the considered model is given by its expression on a proper set of functions (observables) over €.Rd /. With this operator, we can relate two evolution equations. Namely, the backward Kolmogorov equation for observables and the Kolmogorov forward equation on probability measures on the phase space €.Rd / (macroscopic states of the system). The latter equation is known as the Fokker–Planck equation in the terminology of mathematical physics. Comparing with the usual situation in stochastic analysis, there is an essential technical difficulty: the corresponding Markov process in the configuration space may be constructed only in special cases. As a result, a description of Markov dynamics in terms of random trajectories is absent for most of the models under consideration. 1 Projects

A5, A10

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As an alternative approach, we use a concept of the statistical dynamics that substitutes the notion of a Markov stochastic process. A central object now is an evolution of states of the system that shall be defined by mean of the Fokker–Planck equation. This evolution equation with respect to probability measures on €.Rd / may be reformulated as a hierarchical chain of equations for correlation functions of the measures considered. Such kind of evolution equations are well known in the study of Hamiltonian dynamics for classical gases as BBGKY chains, but now they appear as a tool for the construction and analysis of Markov dynamics. As an essential technical step, we consider related pre-dual evolution chains of equations on the so-called quasi-observables. As will be shown below, such hierarchical equations may be analysed in the framework of semigroup theory with the use of powerful techniques of perturbation theory for the semigroup generators etc. Considering the dual evolution for the constructed semigroup on quasi-observables, we then introduce the dynamics on correlation functions. Such a scheme of constructing the dynamics comes as a surprise because one cannot expect any perturbation techniques for the initial Kolmogorov evolution equations. The point is that the states of infinite interacting particle systems are given by measures which are, in general, mutually orthogonal. As a result, we cannot compare their evolutions or apply a perturbative approach. But, under quite general assumptions, they have correlation functions and the corresponding dynamics may be considered in a common Banach space of correlation functions. A proper choice of this Banach space means, in fact, that we find an admissible class of initial states for which the statistical dynamics may be constructed. There, we see again a crucial difference in comparison with the framework of Markov stochastic processes, where the evolution is defined for any initial distribution. Another interesting topic is related to the study of different scalings of the microscopic systems. Among others, the crucial role from the point of view of applications is played by the mesoscopic (Vlasov) description of the microscopic systems mentioned above. Originally, the notion of Vlasov scaling was related to the Hamiltonian dynamics of interacting particle systems. This is a mean field scaling limit when the influence of weak long-range forces is taken into account. Rigorously, this limit was studied by Braun and Hepp in [9] for the Hamiltonian dynamics, and by Dobrushin [11] for more general deterministic dynamical systems. In [14], we proposed a general scheme for a Vlasov-type scaling of stochastic Markovian dynamics. Our approach is based on a proper scaling of the evolutions of correlation functions proposed by Spohn [46] for the Hamiltonian dynamics. In this summary, we apply such an approach to the birth-and-death stochastic dynamics. This gives us a rigorous framework for the study of convergence of the scaled hierarchical equations to a solution of the limiting Vlasov hierarchy, and for the derivation of a resulting non-linear evolutional equation for the density of the limiting system. We consider some special birth-and-death models to show how the general conditions proposed here may be verified in applications. In the last section, we study the kinetic (Vlasov) equation which corresponds to the birth-and-death Bolker–Dieckman–Law–Pacala (BDLP) model [5]. Namely, we consider a non-linear non-local evolution equation with non-local terms, which are convolutions with probability densities. We demonstrate that the long-time behaviour

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of the solution depends on the asymptotic of the birth kernel and the initial condition, where either a constant speed of the propagation or an acceleration may be observed. Under additional assumptions, we also prove existence and uniqueness of travelling waves. The structure of this chapter is as follows. Section 4.2 contains a brief summary of the mathematical description of complex systems. In Section 4.3, we discuss the general concept of statistical dynamics for Markov evolutions in the continuum and introduce the necessary mathematical structures. Then, in Section 4.4, this concept is applied to an important class of Markov dynamics of continuous systems, namely, to birth-and-death models. Here, general conditions for the existence of a semigroup evolution in a space of quasi-observables are obtained. Then, we construct evolutions of correlation functions as dual objects. It is shown how to apply general results to the study of particular models of statistical dynamics coming from mathematical physics and ecology. In Section 4.5, we discuss the Vlasov-type scaling for birth-and-death stochastic dynamics. Finally, in Section 4.6, we study the kinetic (Vlasov) equation for the birth-and-death BDLP model.

4.2 Mathematical description of complex systems Let us proceed to the mathematical realisation of complex systems. Let B .Rd / be the family of all Borel sets in Rd , where d > 1; Bb .Rd / denotes the system of all bounded sets from B .Rd /. The configuration space over Rd consists of all locally finite subsets (configurations) of Rd . Namely, ˇ n o
ˇ € D € Rd WD  Rd ˇ j ƒ j < 1; for all ƒ 2 Bb .Rd / : Here, j
j means the cardinality of a set, and P ƒ WD \ƒ. We may identify each 2 € with the non-negative Radon measure x2 ıx 2 M.Rd /, where ıx is the Dirac P measure with unit mass at x, x2¿ ıx is the zero measure by definition, and M.Rd / denotes the space of all non-negative Radon measures on B .Rd /. This identification allows us to endow € with the topology induced by the vague topology on M.Rd /, i.e. the weakest topology on € with respect to which all mappings X € 3 7! f .x/ 2 R x2

are continuous for any f 2 C0 .Rd /, the set of all continuous functions on Rd with compact supports. It is worth noting that the vague topology is metrisable in such a way that € becomes a Polish space; see [34] and references therein.

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Corresponding to the vague topology, the Borel  -algebra B .€/ appears as the smallest  -algebra for which all mappings € 3 7! Nƒ . / WD j ƒ j 2 N0 WD N [ f0g

(4.2.1)

are measurable for any ƒ 2 Bb .Rd /; compare [2]. N WD R [ f1g, we mark out the set Among all measurable functions F W € ! R F0 .€/ consisting of such of them for which jF . /j < 1 at least for all j j < 1. The important subset of F0 .€/ is formed by cylindric functions on €. Any such function is characterised by a set ƒ 2 Bb .Rd / such that F . / D F . ƒ / for all 2 €. The class of cylindric functions is denoted by Fcyl .€/  F0 .€/. Functions on € are usually called observables. This notion is borrowed from statistical physics and means that typically, in the course of empirical investigation, we may estimate, check or see only some quantities derived from the system as a whole rather than look into the system itself.
We denote the class of all probability measures on €; B .€/ by M1 .€/. Given a distribution  2 M1 .€/, one can consider a collection of random variables Nƒ .
/, ƒ 2 Bb .Rd /, as defined in (4.2.1). They describe random numbers of elements inside bounded regions. The natural assumption is that these random variables should have finite moments. Thus, we consider the class M1fm .€/ of all measures from M1 .€/ such that Z j ƒ jn d. / < 1; ƒ 2 Bb .Rd /; n 2 N: €


Example 4.2.1. Let  be a non-atomic Radon measure on Rd ; B .Rd / . Then, the Poisson measure  with intensity measure  is defined on B .€/ by
n ˚
 .ƒ/ exp  .ƒ/ ; ƒ 2 Bb .Rd /; n 2 N0 :  f 2 €jNƒ . / D j ƒ j D ng D nŠ

In the case of the Lebesgue measure,  .dx/ D dx, one can speak about the homogeneous Poisson distribution (measure)  WD dx with constant intensity 1. The space of (finite) configurations which belong to a bounded domain ƒ 2 Bb .Rd / will be denoted by €.ƒ/. The  -algebra B .€.ƒ// may be generated by a family of mappings €.ƒ/ 3 7! Nƒ0 . / 2 N0 , ƒ0 2 Bb .Rd /, ƒ0  ƒ. A measure  2 M1fm .€/ is called locally absolutely continuous with respect to the Poisson measure  if, for any ƒ 2 Bb .Rd /, the projection of  onto €.ƒ/ is absolutely continuous with respect to (w.r.t.) the projection of  onto €.ƒ/. More precisely, if we consider the projection mapping pƒ W € ! €.ƒ/, pƒ . / WD ƒ , then ƒ WD ıpƒ1 is absolutely continuous w.r.t. ƒ WD  ı pƒ1 . By e.g. [33], for any  2 M1fm .€/ which is locally absolutely continuous w.r.t the Poisson measure, there exists the family of (symmetric) correlation functions k.n/ W .Rd /n ! RC WD Œ0; 1/ which are defined as follows. For any symmetric measurable function f .n/ W .Rd /n ! R with finite support, the following equality

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holds Z €

X

f .n/ .x1 ; : : : ; xn / d. /

fx1 ;:::;xn g

1 nŠ

D

Z

.Rd /n

f .n/ .x1 ; : : : ; xn /k.n/ .x1 ; : : : ; xn / dx1 : : : dxn (4.2.2)

for n 2 N and k.0/ WD 1. The meaning of this notion is the following: the correlation function .n/ k .x1 ; : : : ; xn / describes the non-normalised probability density to have points of our systems in the positions x1 ; : : : ; xn . The symmetric functions of n variables from Rd can be considered as functions on n-point subsets from Rd . We proceed now to the exact constructions. The space of n-point configurations in Y 2 B .Rd / is defined by ˇ ˚ n 2 N: € .n/ .Y / WD   Y ˇ jj D n ; We put € .0/ .Y / WD f¿g. As a set, € .n/ .Y / may be identified with the symmetrisation of ˇ ˚ en D .x1 ; : : : ; xn / 2 Y n ˇ x ¤ x if k ¤ ` : Y k `

Hence, one
can introduce the corresponding Borel  -algebra, which we denote by B € .n/ .Y / . The space of finite configurations in Y 2 B .Rd / is defined as G €0 .Y / WD € .n/ .Y /: n2N0


This space is equipped with the topology of the disjoint union. Let B €0 .Y / denote the corresponding Borel  -algebra. In the case of Y D Rd , we will omit the index Y in the previously defined notations. Namely, €0 WD €0 .Rd /;

€ .n/ WD € .n/ .Rd /;

n 2 N0 :


The restriction of the Lebesgue product measure .dx/n to € .n/ ; B .€ .n/ / is denoted by m.n/ . We set m.0/ WD ıf¿g . The Lebesgue–Poisson measure  on €0 is defined by 1 X 1 .n/  WD m : (4.2.3) nŠ nD0

d

For any ƒ 2 Bb .R /, the restriction of  to €0 .ƒ/ D €.ƒ/ will be also denoted by .
Remark 4.2.2.˚ The space €; B .€/
is the projective limit of the family of mea-
surable spaces €.ƒ/; B .€.ƒ// ƒ2B .Rd / . The Poisson measure  on €; B .€/ b

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from Example 4.2.1 may be defined as the projective limit of the family of measures
f ƒ gƒ2Bb .Rd / , where  ƒ WD e m.ƒ/  is a probability measure on €.ƒ/; B .€.ƒ// and m.ƒ/ is the Lebesgue measure of ƒ 2 Bb .Rd /; compare [2] for details. Functions on €0 will be called quasi-observables . Any B func˚ .€0/-measurable tion G on €0 , is in fact defined by a sequence of functions G .n/ n2N where G .n/ 0

is a B .€ .n//-measurable function on € .n/ . We preserve the same notation for the function G .n/ considered as a symmetric function on .Rd /n . Note that G .0/ 2 R. A set M 2 B .€0 / is called bounded if there exists ƒ 2 Bb .Rd / and N 2 N such that N G M  € .n/ .ƒ/: nD0

The set of bounded measurable functions on €0 with bounded support is denoted by Bbs .€0 /, i.e., G 2 Bbs .€0 / iff G €0 nM D 0 for some bounded M 2 B .€0/. For any G 2 Bbs .€0 /, the functions G .n/ have finite supports in .Rd /n and may be substituted into (4.2.2). But, additionally, the sequence of G .n/ vanishes for big n. Therefore, one can sum up equalities (4.2.2) over n 2 N0 . This requires the following definition. Let G 2 Bbs .€0 /. Then, we define the function KG W € ! R by X .KG/. / WD G./I (4.2.4) b

compare [33, 37, 38]. The summation in (4.2.4) is taken over all finite subconfigurations  2 €0 of the (infinite) configuration 2 €; we denote this by the symbol  b . The mapping K is linear, positivity preserving, and invertible, with X .K 1 F /./ WD . 1/jnj F ./;  2 €0 : (4.2.5) 

By [33], for any G 2 Bbs .€0 /, we have KG 2 Fcyl .€/, moreover, there exists C D C.G/ > 0, ƒ D ƒ.G/ 2 Bb .Rd /, and N D N.G/ 2 N such that
N jKG. /j 6 C 1 C j ƒ j ; 2 €:

The expression (4.2.4) can be extended to the class of all non-negative measurable G W €0 ! RC , in this case, evidently, KG 2 F0 .€/. Note that the left-hand side (l.h.s.) of (4.2.5) has a meaning for any F 2 F0 .€/. Moreover, in this case .KK 1 F /. / D F . / for any 2 €0 . For G as above, we may sum up (4.2.2) over n and rewrite the result in a compact form: Z Z G./k ./d./: (4.2.6) .KG/. /d. / D €

€0

As was shown in [33], the equality (4.2.4) may be extended on all functions G such that the l.h.s. of (4.2.6) is finite. In this case, (4.2.4) holds for -a.a. 2 €, and (4.2.6) holds, too.

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4.3 Statistical descriptions of Markov evolutions Spatial Markov processes in Rd may be described as stochastic evolutions of configurations  Rd . In the course of such evolutions, points of configurations may disappear (die), move (continuously or with jumps from one position to another), or new particles may appear in a configuration (that is birth). The rates of these random events may depend on entire configurations that reflect an interaction between elements of the system. The construction of a spatial Markov process in the continuum is a highly difficult question which is not solved in full generality at present, see e.g. the review [44] and more detailed references about birth-and-death processes in Section 4.3. Meanwhile, for discrete systems, the corresponding processes have been constructed under quite general assumptions; see [39]. One of the main difficulties for continuous systems includes the necessity to control the number of elements in a bounded region. Note that the construction of spatial processes on bounded sets from Rd is typically well understood; compare [26]. The existing Markov process € 3 7! Xt 2 €, t > 0, provides a solution for the backward Kolmogorov equation for bounded continuous functions, d F D LFt ; dt t where L is the Markov generator of the process Xt . The question about the existence of a Markov process with a generator L is still open. On the other hand, the evolution of a state in the course of stochastic dynamics is an important question in its own right. A mathematical formulation of this question may be realised through the forward Kolmogorov equation for probability measures (states) on the configuration space €. Namely, we consider the pairing between functions and measures on € given by Z F . / d. /: (4.3.1) hF; i WD €

Then, we consider the initial value problem d hF; t i D hLF; t i; dt

t > 0;

ˇ t ˇt D0 D 0 ;

(4.3.2)


where F is an arbitrary function from a proper set, e.g. F 2 K Bbs .€0 /  Fcyl .€/. In fact, the solution to (4.3.2) describes the time evolution of distributions instead of the evolution of initial points in the Markov process. We rewrite (4.3.2) in the heuristic form, d  D L t ; (4.3.3) dt t where L is the (informally) adjoint operator of L with respect to the pairing (4.3.1). In the physics literature, (4.3.3) is referred to as the Fokker–Planck equation. The Markovian property of L yields that (4.3.3) might have a solution in the class of probability measures. However, the mere existence of the corresponding Markov process

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will not give us much information about properties of the solution to (4.3.3), in particular, about its moments or correlation functions. To get it, we suppose now that a solution t 2 M1fm .€/ to (4.3.2) exists and remains locally absolutely continuous with respect to the Poisson measure  for all t > 0 provided 0 has such a property. Then, one can consider the correlation function kt WD kt , t > 0. If we suppose that LF 2 F0 .€/ one can calculate K d hhK dt

1

1

for all F 2 Fcyl .€/;

(4.3.4)

LF using (4.2.5), and, by (4.2.6), we may rewrite (4.3.2) as

F; kt ii D hhK

1

ˇ t > 0; kt ˇt D0 D k0 ;

LF; kt ii;

(4.3.5)


for all F 2 K Bbs .€0 /  Fcyl .€/. Here, the pairing between functions on €0 is given by Z G./k./ d./: (4.3.6) hhG; kii WD €0

Let us recall that then, by (4.2.3), hhG; kii D

Z 1 X 1 G .n/ .x1 ; : : : ; xn /k .n/ .x1 ; : : : ; xn / dx1 : : : dxn : d /n nŠ .R nD0

Next, if we substitute F D KG, G 2 Bbs .€0 / in (4.3.5), we derive d hhG; kt ii D hhb LG; kt ii; dt

for all G 2 Bbs .€0 /. Here, the operator .b LG/./ WD .K

1

ˇ t > 0; kt ˇt D0 D k0 ;

LKG/./;

(4.3.7)

 2 €0 ;

is defined point-wise for all G 2 Bbs .€0 / under conditions (4.3.4). Consequently, we are interested in a weak solution to the equation d k D b L kt ; dt t

ˇ t > 0; kt ˇt D0 D k0 ;

(4.3.8)

where b L is the dual operator to b L with respect to the duality (4.3.6), namely, Z Z G./.b L k/./ d./: (4.3.9) .b LG/./k./ d./ D €0

€0

The procedure of deriving the operator b L for a given L is fully combinatorial. Meanwhile, to obtain the expression for the operator b L , we need an analogue of the integration by parts formula.

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We recall that any function on €0 may be identified with an infinite vector of symmetric functions of a growing number of variables. In this
approach, the operator b L in (4.3.8) will be realised as an infinite matrix b Ln;m n;m2N , where b Ln;m 0 is a mapping from the space of symmetric functions of n variables into the space of symmetric functions of m variables. As a result, instead of equation (4.3.2) for infinite-dimensional objects, we obtain an infinite system of equations for functions kt.n/ , each of which is a function of a finite number of variables, namely X d .m/ .n/
b kt .x1 ; : : : ; xm / D Ln;m kt .x1 ; : : : ; xm /; dt n ˇ .m/ .m/ kt .x1 ; : : : ; xm /ˇt D0 D k0 .x1 ; : : : ; xm /:

t > 0; m 2 N0 ;

(4.3.10)

Of course, in general, for a fixed n, any equation from (4.3.10) itself is not closed and includes functions kt.m/ of other orders m ¤ n. Nevertheless, the system (4.3.10) is a .n/ closed linear system. The chain evolution equations for kt constitutes the so-called hierarchy which is an analogue of the BBGKY hierarchy for Hamiltonian systems; compare [12]. Here, we restrict our attention to the so-called sub-Poissonian correlation functions. Namely, for a given C > 0, we consider the Banach space ˇ ˚ KC WD k W €0 ! R ˇ k
C j
j 2 L1 .€0 ; d / (4.3.11) with the norm

kkkK WD kC C

j
j

It is clear that k 2 KC implies ˇ ˇ ˇk./ˇ 6 kkk C jj K C

k.
/kL1 .€

0 ;d/

:

for -a.a.  2 €0 :

(4.3.12)

In what follows, we study the initial value problem (4.3.8) using the following scheme. We solve this equation in the space KC . The well-posedness of the initial value problem in this case is equivalent with an existence of the strongly continuous semigroup (C0 -semigroup in the sequel) in the space KC with a generator b L . How1 j
j ever, the space KC is isometrically isomorphic to the space L .€0 ; C d/ whereas, by the Lotz theorem [40, 3], in an L1 space any C0 -semigroup is uniformly continuous, that is, it has a bounded generator. Typically, for the operator L, any operator b Ln;m , cf. (4.3.10), might be bounded as an operator between two spaces of bounded symmetric functions of n and m variables, whereas the whole operator b L is unbounded in KC . To avoid these difficulties, we use a trick which goes back to Phillips [45]. The main idea is to consider the semigroup in the L1 -space not directly but as a dual semigroup T  .t/ to a C0 -semigroup T .t/ with a generator A in the pre-dual L1 space. In this case, T  .t/ appears as a strongly continuous semigroup not on the whole L1 , but on the closure of the domain of A only.

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In our case, this leads to the following scheme. We consider the pre-dual Banach space to KC , namely, for C > 0,
LC WD L1 €0 ; C j
j d : (4.3.13)

The norm in LC is given by kGkC

Z

Z 1 X ˇ jj ˇ ˇ ˇ .n/ Cn ˇ ˇ ˇG .x ; : : : ; x /ˇ dx : : : dx : G./ C d./ D WD 1 n 1 n nŠ .Rd /n €0 nD0

Consider the initial value problem, cf. (4.3.7), (4.3.8), d G D b LGt ; dt t

t > 0;

ˇ Gt ˇt D0 D G0 2 LC :

(4.3.14)

As long as (4.3.14) is well-posed in LC , there exists a C0 -semigroup b T .t/ in LC . T  .t/ Then, using Philips’ result, we see that the restriction of the dual semigroup b L / will be a C0 -semigroup with generator which is a part of b L (see onto Dom.b Section 4.4 below for details). This provides a solution to (4.3.8) which depends continuously on the initial data from Dom.b L /. Later, we would like to find a more b . The useful universal subspace of KC which does not depend on the operator L realisation of this scheme for a birth-and-death operator L is presented in Section 4.4 below. As a result, we obtain the classical solution to (4.3.8) for t > 0 in a class of sub-Poissonian functions which satisfy the Ruelle-type bound (4.3.12). Of course, after this we need to verify existence and uniqueness of measures whose correlation functions are solutions to (4.3.8). This can usually be done via proper approximation schemes; see Section 4.5.

4.4 Birth-and-death evolutions in the continuum 4.4.1 Microscopic description One of the most important classes of Markov evolution in the continuum is given by the birth-and-death Markov processes in the space € of all configurations in Rd . These are processes in which an infinite number of individuals exist at each instant, and the rates at which new individuals appear and some old ones disappear depend on the current configuration of existing individuals [31]. The corresponding Markov generators have a natural heuristic representation in terms of birth-and-death intensities. The birth intensity b.x; / > 0 characterises the appearance of a new point at x 2 Rd in the presence of a given configuration

2 €. The death intensity d.x; / > 0 characterises the probability of the event that the point x of the configuration disappears, depending on the location of the remaining points of the configuration n fxg (in the sequel n x). Heuristically, the

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corresponding Markov generator is described by the following expression, X .LF /. / WD d.x; n x/ ŒF . n x/ F . / x2

C

Z

Rd

b.x; / ŒF . [ x/

(4.4.1)

F . / dx;

for proper functions F W € ! R. O and L O  . Examples of rates b and d We always sup4.4.2 Expressions for L d pose that the rates d; b W R  € ! Œ0I C1 from (4.4.1) satisfy the following assumptions  2 €0 n f¿g, x 2 Rd n ;

d.x; /; b.x; / > 0; d.x; /; b.x; / < 1; Z
d.x; / C b.x; / d./ < 1; ZM
d.x; / C b.x; / dx < 1;

 2 €0 , x 2 Rd n ;

M 2 B .€0 / bounded, a.a. x 2 Rd ;  2 €0 , ƒ 2 Bb .Rd /:

ƒ

We start with the expression for b LDK

1

LK.

Proposition 4.4.1 ([19, Prop. 5]). For any G 2 Bbs .€0 /, the following formula holds: X X
.b LG/./ D G./ K 1 d.x;
[  n x/ . n / 

C

XZ



x2

Rd

G. [ x/ K

1


b.x;
[ / . n / dx;

(4.4.2)

 2 €0 :

Using this, one can derive the explicit form of b L .

Proposition 4.4.2 ([19, Cor. 9]). For any k 2 Bbs .€0 /, the following formula holds: XZ
 b k. [ / K 1 d.x;
[  n x/ ./ d./ .L k/./ D x2

C

€0

XZ x2

€0

k. [ . n x// K

where b L k is defined by (4.3.9).

1


b.x;
[  n x/ ./ d./;

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Yu. Kondratiev, O. Kutovyi, P. Tkachov

4.4.3 Semigroup evolutions in the space of quasi-observables We proceed now to the construction of a semigroup in the space LC , C > 0, see (4.3.13), which has a generator, given by b L, with a proper domain. To define such a domain, let us set X D ./ WD d .x;  n x/ > 0;  2 €0 I (4.4.3) x2

˚ D WD G 2 LC j D .
/ G 2 LC :

(4.4.4)

Note that Bbs .€0 /  D and Bbs .€0 / is a dense set in LC . Therefore, D is also a dense set in LC . We will now show that .b L; D/ given by (4.4.2), (4.4.4) generates a C0 -semigroup on LC if only ‘the full energy of death’, given by (4.4.3), is big enough. Theorem 4.4.3 ([17, Thm. 3.2]). Suppose that there exists a1 > 1, a2 > 0 such that, for all  2 €0 and a.a. x 2 Rd , XZ ˇ ˇ ˇK 1 d .x;
[  n x/ˇ ./ C jj d ./ 6 a1 D./; (4.4.5) x2

€0

XZ x2

and, moreover,

€0

ˇ ˇK

1

ˇ b .x;
[  n x/ˇ ./ C jj d ./ 6 a2 D./ a1 C

a2 3 < : C 2

(4.4.6)

(4.4.7)

b .t/ on LC . Then, .b L; D/ is the generator of a holomorphic semigroup T

4.4.4 Evolution in the space of correlation functions In this section, we will use b.t/ acting on the space of quasi-observables for the construction of a the semigroup T solution to the evolution equation (4.3.8) on the space of correlation functions.
0 We denote dC WD C j
j d; and the dual space .LC /0 D L1 .€0 ; d C / D 1 0 L .€0 ; dC /. As was mentioned before, the space .LC / is isometrically isomorphic to the Banach space KC considered in (4.3.11)–(4.3.12). The isomorphism is given by the isometry RC .LC /0 3 k 7 ! RC k WD k
C j
j 2 KC :

(4.4.8)

Recall that one may consider the duality between the Banach spaces LC and KC given by (4.3.6) with jhhG; kiij 6 kGkC
kkkK : C
0 0 0 Let b L ; Dom.b L / be an operator in .LC / which is dual to the closed operator
b L; D . We consider also its image on KC under the isometry RC . Namely, let   0 b b0 RC 1 with the domain Dom.b L D RC L L / D RC Dom.b L /. Similarly, one can b .t/ in KC . consider the adjoint semigroup b T 0 .t/ in .LC /0 and its image T

Markov evolutions in spatial ecology

87

b .t/ is not a C0 -semigroup in the whole The space LC is not reflexive, hence T KC . The last semigroup will be weak*-continuous, weak*-differentiable at 0, and   b L will be a weak*-generator of b T .t/. Therefore, one has an evolution in the space of correlation functions. In fact, we have a solution to the evolution equation (4.3.8) in a weak*-sense. This subsection is devoted to the study of a classical solution to bˇ .t/ of the semigroup T b .t/ onto its invariant Banach this equation. The restriction T  subspace Dom.b L / (here and below all closures are in the norm of the space KC ) is ˇ  a strongly continuous semigroup. Moreover, its generator b L will be a part of b L , namely, ˇ n o  ˇ  ˇ  L / (4.4.9) Dom.b L / D k 2 Dom.b L /ˇb L k 2 Dom.b

ˇ ˇ b k for any k 2 Dom.b and b L kDL L /. bˇ˛ .t/ of the semigroup T bˇ .t/ onto K˛C . It will One can consider the restriction T ˇ˛ be a strongly continuous semigroup with the generator b L , which is a restriction of ˇ b L onto K˛C . Namely, cf. 4.4.9, ˇ  n o ˇ˛ ˇ Dom.b L / D k 2 K˛C ˇ b L k 2 K˛C ;

ˇ˛  ˇ˛ bˇ k D b and b L k DL L k for any k 2 K˛C . In the other words, b L is a part of  b L . Now, we may proceed to the main statement of this section.

Theorem 4.4.4 ([18, Thm. 3.16]). Let (4.4.5), (4.4.6) hold together with the following assumptions, d .x; / 6 A.1 C jj/N  jj ;   C 3 16 < a1 ; a2 2

(4.4.10) (4.4.11)

and let ˛ be chosen such that a2 C

3 2

a1


< ˛
0.

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Yu. Kondratiev, O. Kutovyi, P. Tkachov

Example 4.4.5. (BDLP model) This example describes a generalisation of the model of plant ecology (see [13] and references therein). Let L be given by (4.4.1) with X d.x; n x/ D m.x/ C ~ .x/ a .x y/; x 2 ; 2 €; C

b.x; / D ~ .x/

X

C

y2 nx

a .x

x 2 Rd n ; 2 €;

y/;

y2

where 0 < mR 2 L1 .Rd /, 0 6 ~ ˙ 2 L1 .Rd /, 0 6 a˙ 2 L1 .Rd ; dx/ \ L1 .Rd ; dx/, Rd a˙ .x/dx D 1. Let us suppose, cf. [13], that there exists ı > 0 such that .4 C ı/C ~ .x/ 6 m.x/; C

.4 C ı/~ .x/ 6 m.x/; C

C

x 2 Rd ;

(4.4.12)

d

x2R ;

4~ .x/a .x/ 6 C ~ .x/a .x/;

(4.4.13) d

x2R :

(4.4.14)

Then,  m.x/ 1  d.x; /; 6 1C 4Cı 4Cı X C C m.x/ C ~ .x/ < d.x; /: a .x y/ C b.x; / C C ~ C .x/ 6 4 4Cı 4 d.x; / C C ~ .x/ 6 d.x; / C

y2

1 Hence, (4.4.5), (4.4.6) hold with a1 D 1 C 4Cı ; a2 D under conditions (4.4.12), (4.4.14), we have

C 4

, which fulfills (4.4.7). Next,

d.x; / 6 kmkL1 .Rd / C k~ kL1 .Rd / ka kL1 .Rd / jj;

 2 €0 ;

and hence (4.4.10) holds with  D 1, which makes (4.4.11) obvious. Remark 4.4.6. It was shown in [13] that, for the case of constant m; ~ ˙ , a condition like (4.4.12) is essential. Namely, if m > 0 is arbitrarily small, the operator b L will not even be accretive in LC .

4.5 Vlasov-type scalings For the reader’s convenience, we first explain the idea of Vlasov-type scaling. The general scheme for the birth-and-death dynamics as well as for the conservative ones may be found in [14]. The realisations of this approach for the Glauber dynamics (Example 1 with s D 0) and for the BDLP dynamics (Example 2) were considered in [15, 16], respectively. The idea of Vlasov-type scaling consists in the following.

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We would like to construct some scaling L" , " > 0, of the generator L such that the following scheme holds. Suppose that we have a semigroup UO " .t/ with the generator LO " in some LC" , " > 0. Consider the dual semigroup UO " .t/. Let us choose an initial function of the corresponding Cauchy problem with a singularity in ". Namely, "jj k0."/ ./  r0 ./, " ! 0,  2 €0 for some function r0 which is independent of ". The scaling L 7! L" should be chosen in such a way that, first of all, the corresponding semigroup UO " .t/ preserves the order of the singularity, "jj .UO " .t/k0."/ /./  rt ./; " ! 0;  2 €0 ; and, secondly, the dynamics r0 7! Q rt preserves the Lebesgue–Poisson exponents. Namely, if r0 ./ D e .0 ; / WD x2 0 .x/, then rt ./ D e .t ; /. There exists an explicit (in general nonlinear) differential equation for t , d  .x/ D .t /.x/; dt t

(4.5.1)

which will be called the Vlasov-type equation. Now, we explain an informal way to realise such a scheme. Let us consider, for any " > 0, the following mapping (cf. (4.4.8)) defined for functions on €0 : .R" r/./ WD "jj r./: This mapping is “self-dual” with respect to the duality (4.3.6). Moreover, R" 1 D ."/ ."/ R" 1 . Having R" k0  r0 , " ! 0, we need rt  R" UO " .t/k0  R" UO " .t/R" 1 r0 , " ! 0. Therefore, we have to show that, for any t > 0, the operator family R" UO " .t/R" 1 , " > 0, has limiting (in a proper sense) operator U.t/ and U.t/e .0 / D e .t /: But, heuristically, UO " .t/ D exp ft LO " g and R" UO " .t/R" us consider the “renormalised” operator LO "; ren WD R" LO " R" 1 :

(4.5.2) 1

D exp ftR" LO " R" 1 g. Let (4.5.3)

In fact, we need that there exists an operator LO V such that exp ftR" LO " R" 1 g ! exp ft LO V g DW U.t/ with U.t/ satisfying (4.5.2). Therefore, a heuristic way to produce a scaling L 7! L" is to demand that   d lim e .t ; / LO "; ren e .t ; / D 0;  2 €0 ; "!0 dt

provided t satisfies (4.5.1). The point-wise limit of LO "; ren will be a natural candidate for LO V . Note that (4.5.3) implies informally that LO "; ren D R" 1 LO " R" . Below we propose a scheme to give a rigorous meaning to the idea introduced above. We consider, for

90

Yu. Kondratiev, O. Kutovyi, P. Tkachov

O "; ren and prove that it is a generator a proper scaling L" , the “renormalised” operator L O of a strongly continuous contraction semigroup U"; ren .t/ in LC . Next, we show that the formal limit LO V of LO "; ren is a generator of a strongly continuous contraction semigroup UO V .t/ in LC . Finally, we prove that UO "; ren .t/ ! UO V .t/ strongly in LC . This implies weak*-convergence of the dual semigroups UO "; ren .t/ to UO V .t/. We also explain in which sense UO V .t/ satisfies the properties above. Let us consider, for any " 2 .0I 1, the following scaling of the operator L defined in Equation (4.4.1): X .L" F /. / WD d" .x; n x/ ŒF . n x/ F . / C x2 1

"

Z

Rd

b" .x; / ŒF . [ x/

F . / dx;

where d" and b" are defined for each model explicitly (see e.g. Example 4.5.2). We define the renormalised operator LO ";ren WD R" 1 K 1 L" KR" . Using the same arguments as in the proof of Proposition 4.4.1, we get X X
.LO ";ren G/./ D G./" jnj K0 1 d" .x;
[  n x/ . n / 

C

XZ 

Rd

x2

G. [ x/"

jnj


K0 1 b" .x;
[ / . n / dx:

Suppose that there exist a1 > 1, a2 > 0 such that, for all  2 €0 , for a.a. x 2 Rd , and for any " 2 .0I 1, XZ ˇ ˇ ˇK 1 d .x;
[  n x/ˇ ./ " jj C jj d ./ 6 a D ./ ; (4.5.4) 1 " 0 " x2

€0

XZ x2

€0

ˇ ˇK

1

0

ˇ b" .x;
[  n x/ˇ ./ " a1 C

jj

C jj d ./ 6 a2 D" ./ ;

a2 3 < : C 2

(4.5.5)

(4.5.6)

For all ,  2 €0 and a.a. x 2 Rd , the following limits exist and coincide:

lim " jj K0 1 d" .x;
[ / ./ D lim " jj K0 1 d" .x;
/ ./ DW DxV ./I "!0

lim "

"!0

"!0

jj


K0 1 b" .x;
[ / ./ D lim " "!0

jj




(4.5.7)

K0 1 b" .x;
/ ./ DW BxV ./: (4.5.8)

We would like to emphasise that above limits should not depend on . A collection of examples for such d" , b" can be found in [14]. Now we are able to state a result about convergence in LC .

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Markov evolutions in spatial ecology

Theorem 4.5.1 ([17, Thm. 4.4]). Let conditions (4.5.4), (4.5.5), and (4.5.6) be satisfied. Suppose that convergence in (4.5.7), (4.5.8) holds for all  2 €0 as well as in the sense of LC . Assume also that there exists  > 0 such that either d" .x; / 6 DxV .¿/

or

d" .x; / > DxV .¿/

is satisfied for all  2 €0 and for a.a. x 2 Rd . Then, UO " .t/ uniformly on finite time intervals.

s ! UO V .t/ in LC

Example 4.5.2 (BDLP model, revisited). Let X X d" .x; n x/ D m C "~ a .x y/; b" .x; / D "~ C aC .x

y/:

y2

y2 nx

In comparison with the previous notation, we have changed ~ ˙ into "~ ˙ . Clearly, conditions (4.4.12), (4.4.14) imply the same inequalities for "~ ˙ . Note also that d" is decreasing in " ! 0. Therefore, to apply all results of this section to the BDLPmodel, we should prove the convergence (4.5.7), (4.5.8) in LC . Note that X " jj K0 1 d" .x;
[ / ./ D d" .x; /0jj C 1€ .1/ ./ a .x y/ ! m0

jj

C 1€ .1/ ./

X

y2

y/ DW DxV ./

a .x

y2

and, analogously, "

jj

K0 1 b" .x;
[ / ./ D b" .x; /0jj C 1€ .1/ ./ ! 1€ .1/ ./

X y2

aC .x

X

aC .x

y/

y2

y/ DW BxV ./:

The convergence in LC is now obvious. The kinetic (Vlasov) equation has the following form d  .x/ D ~ C .aC  t /.x/ dt t

~ t .x/.a  t /.x/

mt .x/:

(4.5.9)

We study the equation so obtained in the following section. Remark 4.5.3. By duality (4.3.6), Theorem 4.5.1 yields weak*-convergence of the semigroups UO "ˇ˛ .t/ to UO Vˇ˛ .t/ in K˛C . To prove such convergence in the strong sense, we need additional analysis of their generators. The problem concerns the fact that we have an explicit expression for the generator LO ˇ˛ D LO V only on the core V ˇ  ˚ k 2 K˛C ˇ LO V k 2 K˛C . However, we are able to show such a convergence for the Glauber dynamics described in Example 1 for s D 0 using some modified technique (see [15]).

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4.6 Kinetic equation of a spatial ecology model 4.6.1 Introduction In this section, we study the mesoscopic equation of the BDLP model (4.5.9) from different perspectives. Namely, we will deal with the following nonlinear nonlocal evolution equation, for x 2 Rd , ( du

.x; t/ D ~ C .aC  u/.x; t/ dt u.x; 0/ D u0 .x/;

mu.x; t/

~ u.x; t/.a  u/.x; t/;

t > 0;

(4.6.1) which we will study in a class of non-negative functions bounded in x. The solution u D u.x; t/ to (4.6.1) describes approximately a density (at the moment of time t and at the position x of the space Rd ) for a particle system evolving in the continuum. During the evolution, particles might reproduce themselves, die, and compete (say, for resources). Namely, a particle located at a point y 2 Rd may produce a ‘child’ at a point x 2 Rd with the intensity ~ C and according to the dispersion kernel aC .x y/. Next, any particle may die with the constant intensity m. And, additionally, a particle located at x may die according to the competition with the rest of particles; the intensity of the death because of a competitive particle located at y is equal to ~ and the distribution of the competition is described by a .x y/. This model was originally proposed in mathematical ecology [5]. Rigorous mathematical constructions were done in [26, 13]. In [13], the mathematical approach was realised using the theory of Markov statistical dynamics on the so-called configuration spaces expressed in terms of evolution of time-dependent correlation functions of the system, compare [19, 35, 33]. Here, m > 0, ~ ˙ > 0 are constants, and the functions 0 6 a˙ 2 L1 .Rd / are probability densities, Z Z a .y/dy D 1: aC .y/dy D Rd

Rd

Here and below, for a function u D u.y; t/, which is (essentially) bounded in y 2 Rd , and a function (a kernel) a 2 L1 .Rd /, we denote Z a.x y/u.y; t/dy: .a  u/.x; t/ WD Rd

We assume that u0 is a bounded function on Rd . For technical reasons, we will consider two Banach spaces of bounded real-valued functions on Rd : the space Cub .Rd / of bounded, uniformly continuous functions on Rd with sup-norm and the space L1 .Rd / of essentially bounded (with respect to the Lebesgue measure) functions on Rd with ess-sup-norm. Let also Cb .Rd / and C0 .Rd / denote the spaces of continuous functions on Rd which are bounded or have compact supports, respectively.

Markov evolutions in spatial ecology

93

Let E be either Cub .Rd / or L1 .Rd /. Consider the Equation (4.6.1) in E; in particular, u must be continuously differentiable in t, for t > 0, in the sense of the norm in E. Moreover, we consider u as an element from the space Cb .I ! E/ of continuous bounded functions on I (including 0) with values in E and with the norm kukC

b .I !E /

D sup ku.
; t/kE : t 2I

Such a solution is said to be a classical solution to (4.6.1); in particular, u will continuously (in the sense of the norm in E) depend on the initial condition u0 . We will also use the space Cb .I ! E/ with I D ŒT1 ; T2 , T1 > 0. For simplicity, we denote
T2 > T1 > 0; XT1 ;T2 WD Cb ŒT1 ; T2  ! E ;

and the corresponding norm will be denoted by k:kT1 ;T2 . We set also XT WD X0;T , k:kT WD k:k0;T , and
X1 WD Cb RC ! E

with the corresponding norm k:k1 . The upper index ‘+’ will denote the cone of non-negative functions in the corresponding space, namely,

X]C WD fu 2 X] j u > 0g; where ] is one of the subscripts above. We will also omit the subscript for the norm k:kE in E, if it is clear whether we are working with sup- or ess-sup-norm. 4.6.2 Basic properties The following theorem yields existence and uniqueness of a solution. Theorem 4.6.1 ([20, Thm. 2.2]). Let u0 2 E and u0 .x/ > 0, x 2 Rd . Then, for any T > 0, there exists a unique non-negative solution u to the equation (4.6.1) in E such that u 2 XT . Below, j:j D j:jRd denotes the Euclidean norm in Rd , Br .x/ is a closed ball in R with the centre at x 2 Rd and radius r > 0; and br is the volume of this ball. The following theorem is an extension of Theorem 4.6.1 for E D Cub .Rd /, in which case the global boundedness of the solutions holds in both space and time under additional weak assumptions. d

Theorem 4.6.2 ([20, Thm. 2.8]). Suppose that there exists r0 > 0 such that ˛ WD inf a .x/ > 0; jxj6r0

and, for some "; A > 0, one has aC .x/ 6

A , 1Cjxjd C" d

for all x 2 Rd . Then, the

solution u > 0 to (4.6.1), with 0 6 u0 2 Cub .R /, belongs to Cub .Rd /.

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The main difficulty in studying non-local monostable evolution equations is the lack of techniques for this class of equations. In particular, variational methods may be hardly applied here because of the type of the nonlinear (‘reaction’) term. Nevertheless, under restrictions on the kernels aC ; a , a version of the comparison principle may be proved. This result will be needed in the rest of the article. Let T > 0 be fixed. Define the sets XT1 of functions in XT that are continuously differentiable on .0; T  in the sense of the norm in E. Here and below, we consider the left derivative at t D T only. For any u 2 XT1 , one can define the function

F u WD du dt

~ C aC  u C mu C ~ u.a  u/;

t 2 .0; T ; x 2 Rd : (4.6.2)

Theorem 4.6.3 ([20, Thm. 3.1]). Let there exist c > 0 such that ~ C aC .x/ > c~ a .x/

for a.a. x 2 Rd :

Let T 2 .0; 1/ be fixed and functions u1 ; u2 2 XT1 be given such that, for any .x; t/ 2 Rd  .0; T , u1 .x; t/ > 0;

.F u1 /.x; t/ 6 .F u2 /.x; t/; 0 6 u2 .x; t/ 6 c; u1 .x; 0/ 6 u2 .x; 0/:

(4.6.3)

Then, u1 .x; t/ 6 u2 .x; t/, for all .x; t/ 2 Rd  Œ0; T . In particular, u1 6 c. For E D Cub .Rd /, one can prove a refined version of Theorem 4.6.3 for nondifferentiable (in time) functions. For any T 2 .0; 1, define the set DT of all functions u W Rd  RC ! R, such that, for all t 2 Œ0; T /, u.
; t/ 2 Cub .Rd /, and, for all x 2 Rd , the function f .x; t/ is absolutely continuous in t on Œ0; T /. Then, for any u 2 DT , one can define the function (4.6.2), for all x 2 Rd and a.a. t 2 Œ0; T /. Proposition 4.6.4 ([20, Prop. 3.3]). The statement of Theorem 4.6.3 remains true if we assume that u1 ; u2 2 DT and, for any x 2 Rd , the inequality (4.6.3) holds for a.a. t 2 .0; T / only. We introduce a notation for the non-zero constant solution  WD

~C m 2 R: ~

(4.6.4)

Using Duhamel’s principle, it is easy to show that, if ~ C < m, then the solution to (4.6.1) converges to 0 exponentially fast and uniformly in space, as time tends to infinity. The case ~ C D m was partially considered by Terra and Wolanski (see [48, 49]), and we omit it in the present article. Hence, we make the following assumption in the rest of the article: ~ C > m: (A1) It implies in particular that the constant solution  is greater than zero. We will study solutions with initial conditions that are non-negative and bounded by  .

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Definition 4.6.5. For  > 0, given by (4.6.4), consider the following sets U WD ff 2 Cub .Rd / j 0 6 f .x/ 6 ; x 2 Rd g;

L WD ff 2 L1 .Rd / j 0 6 f .x/ 6 ; for a.a. x 2 Rd g;

E WD ff 2 E j 0 6 f .x/ 6 ; x 2 Rd g: Hence E is either U or L .

Having in mind the conditions of Theorem 4.6.3, we assume, ~ C aC .x/ > .~ C

m/a .x/;

a.a. x 2 Rd :

(A2)

Let us mention an important consequence of Theorem 4.6.3. Proposition 4.6.6 ([20, Prop. 3.4]). Suppose that (A1) and (A2) hold. Let 0 6 u0 2 E be an initial condition to (4.6.1), let u 2 XT be the corresponding solution on any Œ0; T , T > 0. Then u 2 X1 , with kuk1 6  . Let v0 2 E be another initial condition to (4.6.1) such that u0 .x/ 6 v0 .x/, x 2 Rd ; and let v 2 X1 be the corresponding solution. Then, u.x; t/ 6 v.x; t/;

x 2 Rd ; t > 0:

Let us show that, if u0 6 0, the solutions to (4.6.1) are strictly positive; this is quite a common feature of linear parabolic equations. However, in general, it may fail for nonlinear ones. It is required that there exists ; ı > 0 such that aC .x/ > ; for a.a. x 2 Bı .0/:

(A3)

Proposition 4.6.7 ([20, Prop. 3.8]). Let (A1), (A2), (A3) hold. Let u0 2 U , u0 6 0, u0 6  , be the initial condition to (4.6.1), and let u 2 X1 be the corresponding solution. Then, u.x; t/ > inf u.y; s/ > 0; y2Rd s>0

x 2 Rd ; t > 0:

As a matter of fact, under (A4), a much stronger statement than unattainability of  holds. To state this, we assume that there exists ; ı > 0, such that C C

J .x/ D ~ a .x/

.~ C

m/a .x/ > ; for a.a. x 2 Bı .0/:

(A4)

Theorem 4.6.8 ([20, Thm. 3.9]). Let (A1), (A2), (A4) hold. Let u1 ; u2 2 X1 be two solutions to (4.6.1) such that u1 .x; 0/ 6 u2 .x; 0/, x 2 Rd , are in U . Then, either u1 .x; t/ D u2 .x; t/, x 2 Rd , t > 0 or u1 .x; t/ < u2 .x; t/, x 2 Rd , t > 0.

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By choosing u2   in Theorem 4.6.8, we immediately get the following. Corollary 4.6.9 ([20, Cor. 3.10]). Let (A1), (A2), (A4) hold. Let u0 2 U , u0 6  , be the initial condition to (4.6.1), and let u 2 X1 be the corresponding solution. Then, u.x; t/ <  , x 2 Rd , t > 0. 4.6.3 Travelling waves For simplicity, we consider a one-dimensional space (d D 1) in the following. For higher-dimensional analogues of the statements, see [20, 21, 22, 23, 24]. Travelling waves were studied intensively for the original Fisher–KPP equation, compare [25, 32, 8]; for locally nonlinear equations with nonlocal diffusion, see [10, 50, 47]; and for nonlocal nonlinear equations with local diffusion, see [1, 4, 30, 43]. Throughout this section, we will mainly work in the L1 -setting. Recall that we will always assume that (A1) and (A2) hold, and that  > 0 is given by (4.6.4). Let us give a brief overview of the results of this section. The existence and properties of the travelling wave solutions will be considered under the so-called Mollison condition (A5); compare [42, 41]. Namely, in Theorem 4.6.12, we will prove that, for any  2 f 1; 1g, there exists c ./ 2 R, such that, for any c > c ./, there exists a travelling wave with the speed c in direction , and, for any c < c ./, such a travelling wave does not exist. Moreover, we will find an expression for c ./, see (4.6.6). We will prove that the profile of a travelling wave with a non-zero speed is smooth, whereas the zero-speed travelling wave (provided it exists, i.e. if c ./ 6 0) has a continuous profile (Proposition 4.6.13, Corollary 4.6.14). Next, we will demonstrate the uniqueness (up to shifts) of a travelling wave profile with a given speed c > c ./ (Theorem 4.6.18). Definition 4.6.10. Let M .R/ denote the set of all decreasing and right-continuous functions f W R ! Œ0;  . 1 1 Definition 4.6.11. Let X1 WD X1 \ C 1 ..0; 1/ ! L1 .R//. A function u 2 X1 is said to be a travelling wave solution to the equation (4.6.1) with a speed c 2 R and in a direction  2 f 1; 1g if and only if (iff, in the sequel) there exists a function 2 M .R/, such that, for all t > 0; a:a: x 2 R,

u.x; t/ D

.x


ct/;

Here and below, the function whereas c is its speed.

. 1/ D ;

.C1/ D 0:

(4.6.5)

is said to be the profile for the travelling wave,

For a given  2 f 1; 1g, consider the following assumption on aC : Z There exists  D ./ > 0 such that a ./ WD aC .x/e
x dx < 1: R

(A5)

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Theorem 4.6.12 ([20, Thm. 4.9]). Let (A1) and (A2) hold and  2 f 1; 1g be fixed. Suppose also that (A5) holds. Then, there exists c ./ 2 R such that 1. for any c > c ./, there exists a travelling wave solution, in the sense of Definition 4.6.11, with a profile 2 M .R/ and speed c,

2. for any c < c ./, such a travelling wave does not exist.

The next statements describe the properties of a travelling wave solution. Proposition 4.6.13 ([20, Prop. 4.11]). Let 2 M .R/ and c 2 R be such that 1 Q there exists a solution u 2 X1 to Equation (4.6.1) such that (4.6.5) holds, for some  2 f 1; 1g. Then, 2 C 1 .R ! Œ0;  /, for c ¤ 0, and 2 C.R ! Œ0;  /, otherwise. Corollary 4.6.14 ([20, Cor. 4.12, Prop. 4.13]). In the conditions and notations of Proposition 4.6.13, is a strictly decaying function, for any speed c, and for any speed c ¤ 0, the profile is in Cb1 .R/. We assume that the first moment of aC in direction  2 f 1; 1g exists, namely, Z jx
j aC .x/ dx < 1: (A6) R

The following assumption is a weaker form of (A3). There exist r D r./ > 0,  D ./ > 0, ı D ı./ > 0, such that aC .x/ > ; for a.a. x 2 Bı .r/.

(A7)

There exists a critical situation, when the abscissa of the travelling wave coincides with the abscissa of the kernel aC . In this case, properties of the travelling waves may be different from the ‘generic’ case. Therefore, we introduce the following two classes of functions to simplify the further statements. Definition 4.6.15. Let m > 0, ~ ˙ > 0, 0 6 a 2 L1 .R/ be fixed, and assume that (A1) holds. For an arbitrary  2 f 1; 1g, we denote by V the the class of all kernels 0 6 aC 2 L1 .R/ such that (A2), (A5)–(A7) and one of the following assumptions holds: 1. 0 WD supf 2 R W a ./ < 1g D 1;

2. 0 < 1 and a .0 / D 1;

3. 0 < 1, a .0 / < 1 and t .0 / 2 Π1; m/, where t ./ is given by t ./ WD ~ C

Z

R

.1

s/aC .s/es ds 2 Π1; ~ C /;

 2 Œ0; 0 /:

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Similarly, we denote by W the class of all kernels such that 0 < 1, a .0 / < 1, and t .0 / 2 Œm; ~ C/ instead of .1/ .3/. For aC 2 V [ W , denote by I  .0; 1/ the interval

8 .0; 1/; if 0 D 1; ˆ ˆ <

I WD 0; 0 ; if 0 < 1 and a1 0 D 1; ˆ ˆ 
: 0; 0 ; if 0 < 1 and a1 0 < 1:

Proposition 4.6.16. Let  2 f 1; 1g be fixed and aC 2 V [ W . Then, there exists a unique  D  ./ 2 I such that inf G ./ D min G ./ D G . / > ~ C m :

>0

2I

Moreover, G is strictly decreasing on .0;   and strictly increasing on I n .0;  , where the latter interval may be empty. The following result provides expressions of for the minimal speed of travelling waves. Theorem 4.6.17 ([20, Thm. 4.23]). Let  2 f 1; 1g be fixed and aC 2 V [ W . Let c ./ be the minimal travelling wave speed according to Theorem 4.6.12, and let, for any c > c ./, the function D c 2 M .R/ be a travelling wave profile corresponding to the speed c. Let  2 I be the same as in Proposition 4.6.16. 1. The following relations hold: c ./ D min >0

~ C a ./ 

m

D

~ C a . / 

m

> ~ C m :

(4.6.6)

2. For aC 2 V , there exists another representation for the minimal speed, Z Z c ./ D ~ C x
 aC .x/e x
 dx D ~ C s aL C .s/e s ds > ~ C m : R

R

Now, we will formulate the uniqueness (up to shifts) of a profile wave with given speed c > c ./, c ¤ 0.

for a travelling

Theorem 4.6.18 ([20, Thm. 4.33]). Let  2 f 1; 1g be fixed and aC 2 V [ W . Suppose, additionally, that (A4) holds. Let c ./ be the minimal travelling wave speed according to Theorem 4.6.12. For the case aC 2 W with m D t .0 /, we will R assume, additionally, that R s 2 aL C .s/e0 s ds < 1. Then, for any c > c such that c ¤ 0, there exists a unique, up to a shift, travelling wave profile for (4.6.1).

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4.6.4 Propagation with a constant speed We will study here the behaviour of u.tx; t/, where u solves (4.6.1), for big t > 0. The results of Section 4.6.3 together with the comparison principle imply that if an initial condition u0 .x/ to (4.6.1) has a minorant/majorant which has the form .x
/,  2 f 1; 1g, where 2 M .R/ is a travelling wave profile in the direction  with a speed c > c ./, then, for the corresponding solution u to (4.6.1), the function u.tx; t/ will have the minorant/majorant .t.x
 c//, respectively. In particular, if the initial condition is “below” any travelling wave in a given direction, then one can estimate the corresponding value of u.tx; t/ (Theorem 4.6.19). Considering such a behaviour in different directions, one can obtain a (bounded) set outside of which the solution exponentially decays to 0 (Theorem 4.6.20). Inside this set, the solution will uniformly converge to  (Theorem 4.6.21). Here and below, for any measurable A  R, we define tA WD ftx j x 2 Ag  R, and ( ) ˇ d 1 d ˇ x
 WD sup jf .x/je E .R / WD f 2 L .R / kf k 0.

Theorem 4.6.19 ([20, Thm. 5.4]). Let  2 f 1; 1g and aC 2 V [ W ; i.e. all conditions of Definition 4.6.15 hold. Let  D  ./ 2 I be the same as in Proposition 4.6.16. Suppose that u0 2 E ; .R/ \ E and let u 2 X1 be the corresponding solution to (4.6.1). Let O  R be an open set, such that ‡1;  O and ı WD dist.‡1; ; R n O / > 0. Then, the following estimate holds,  ıt

sup u.x; t/ 6 ku0 k ; e

;

t > 0:

x…t O

Next we consider the global long-time behaviour in both directions  2 f 1; 1g simultaneously. Obviously \ \ ‡T D ‡T; D T ‡1; D T ‡1 ; T > 0: 2f 1;1g

2f 1;1g

We are now ready to state a result about the long-time behaviour at infinity in space. We consider the assumptions aC 2 L1 .R/: There exists d > 0 such that

Z

R

Clearly, (A9) implies

Z

R

aC .x/ed jxj dx < 1:

jxjaC .x/ dx < 1:

(A8) (A9)

(4.6.7)

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Theorem 4.6.20 ([20, Thm. 5.9]). Let the conditions (A1), (A2), (A3), (A8) and (A9) hold. Let u0 2 E be such that jjju0 jjj WD max ku0 k ./; < 1; 2f 1;1g

and let u 2 X1 be the corresponding solution to (4.6.1). Then, for any open set O  ‡1 , there exists  D .O/ > 0 such that sup u.x; t/ 6 jjju0 jjje

t

;

t > 0:

x…t O

Our second main result about the long-time behaviour states that the solution u 2 X1 uniformly converges to  inside the set t‡1 D ‡t . For a closed set A  R, we denote by int.A/ the interior of A. Theorem 4.6.21 ([20, Thm. 5.10]). Let the conditions (A1), (A2), (A4), (A8) and (A9) hold. Let u0 2 U , u0 6 0, and u 2 X1 be the corresponding solution to (4.6.1). Then, for any compact set C  int.‡1/, lim min u.x; t/ D :

t !1 x2t C

(4.6.8)

All results above about travelling waves and long-time behaviour of the solutions were obtained under exponential integrability assumptions, cf. (A5) or (A9). In [29], it was proved, in the case of the local competition (e.g. a D ı0 ), on R with local nonlinear term, that the case with aC which does not satisfy such conditions leads to ‘accelerating’ solutions, i.e. in this case an equality like (4.6.8) holds for arbitrarily big compact C  R. The detailed analysis of the propagation for the slowly decaying aC is done in the following section. We will formulate an analogue of the first statement in [29, Thm. 1]. Theorem 4.6.22 ([20, Thm. 5.21]). Let the conditions (A1), (A2), (A4), (A8) and (4.6.7) hold. Suppose also (cf. (A9)) that a ./ D 1 for any  > 0 and for any  2 f 1; 1g. Let u0 2 E be such that there exist x0 2 R,  > 0, r > 0, with the property u0 > , for a.a. x 2 Br .x0 /. Let u 2 X1 be the corresponding solution to (4.6.1). Then, for any compact set K  R, lim inf u.x; t/ D :

t !1 x2t K

4.6.5 Accelerating propagation The main result of this section is Theorem 4.6.25, where we demonstrate the accelerated propagation of solutions to (4.6.1) in the case when either of the dispersion kernel or the initial condition has regularly heavy tails at both ˙1, perhaps different. We show that, in such a case, the propagation to the right direction is fully determined by the right tails of either the kernel or the initial condition. Our approach in this section is based, in particular, on the extension of the theory of sub-exponential distributions, which we introduced earlier in [23].

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To state our main result, we start with the following definition. Definition 4.6.23. Let ˇ D ~ C m. 1. Let b W R ! RC be continuous and decreasing on .; 1/, for some  > 0, with limx!1 b.x/ D 0. Then, for some  > 0, there exists a function r.t/ D r.t; b/, t >  , which uniquely solves the equation
b r.t/ D e ˇ t ; t > ; (4.6.9) and r.t/ ! 1 as t ! 1. 2. Similarly, if the function b is continuous and increasing on . 1; / with limx! 1 b.x/ D 0, then we define l.t/ D l.t; b/ ! 1, t ! 1, as the unique solution to the equation
b l.t/ D e ˇ t ; t > :

In other words, r.t/ and l.t/ are given through the inverse functions to log b, namely, for t >  ,
1
1 r.t; b/ D log b R .ˇt/; l.t; b/ D log b R .ˇt/: (4.6.10) C

We are going to find sufficient conditions on a and u0 such that the corresponding solution u to (4.6.1) becomes arbitrarily close to  (as t goes to 1) inside the (expanded) interval . l.t/; r.t// and becomes arbitrarily close to 0 outside of this interval. Here, l.t/ D l.t; b/ and r.t/ D r.t; b/, where, cf. (4.6.10), ˚ log b.s/  log max a.x/; u0 .x/ ; x ! 1;

and we suppose that the function b has regularly heavy tails at 1, see Definition 4.6.24 below. In particular, for any small "; ı > 0, we will have that ˇ ˚
x > 0 ˇ u.x; t/ 2 .ı;  ı/  r.t t "/; r.t C t "/ for all t big enough; the corresponding result also holds for negative values of x and the function l.t/ instead.

Definition 4.6.24. A bounded function b W R ! RC is said to have a regularly heavy tail at 1 (in the sense of densities), if b 2 L1 .RC /, b is decreasing to 0 and convex on .; 1/ for some  > 0, and
b x C y  b.x/; y 2 R; x ! 1;  Z Z x b.y/ dy b.x/; x ! 1: b.x y/b.y/ dy  2 0

RC

A bounded function b W R ! RC is said to have a regularly heavy tail at 1 in the sense of densities, if the function b. x/ has a regularly heavy tail at 1 in the sense of densities.

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Examples of functions with regularly heavy tails at 1 are the following:
.log x/ x q ; .log x/ x  exp p.log x/q ; 
x  (4.6.11) .log x/ x  exp x ˛ ; .log x/ x  exp ; q .log x/

where p > 0, q > 1, ˛ 2 .0; 1/, ;  2 R. Note that any b with a regularly heavy tail at 1 in the sense of densities is such that, for each k > 0, x!1 ekx b.x/ ! 1;

this explains the name: the tail of b at 1 is ‘heavier’ than the tail of an exponential function. If (4.6.7) does not hold, then we assume that,

06

an˙

for each n 2 N, there exist 2 L1 .R/;

~n˙ > 0;

n 2 .0;  

which satisfy (A1)–(A4) and (4.6.7) instead of a˙ , ~ ˙ ,  , respectively, such that Z 1 ; n 2 N; xan .x/dx 2 R; n >  d n R ~n an  w wGn w 6 ~a  w wGw for 0 6 w 6 n ; n 2 N:

(A10)

Now one can formulate our main result. Theorem 4.6.25 ([24, Thm. 1]). Let either (A1)–(A4) and (4.6.7) hold or (A10) hold. Let 0 6 u0 6  , u0 6 0 and u be the corresponding solution to (4.6.1). Let u0 2 L1 .R/ and functions b; b1 ; b2 W R ! RC have regularly heavy tails at both ˙1 in the sense of densities, and let the following assumptions hold, either u0 .x/ > b1 .x/ or a.x/ > b1 .x/; ˚ max a.x/; u0 .x/ 6 b2 .x/; x 2 R:

x 2 R;

Suppose also that

log b1 .x/  log b2 .x/  log b.x/;

(4.6.12)

as x ! ˙1. Then, for each " 2 .0; 1/, lim

inf

t !1 Œ l.t "t;b/;r.t "t;b/

lim

sup

u.x; t/ D ;

t !1 . 1; l.t C"t;b/[Œr.t C"t;b/;1/

u.x; t/ D 0:

(4.6.13) (4.6.14)

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Note that the convergence in (4.6.13)–(4.6.14) is indeed ‘accelerated’ in t, since each b W R ! RC with regularly heavy tail satisfies, for each k > 0, r.t; b/

kt ! 1;

l.t; b/

kt ! 1;

as t ! 1:

The reason to introduce the function b in Theorem 4.6.25 is two-fold. First, we allow some flexibility in the choice of b1 and b2 and hence of a and u0 . Secondly, choosing such b, one can find r.t; b/ explicitly (i.e. (4.6.9) can be solved). Namely, cf. [21, Example 2.18], one has the following values of r.t/ D r.t; b/:   ˇ q b.x/ D x ; r.t/ D exp t I q   1 
ˇ q q t b.x/ D exp p.log x/ ; r.t/ D exp I p
1 r.t/ D .ˇt/ ˛ I b.x/ D exp x ˛ ;   x ; r.t/  ˇt.log t/˛ ; t ! 1: b.x/ D exp .log x/q

(Recall that here q > 1, ˛ 2 .0; 1/, p > 0.) Note that, for b. x/ and l.t/ D l.t; b/, the same examples hold.

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[10] J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Diff. Equ. 244 (2008), 3080–3118. [11] R.L. Dobrushin, Vlasov equations, Functional Anal. Appl. 13 (1979), 115–123. [12] R.L. Dobrushin, Y.G. Sinai and Y.M. Sukhov, Dynamical systems of statistical mechanics, in: Dynamical Systems, volume II of Encyclopaedia Mathematical Sciences, (Y.G. Sinai, ed.), Springer, Berlin (1989), pp. 208–254. [13] D. Finkelshtein, Y. Kondratiev and O. Kutoviy, Individual based model with competition in spatial ecology, SIAM J. Math. Anal. 41 (2009), 297–317. [14] D. Finkelshtein, Y. Kondratiev and O. Kutoviy, Vlasov scaling for stochastic dynamics of continuous systems, J. Stat. Phys. 141 (2010), 158–178. [15] D. Finkelshtein, Y. Kondratiev and O. Kutoviy, Vlasov scaling for the Glauber dynamics in continuum, Infin. Dim. Anal. Quantum Probab. Relat. Top. 14 (2011), 537–569. [16] D. Finkelshtein, Y. Kondratiev and O. Kutoviy, An operator approach to Vlasov scaling for some models of spatial ecology, Methods Funct. Anal. Top. 19 (2013), 108–126. [17] D. Finkelshtein, Y. Kondratiev and O. Kutoviy, Semigroup approach to birth-and-death stochastic dynamics in continuum, J. Funct. Anal. 262 (2012), 1274–1308. [18] D. Finkelshtein, Y. Kondratiev and O. Kutoviy, Statistical dynamics of continuous systems: perturbative and approximative approaches, Arab. J. Math. 4 (2015), 255–300. [19] D. Finkelshtein, Y. Kondratiev and M. J. Oliveira, Markov evolutions and hierarchical equations in the continuum. I. One-component systems, J. Evol. Equ. 9 (2009), 197–233. [20] D. Finkelshtein, Y. Kondratiev and P. Tkachov, Traveling waves and long-time behavior in a doubly nonlocal Fisher–KPP equation, preprint; arXiv:1508.02215. [21] D. Finkelshtein, Y. Kondratiev and P. Tkachov, Accelerated front propagation for monostable equations with nonlocal diffusion, preprint; arXiv:1611.09329. [22] D. Finkelshtein and P. Tkachov, The hair-trigger effect for a class of nonlocal nonlinear equations, preprint; arXiv:1702.08076. [23] D. Finkelshtein and P. Tkachov, Kesten’s bound for sub-exponential densities on the real line and its multi-dimensional analogues, preprint; arXiv:1704.05829. [24] D. Finkelshtein and P. Tkachov, Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line, preprint; arXiv:1706.09647. [25] R. Fisher, The advance of advantageous genes, Ann. Eugenics 7 (1937), 335–369. [26] N. Fournier and S. Méléard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations, Annals Appl. Probab. 14 (2004), 1880–1919. [27] N.L. Garcia and T.G. Kurtz, Spatial birth and death processes as solutions of stochastic equations, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), 281–303. [28] N.L. Garcia and T.G. Kurtz, Spatial point processes and the projection method, Progress Probab. 60 (2008), 271–298. [29] J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal. 43 (2011), 1955–1974. [30] F. Hamel and L. Ryzhik, On the nonlocal Fisher–KPP equation: steady states, spreading speed and global bounds, Nonlinearity 27 (2014), 2735–2753.

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[31] R.A. Holley and D.W. Stroock, Nearest neighbor birth and death processes on the real line, Acta Math. 140 (1978), 103–154. [32] A.N. Kolmogorov, I.G. Petrovsky and N.S. Piskunov, Étude de l’équation de la diffusion avec croissance de la quantité de matiere et son application a un probleme biologique, Bull. Univ. État Moscou Sér. Inter. A 1 (1937), 1–26. [33] Y. Kondratiev and T. Kuna, Harmonic analysis on configuration space. I. General theory, Infin. Dim. Anal. Quantum Probab. Relat. Top. 5 (2002), 201–233. [34] Y. Kondratiev and O. Kutoviy, On the metrical properties of the configuration space, Math. Nachr. 279 (2006), 774–783. [35] Y. Kondratiev, O. Kutoviy and R. Minlos, On non-equilibrium stochastic dynamics for interacting particle systems in continuum, J. Funct. Anal. 255 (2008), 200–227. [36] Y. Kondratiev, O. Kutoviy and S. Pirogov, Correlation functions and invariant measures in continuous contact model, Infin. Dim. Anal. Quantum Probab. Relat. Top. 11 (2008), 231– 258. [37] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Rat. Mech. Anal. 59 (1975), 219–239. [38] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures, Arch. Rat. Mech. Anal. 59 (1975), 241–256. [39] T.M. Liggett, Interacting Particle Systems, Springer, New York, 1985. [40] H.P. Lotz, Uniform convergence of operators on L1 and similar spaces, Math. Z. 190 (1985), 207–220. [41] D. Mollison, Possible velocities for a simple epidemic, Adv. Appl. Probab. 4 (1972), 233–257. [42] D. Mollison, The rate of spatial propagation of simple epidemics, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III: Probability theory, (L.M. Le Cam, J. Neyman, and E.L. Scott, eds.), Univ. California Press, Berkeley, CA (1972), pp. 579–614. [43] G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris 349 (2011), 553–557. [44] M.D. Penrose, Existence and spatial limit theorems for lattice and continuum particle systems, Probab. Surveys 5 (2008), 1–36. [45] R.S. Phillips, The adjoint semi-group, Pacific J. Math. 5 (1955), 269–283. [46] H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Mod. Phys. 52 (1980), 569–615. [47] Y.-J. Sun, W.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal. 74 (2011), 814–826. [48] J. Terra and N. Wolanski, Asymptotic behavior for a nonlocal diffusion equation with absorption and nonintegrable initial data. The supercritical case, Proc. Amer. Math. Soc. 139 (2011), 1421–1432. [49] J. Terra and N. Wolanski, Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data, Discr. Contin. Dyn. Syst. 31 (2011), 581–605. [50] H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci. 45 (2009), 925–953.

Chapter 5

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory B. Gentz We address the question of noise-induced transitions in continuous-time dynamical systems with special emphasis on aspects which go beyond standard large-deviation theory. After reviewing the classical Wentzell–Freidlin theory, we discuss the subexponential asymptotics of transition times between potential wells, transitions between stationary states in parabolic stochastic partial differential equations (SPDEs), firstexit from a domain with characteristic boundary, and the effect of noise on so-called mixed-mode oscillations.1

5.1 Introduction: Large deviations and the exit problem Physical systems are often described by ordinary differential equations (ODEs) in which unresolved degrees of freedom are modelled by noise, so that we are led to study stochastic differential equations (SDEs). We will consider the solutions to these SDEs as dynamical systems subject to a random perturbation, and we are interested in noise-induced phenomena, i.e. sample-path behaviour which differs significantly from the underlying deterministic dynamics. A typical example are noise-induced transitions between metastable equilibria. These transitions can generally be described in the framework of the exit problem for a diffusion process from a domain. Consider the overdamped motion of a particle in a potential U , described by gradient dynamics xP tdet D rU.xtdet / : (5.1.1) Taking a small random perturbation by Gaussian white noise into account, the now stochastic dynamics is given by an SDE p (5.1.2) dxt" D rU.xt" / dt C 2" dWt ; where U W Rd ! R is a confining potential, the noise intensity " > 0 is a small parameter, and .Wt /t >0 denotes a d -dimensional standard Brownian motion. We will interpret (5.1.2) as an Itô SDE.
The long-time behaviour of solutions xt" .!/ t to (5.1.2) is easy to understand in this case. The solutions are given by a (time-homogenous) Markov process whose 1 Projects

A3 and A9

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(a)

q1=2 .x/

(b) U.x/

-

-

q1=4 .x/

(c)

-

x

-

-

-

-

q1=10 .x/

(d)

x

-

-

x

-

-

-

x

Figure 5.1.1. (a) Asymmetric double-well potential U as an example and the corresponding invariant density q" for (b) " D 1=2, (c) " D 1=4, (d) " D 1=10.

transition probability densities .x; t/ 7! p" .x; yI t/ satisfy Kolmogorov’s forward (or Fokker–Planck) equation   @ p" .x; yI t/ D L" p" .x; yI t/ D rx
rx U.x/p" .x; yI t/ C "x p" .x; yI t/ : @t
If xt" .!/ t admits an invariant density q" , then L" q" D 0. For gradient systems with confining potential, q" exists and the invariant measure (or equilibrium distribution) is given by 1 " .dx/ D q" .x/ dx D e U.x/=" dx ; (5.1.3) Z" with normalisation Z e U.x/=" dx : Z" D Rd

As a consequence, for small noise intensities ", the invariant measure " concentrates in the minima of the potential U , cf. Figure 5.1.1.
At this point, we need to ask how long it actually takes until xt" t is well described by its invariant distribution (5.1.3). To illustrate the importance of the different timescales present in the system (5.1.2), consider a one-dimensional set-up with
a double-well potential U , and assume that xt" t starts at the bottom of the left-hand well, i.e., x0" D x ? , using the notation from Figure 5.1.2. First, in a time of order Trelax D 1=U 00 .x ? /, where U 00 .x ? / is the curvature of the potential U at the bottom

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 109

x0? H x? ? xC ? and a saddle Figure 5.1.2. Example of a double-well potential with local minima at x ? and xC at x0? . The dashed line indicates the barrier height H which needs to be surmounted in a transition from the left-hand well to the right-hand well.


of its left-hand well, xt" t approaches a Gaussian distribution, centred in x ? . From large-deviation theory we know that, with overwhelming probability, sample paths will remain inside the left-hand well for all times significantly shorter than Kramers’ time TKramers
D eH=" , where H D U.x0? / U.x ? / is the barrier height the stochastic process xt" t has to surmount to reach the other well. Thus, for small noise intensities ", the paths will stay in the left-hand well for a long time, and only for t  TKramers , the distribution of xt" approaches the bimodal stationary density q" . The transition from one well to the other is crucial here—until such a transition occurs, the system behaves as if the underlying deterministic dynamics, given by (5.1.1), had a unique stable equilibrium at x ? . The necessary transition to the other well can be viewed as the exit from a suitably chosen domain, which leads us to the more general question where and when the solution of an SDE exits from a given domain. Consider a general SDE p x0" D x0 2 Rd ; (5.1.4) dxt" D b.xt" / dt C 2"g.xt" / dWt ; where we will assume that b und g satisfy the usual local Lipschitz and boundedgrowth conditions. For the sake of brevity of the presentation, let us also assume that 1 a.x/ D g.x/g.x/T > M Id for a constant M > 0, i.e., uniform ellipticity for the infinitesimal generator A" ,

A" v.x/ D "
"

d X

i;j D1

aij .x/

@2 v.x/ C h b.x/; rv.x/i ; @xi @xj

of the diffusion xt t . Note that we do not assume that b derives from a potential. For a bounded domain D with smooth boundary and initial condition x0 2 D , define the first-exit time  D D" D infft > 0W xt" 62 D g : The
first-exit location is then given by x" 2 @D . The following questions arise: Does xt" t leave D ? If so: What do we know about the expected first-exit time or, more precisely, the distribution of the first-exit time  , and the first-exit location x" ?

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If the solution xtdet t of the corresponding deterministic ODE xP tdet D b.xtdet /
leaves D in finite time, xt" t will do the same, because, in finite time and for small noise intensity, the deviation xt" xtdet will remain small. If xtdet does not leave D , i.e., if D is positively invariant under the deterministic flow, we want to study the problem of diffusion exit, i.e., an exit which is noise induced and would not occur in the absence of noise. Expected first-exit times and the distribution of first-exit locations can be characterised by means of partial differential equations (PDEs). Below, we indicate the initial condition x0 for the SDE
(5.1.4) by writing P x0 and Ex0 for probabilities and expectations relating to xt" t , respectively.

Theorem 5.1.1 ([23, Chapter 1.5], [28, Chapter 5.7]). (a) x 7! Ex fD" g is the unique solution of the Poisson problem ( A" u D 1 in D ; uD0 on @D :

(5.1.5)

(b) Let f W @D ! R be continuous. Then, x 7! Ex ff .x"" /g is the unique D solution of the Dirichlet problem ( A" w D 0 in D ; (5.1.6) w D f on @D : This classical theorem provides answers to all our questions above—provided we can actually solve the two PDEs. As an example, consider the overdamped motion of a Brownian particle in a quadratic single-well potential in R. Example 5.1.2. The overdamped motion of a Brownian particle in a quadratic singlewell potential in R is given by the SDE (5.1.4) with d D 1, a drift term b.x/ D rU.x/ deriving from a potential U.x/ D ax 2 =2 with a > 0, and g.x/ D 1. The drift will always push the particle towards the bottom of the well located at x ? D 0, thus counteracting the effect of the noise. We consider the one-dimensional Dirichlet problem (5.1.6) for the domain D D .˛1 ; ˛2 /, with ˛1 < x ? D 0 < ˛2 , and f W f˛1 ; ˛2 g ! R defined by f .˛1 / D 1 and f .˛2 / D 0. Solving
the PDE (5.1.6) and applying Theorem 5.1.1(b), we see that the probability that xt" t leaves the interval D D .˛1 ; ˛2 / through ˛1 is given by  Z ˛2 Z ˛2 ˚ eU.y/=" dy : eU.y/=" dy P x x"" D ˛1 D Ex f .x"" / D D

D

x

˛1

In the small-noise limit " ! 0, the exit location concentrates in ˛1 if ˛1 is the lower of the two boundary points of D , i.e., if U.˛1 / < U.˛2 /, and in ˛2 if U.˛2 / < U.˛1 /. If U.˛1 / D U.˛2 /, the geometry of the well starts playing a role as we can see from   1 1 1 : (5.1.7) C lim P x fx"" D ˛1 g D D "!0 jU 0 .˛1 /j jU 0 .˛1 /j jU 0 .˛2 /j

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 111

The noise-induced exit is more easily achieved on the side on which the wall of the well is less steep. Later on, we will encounter other phenomena in which the geometry of wells also becomes important. In general, the PDEs (5.1.5) and (5.1.6) cannot be solved explicitly, and a largedeviations approach, developed by Wentzell and Freidlin, can be used to obtain the exponential asymptotics of first-exit times and concentration results for first-exit locations. More subtle details as in (5.1.7) can typically not be obtained by Wentzell– Freidlin theory, and we will content ourselves with a short review of the basic results before we return to questions which go beyond this theory. The key ingredient to all large-deviation results is the rate function or action functional 8 Z T

2 1 ˆ

1=2 ˆ < x0 2 H1 , / Œ ' P b.' /

a.'s s s ds ; for ' 2 0 I.'/ D IŒ0;T ;x0 .'/ D ˆ ˆ :C1; otherwise ,

where H1 denotes the space of functions ' W Œ0; T  ! Rd , '.0/ D 0, which possess an L2 -derivative. Recall that we introduced the notation a.x/ D g.x/g.x/T . The large-deviation principle (LDP) shows that, in the small-noise limit, the probability of a solution of the SDE (5.1.4) being close to a given path ' W Œ0; T  ! Rd behaves like  expf I.'/=2"g. More precisely, the LDP states that, for any set € of paths on Œ0; T , infı I 6 lim inf 2" log Pf.xt" /t 2 €g 6 lim sup 2" log Pf.xt" /t 2 €g 6 €

"!0

"!0

inf I ; €

where € ı and € denote the interior and closure of €, respectively. The LDP reduces estimates of probabilities of rare events to a variational principle. Assume that the domain D contains a unique, asymptotically stable equilibrium point x ? for the deterministic dynamics xP tdet D b.xtdet / associated with (5.1.4). Then the quasipotential with respect to x ? is defined as z 7! V .x ? ; z/ D inf inffIŒ0;t  .'/W ' 2 C .Œ0; t; Rd /; '0 D x ? ; 't D zg : t >0

It can be interpreted as the cost to reach z against the deterministic flow, and V D inffV .x ? ; z/W z 2 @D g reflects the cost of leaving D . With the help of the quasipotential, the key result of the Wentzell–Freidlin theory can be expressed as follows.

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Theorem 5.1.3 ([23, Chapter 4]). For any initial condition x0 2 D , (a) the mean first-exit time satisfies Ex0 D"  eV =2" as " ! 0;

(b) the first-exit time concentrates around its mean in the sense that, for any ı > 0, ˚ one has P x0 e.V ı/=2" 6 D" 6 e.V Cı/=2" ! 1 as " ! 0; (c) the first-exit locations concentrate near minima of quasipotential V .x ? ;
/ on the boundary @D . In particular, ˚if the infimum of V .x ? ;
/ is attained in a unique point z ? 2 @D , then, P x0 kx"" z ? k > ı ! 0 as " ! 0, for any D choice of ı > 0.

At first glance, it might be surprising that these results do not depend on the initial condition x0 . This is a question of timescales. The deterministic dynamics brings the process to a small neighbourhood of the unique stable equilibrium point in a time of order one, while the random perturbation enables an exit only on an exponentially long timescale. Thus, the first-exit time is long compared to the relaxation time, so that the initial condition is “forgotten” long before an exit becomes likely. In the gradient case with isotropic noise, i.e., for b D rU and g D IdRd , cf. (5.1.2), the variational principle defining the quasipotential with respect to a unique, asymptotically stable equilibrium point x ? can be solved. We find that V .x ? ; z/ D 2ŒU.z/ U.x ? / for all z 2 D such that U.z/ 6 inffU.z/W Q zQ 2 @D g. This implies that the cost for leaving D is given by the minimal height to be surmounted, i.e., V D infz2@D V .x ? ; z/ D 2ŒU.z ? / U.x ? /. Here z ? denotes the “lowest point on @D ” in the sense that U.z/ is minimal for z D z ? . We also know that the infimum is attained for paths going against the deterministic flow, i.e., for .'t /t satisfying 'Pt D CrU.'t /, cf. [23, Chapter 4]. In the gradient case with isotropic noise, transitions between wells are described by the Arrhenius Law [1]. Using again the notations from Figure 5.1.2, we define the first-hitting time " ? ? D infft > 0W x 2 Bı .x /g  xC t C ? ? of a small ball Bı .xC / around the local minimum xC . The Arrhenius Law states ? behaves like that, in the small-noise limit, the expected first-hitting time Ex ? xC ˚ ? ? ? ' const exp ŒU.x / Ex ? xC U.x /=" , provided the process starts in x ? . This 0 can be seen as a direct consequence of Theorem 5.1.3. It is easy to understand why ? the exponential asymptotics does not depend on the size ı of the small ball around xC : ? It takes an exponentially long time to exit from the initial well containing x and to cross the saddle, while a time of order jlog ıj suffices to reach the ı-neighbourhood ? of xC , once the saddle is crossed. We observe that the exponential asymptotics described by the Arrhenius Law depends only on the barrier height and not on the curvature of wells and saddle, and that the approach based on large-deviation theory does not yield information on the prefactor. But the LDP does provide information on optimal transition paths, and we see that only 1-saddles are relevant for transitions between wells. The multiwell case with more than two wells can be described by a hierarchy of “cycles”, see [23, Chapter 6].

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 113

5.2 Subexponential asymptotics: Kramers’ Law and nonquadratic saddles Refined results in the gradient case (5.1.2) go back to the work of Eyring [21] and Kramers [29]. The so-called Kramers’ Law states that, in the small-noise limit " ! 0, ? ' p Ex ? xC ? Ex ? xC

2

?

U 00 .x ? /jU 00 .x0? /j

2 ' j1 .x0? /j

s

eŒU.x0 /

U.x ? /="

jdet r 2 U.x0? /j ŒU.x ? / 0 e det r 2 U.x ? /

U.x ? /="

for d D 1 ,

(5.2.1)

for d > 2 ,

(5.2.2)

where we once more recall the notations from Figure 5.1.2. Here 1 .x0? / denotes the unique negative eigenvalue of r 2 U at the saddle x0? . Note that the geometry of the potential well and saddle only affects the subexponential prefactor. The prefactor is larger if (i) the saddle is flatter in the direction associated with the unique negative eigenvalue, if (ii) the confining directions of the saddle are steeper, or (iii) if the well in which the dynamics starts is shallower. The first mathematically rigorous proof of (5.2.2) in finite dimension, including a generalisation to more than two wells, is due to Bovier, Eckhoff, Gayrard and Klein [15, 16]. It is based on potential theory. Finally, a full asymptotic expansion of the prefactor in powers of " was proved in [26, 27], using analytical methods. Obviously, Kramers’ Law in the form (5.2.1) or (5.2.2), respectively, can only hold if none of the eigenvalues of r 2 U.x0? / and r 2 U.x ? / vanishes since it would predict a vanishing or an infinite transition time. Vanishing eigenvalues correspond to non-quadratic saddles or wells. One might wonder why this non-generic situation is of interest. The immediate answer lies in the study of parameter-dependent systems. As a parameter varies, a parameter-dependent system may undergo bifurcations, and a quadratic saddle or well may not be quadratic any longer and even cease to exist as the following simple example illustrates. Example 5.2.1. Consider two harmonically coupled particles, each with (individual) dynamics given by the gradient dynamics in a symmetric double-well potential 4 x2 , so that the global dynamics is determined by the potential U.x/ D x4 2 U .x1 ; x2 / D U.x1 / C U.x2 / C

.x1 2

x2 / 2 ;

(5.2.3)

where > 0 denotes the coupling strength. A change of variable, corresponding to a rotation by =4, allows
to rewrite b .y1 ; y2 / D 1 y 2 1 2 y 2 C 1 y 4 C 6y 2 y 2 C y 4 , yielding the potential as U 2 1 1 2 2 2 1 2 8 b .0; 0/ D 1 2 . This shows that the 1-saddle at .0; 0/, present at large det r 2 U coupling strength , ceases to be a 1-saddle at D 1=2 and turns into a global maximum, while two new saddles are created in a transversal pitchfork bifurcation. If the coupling strength is further decreased, another bifurcation occurs at D 1=3. In

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D

3 4

2 . 12 ; 1/

D

1 3

D

D

1 4

1 2

2 .0; 31 /

D

3 8

2 . 31 ; 21 /

D0

Figure 5.2.1. Level lines of the global potential landscape (5.2.3). Coupling strength D 34 corresponds to the strong-coupling regime, exhibiting two global minima and one saddle. At critical coupling strength D 21 the saddle at the origin is no longer quadratic. In the regime

2 . 31 ; 21 /, represented by D 83 , the origin is no longer a 1-saddle, and two new saddles have formed in a transversal pitchfork bifurcation. Coupling strength D 31 is critical again, and the two saddles which formed at D 12 undergo a longitudinal pitchfork bifurcation, splitting into two saddles and a well each. The resulting potential landscape for < 31 is represented by D 14 . Finally, in the uncoupled case D 0, there are four global minima and four 1-saddles.

this longitudinal pitchfork bifurcation, the saddles which were created at D 1=2 split into two new saddles and a well. As a consequence, the optimal transition paths for the stochastic dynamics from the global minimum at . 1; 1/, expressed in the original variables, to the other global minimum at .C1; C1/ are not the same in the different regimes. For strong coupling, the optimal transition path takes both particles simultaneously over the unique 1-saddle at the origin to the other well. For intermediate coupling strength, there are two optimal transition paths of equal cost (in terms of the large-deviation rate function), crossing one of the two saddles created at D 1=2. For weak coupling, there are still two optimal transition paths, but now they involve crossing a saddle, followed by a well and another saddle. This can be understood as an Ising-model-like behaviour, with the first particle entering the other well and, thus, reaching a local minimum, before “pulling” the second particle to the

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 115

other well. Figure 5.2.1 shows the level lines of the global potential landscape in original variables. In the general case of an arbitrary number N of particles, a sequence of symmetrybreaking bifurcations is observed. For a detailed analysis of the potential landscape and the resulting expected transition times between global minima, see [4]. Let us now turn to the general case of non-quadratic saddles. The following result generalises (5.2.2), cf. [15], to non-quadratic saddles. Theorem 5.2.2 ([9, Chapter 3]). Let x ? denote a quadratic local minimum of U , and ? let xC be another local minimum of U . Assume that x0? D 0 is the relevant, i.e. ? lowest, saddle for passage from x ? to xC . The normal form near the saddle x0? D 0 is then of the form d 1X U.y/ D u1 .y1 / C u2 .y2 / C j yj2 C : : : 2 j D3

Under suitable growth conditions on u1 ; u2 , which are satisfied whenever u1 and u2 are monomials of an even power, Z 1 ? e u1 .y1 /=" dy1 Y d r j .2"/d=2 e U.x /=" 1 ? D Z 1 p Ex ? xC 2" det r 2 U.x ? / " (5.2.4) e u2 .y2 /=" dy2 j D3 

1

  1 C O .."jlog "j˛ /ˇ / ;

where ˛; ˇ > 0 depend on the growth conditions and are explicitly known. In the case that the well containing the initial condition x ? is not quadratic, the first factor, which stems from a Gaussian integral, is replaced by the evaluation of a non-Gaussian integral. The theorem easily generalises to the case of more than the two directions y1 ; y2 being non-quadratic, by including those directions in the integrals on the right-hand side of (5.2.4) and having fewer factors in the product accordingly, cf. [9, Chapter 3.3]. As a corollary, let us illustrate Theorem 5.2.2 in a few standard cases. Corollary 5.2.3 ([9, Chapter 3]). Let x ? be a quadratic minimum of U . (a) If x0? is a quadratic saddle, i.e., if all directions are quadratic, we retrieve (5.2.2). (b) If one of the stable directions at the saddle x0? is quartic while all other directions, whether stable or unstable, are quadratic, i.e., if U.y/ D 21 j1 jy12 C P C4 y24 C 21 djD3 j yj2 C : : : , then s 2C41=4 "1=4 .2/3 3 : : : d ŒU.x ? / U.x ? /=" 0 ? D e Œ1 C O ."1=4 jlog "j5=4 / : Ex ? xC €.1=4/ j1 j det r 2 U.x ? /

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(c) If the unique unstable direction at the saddle x0? is quartic, while all stable P directions are quadratic, i.e., if U.y/ D C4 y14 C 21 djD2 j yj2 C : : : , then ? D Ex ? xC

€.1=4/ 2C41=4 "1=4

s

.2/1 2 : : : d ŒU.x ? / 0 e det r 2 U.x ? /

U.x ? /="

Œ1 C O ."1=4 jlog "j5=4 / :

(d) If one of the stable directions at the saddle x0? undergoes a pitchfork bifurcaP tion, i.e., if U.y/ D 21 j1 jy12 C 12 2 y22 C C4 y24 C 12 djD3 j yj2 C : : : , and 2 changes sign from negative to positive, cf. Example 5.2.1, then

(i) for 2 D 2 ."/ > 0, s p ? ? .2 C 2"C4 /3 : : : d eŒU.x0 / U.x /=" ? D 2 p Œ1CR."/ ; Ex ? xC j1 j det r 2 U.x ? / ‰C .2 = 2"C4 / (5.2.5) where the function p
2 (5.2.6) ‰C .˛/ D ˛.1 C ˛/=8 e˛ =16 K1=4 ˛ 2 =16 ; controlling the relative size of the eigenvalue 2 compared to the noise intensity ", satisfies lim˛!1 ‰C .˛/ D 1. Here, K1=4 denotes the modified Bessel function of the second kind. The error term R."/ is bounded by an expression of the form "˛1 jlog "j˛2 and vanishes in the limit " ! 0.

(ii) For 2 D 2 ."/ < 0, a similar expression holds. It involves the eigenvalues at the two newly created saddles near x0? and the modified Bessel functions I˙1=4 of the first kind.

Note that, as opposed to the case of quadratic well and saddle, in all the other cases the prefactor becomes "-dependent to the leading order. The result (5.2.5) on the pitchfork bifurcation illuminates how the crossover between the regimes of a zero eigenvalue and an eigenvalue of order one is realised: When 2 is of smaller order than "1=2 , a saturation effect is observed, and the dynamics behaves as if the potential’s curvature were bounded below by .2"C4 /1=2 . 5.2.3 (d) as a funcFigure 5.2.2 shows the subexponential prefactor in Corollary p tion of 2 . More precisely, we show .0; 1/ 3 2 7! 2 C "1=2 =‰C .2 ="1=2 /, cf. (5.2.5), and the continuation to the interval . 1; 0. The proof of Theorem 5.2.2 uses the potential-theoretic approach from [15]. It is based on expressing expected transition times in terms of Newtonian capacities between sets, which can be estimated by a variational principle involving Dirichlet forms.

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 117 2.0

1.5

1.0

0.5

0.0 -4

-2

0

2

4

Figure 5.2.2. The behaviour of the subexponential prefactor in the expected transition time Ex ? x ? as a function of the eigenvalue 2 2 R in the case that the stable direction C corresponding to y2 undergoes a pitchfork bifurcation at the origin, cf. Corollary 5.2.3 (d). For the sake of definiteness, we fix the parameter values in such a way that 2C4 D 1 and ? 3 : : : d D j1 j det r 2 U.x ? /. For 2 < 0, the eigenvalues ˙ j at the two saddles near x0 satisfy ˙ ˙ 2  22 and j  j for j 6D 2, so that our choice indeed fixes all relevant parameters apart from the noise intensity ". The curves show the values " D 0:5, " D 0:2, " D 0:1, " D 0:02 and " D 0:002. The darker shades correspond to successively smaller values of ".

5.3 Metastability in parabolic SPDEs Let us return for a moment to the example of harmonically coupled Brownian particles, each with (individual) dynamics given by the gradient dynamics in the symmet4 x2 , cf. Example 5.2.1 for a discussion of the ric double-well potential U.x/ D x4 2 case of two particles. For an arbitrary number N of particles and nearest-neighbour coupling, we close the resulting chain to a circle. Then, the global dynamics is described by the gradient dynamics with respect to the global potential X

X U.xi / C U .x/ D .xi C1 xi /2 ; i 2 ƒ D Z=N Z ; 4 i 2ƒ

i 2ƒ

with additive Gaussian white noise. We obtain the SDE p dxt" D rU .xt" / dt C 2"N dWt ;

(5.3.1)

where .Wt /t denotes an N -dimensional standard Brownian motion. This model has been analysed in great detail in [4, 5]. The deterministic dynamics (corresponding to " D 0) exhibits a series of symmetry-breaking bifurcations, ranging from the strong-coupling case in which noise-induced transitions of (5.3.1) between the global

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? minima x ? D . 1; : : : ; 1/ and xC D .C1; : : : ; C1/ are realised by all particles switching wells simultaneously to the weak-coupling regime in which an Ising-like behaviour is observed, with a first particle switching wells, then pulling one of its neighbours to the other well, and so on, until the transition is complete. The optimal transition paths and expected transition times have been derived. Rescaling (5.3.1) appropriately and taking the infinite-particle-number limit N ! 1 yields the Allen–Cahn SPDE on a compact interval Œ0; L, i.e., p   @t u.x; t/ D x u.x; t/ rx U.u.x; t// dt C 2" dW .t; x/ ; x 2 Œ0; L ; (5.3.2)

with real-valued p u.x; t/, periodic boundary conditions and weak space–time Gaussian white noise 2" dW .t; x/, where W .t; x/ is a cylindrical Wiener process compatible with the boundary conditions. The deterministic PDE associated with (5.3.2) admits several stationary solutions, namely the constant functions u.x; t/  C1, u.x; t/  1 and u.x; t/  0. Out of these, u.x; t/  C1 and u.x; t/  1 are stable, and it seems natural to expect an analogue to Kramers’ Law to hold, governing the expected transition time, say from u.x; t/  1 to u.x; t/  C1. To keep the presentation simple, we will focus on the symmetric double-well potential U and Neumann boundary conditions with zero flux, i.e., p   @t u.x; t/ D x u.x; t/ rx U.u.x; t// dt C 2" dW .t; x/ ; u.
; 0/ D '.
/ ; @x u.0; t/ D @x u.L; t/ D 0

(5.3.3)

(Neumann b.c.) :

Existence and uniqueness of a mild solution as well as a large-deviation principle for the solutions and the exponential asymptotics of the transition time have been established by Faris and Jona-Lasinio [22]. As for finite system size, refined results require a detailed understanding of the deterministic system. Recall that the deterministic dynamics minimises the energy functional Z L VL .u/ D ŒU.u.x// C 12 u0 .x/2  dx ; 0

and the stationary states for the deterministic system can be found as solutions to @xx u.x; t/ C u.x; t/

u.x; t/3 D 0 :

(5.3.4) 2

d Their stability is determined by the eigenvalues of the operator AŒu D dx 2 C 1 3u2 which governs the linearisation @t v D AŒuv of the Allen–Cahn equation at the stationary solutions u. We find:  The uniform stationary states u˙ .x/  ˙1. In this case, AŒu˙  has eigenvalues k D .2 C .k=L/2 /, k 2 N0 , and there are no positive eigenvalues. This implies that u˙ are stable states. Actually, u˙ are global minima of VL .  The uniform state u0 .x/  0 which has eigenvalues k D .k=L/2 1, k 2 N0 . Thus, u0 is unstable, and we will need to investigate whether it is a transition state.

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 119

 For sufficiently large L, there are two additional stationary states for each k 2 N which satisfies L > k. These additional states are of the form r  kx  2m sn p uk;˙ .x/ D ˙ C K.m/; m : mC1 mC1 p where K.m/ is defined by 2k m C 1K.m/ D L and sn denotes Jacobi’s elliptic sine. Thus, the number of solutions to (5.3.4) depends on the interval length L, and bifurcations are observed whenever L is a multiple of . Regarding the transition state, we observe the following.  For L < , the uniform state u0 .x/  0 is the unique transition state with activation energy WL D VL .u0 / VL .u˙ / D L=4. The large-deviation result [22] shows that the expected transition time from u to a neighbourhood of uC satisfies 1 eL=4" Eu uC D €0 .L/ for some prefactor €0 .L/ rate

1

. Maier and Stein [32, 33, 37] derive the transition v s p u1 Y u 1 j0 j t sinh. 2L/ k D 3=4 €0 .L/ ' 2 jk j sin L 2  kD0

by a formal computation.

 For L > , u0 .x/  0 remains unstable, but no longer forms the transition state, and u1;˙ .x/ become the new transition states (of instanton shape). Thus a pitchfork bifurcation occurs as L increases through , and the uniform transition state u0 bifurcates into pair of instanton states u1;˙ . The activation energy can be calculated from WL D VL .u1;˙/ VL .u /, and a formal computation, using Gelfand’s method, allows to compute the spectral determinant in the prefactor.  The subsequent bifurcations at L D k with k > 2 do not change the transition state as they have larger activation energy. At this point, two questions come to mind. Firstly, what happens when L %  and u0 ceases to be a transition state? Secondly, is the formal computation correct in infinite dimension? In [10], Kramers’ Law has been proved for a class of parabolic SPDEs with a general double-well potential and periodic or Neumann boundary conditions. Indeed, a spectral Galerkin approximation enables us to draw on results for finitesystem size. Let us truncate the Fourier series for u to obtain d 1 X 2 X 1 yk .t/ cos.kx=L/ D p yQk .t/eikx=L ; ud .x; t/ D p y0 .t/ C p L L kD1 L jkj6d

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and also rewrite the potential in Fourier variables, retaining only modes with k 6 d , VL.d / .y/ D where .d / .y/ D VL;4

1 4L

d 1X .d / k yk2 C VL;4 .y/ 2 kD0

X

k1 Ck2 Ck3 Ck4 D0 ki 2f d;:::;0;:::;Cd g

yQk1 yQk2 yQk3 yQk4 :

The resulting system of d coupled SDEs i h X p 1 yQk1 yQk2 yQk3 dt C 2" dWt.k/ ; dyk D k yk L

(5.3.5)

k1 Ck2 Ck3 Dk


where W .k/ t , k D 1; : : : ; d , are d independent one-dimensional Brownian motions, provides a good approximation of the full system, see [14]. The reduced, finite-dimensional system (5.3.5) can be analysed using Theorem 5.2.2, yielding an estimate of the type .d /

" C.d /eWL

="

Œ1

.d /

Rd ."/ 6 Eu.d / u.d / 6 " C.d /eWL C

="

Œ1 C RdC ."/ ;

(5.3.6) .d / where u˙ D ˙.1; : : : ; 1/ and u.d / denotes the first time, the d -dimensional stochasC / .d / tic process (5.3.5) hits a small ball around u.d C . Furthermore, WL is the activation energy in the d -dimensional case, and C.d / and Rd˙ ."/ denote the subexponential prefactor and the error term from Theorem 5.2.2, respectively. Note that the factor " is only present (i.e., ¤ 0) at bifurcation points or in the presence of non-quadratic saddles. .d / The limits limd !1 C.d / DW C.1/ and limd !1 WL DW WL exist and are finite. The crucial point when taking the limit is that the error terms can be controlled uniformly in the dimension d , i.e., R˙ ."/ D sup Rd˙ ."/ ! 0 d

as " ! 0 :

This fact was first observed in [3] for the so-called synchronised regime of the chain of coupled bistable particles studied in [4, 5], a regime which excludes bifurcation points. From (5.3.6), we therefore obtain " C.1/ eW="Œ1

R ."/ 6 Eu uC 6 " C.1/ eW="Œ1 C RC ."/ ;

which implies the following theorem. Theorem 5.3.1 ([10, Theorem 2.5]). For the Allen–Cahn SPDE (5.3.3) with Neumann boundary conditions, the mean transition time between the uniform stable states u˙ satisfies the following.

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 121 ε =0.0003 10 ε =0.001 ε =0.003 5

ε =0.01

0 0.5

1.0

1.5

Figure 5.3.1. The map L 7! €0 .L/, cf. Theorem 5.3.1, for the SPDE (5.3.3) with Neumann boundary conditions. We show the subexponential prefactor in the expected transition time Eu uC as a function of the interval length L for different values of the noise intensity ". Note that we rescaled the interval length L in such a way that the critical interval length L D  coincides with the value 1 on the horizontal axis.

 For L < , the mean transition time satisfies Eu uC D

1 €0 .L/

eL=4" , and

s

s p    sinh. 2L/ 1 1 €0 .L/ ' 3=4 p ‰C p sin L 2  1 C 3"=4L 3"=4L q p €.1=4/ sinh. 2/ " 1=4 as L % . ! 2.3 7 /1=4 1

The function ‰C has been defined in (5.2.6).

 For L > , the mean transition time satisfies Eu uC D €01.L/ eWL =" , with explicitly known activation energy WL and a similar expression for €0 .L/, involving the eigenvalues at the saddles which are created at the critical interval length L D , see Figure 5.3.1 for the behaviour of €0 .L/ as L increases through . This result covers all finite positive values of the interval length L, and thus includes bifurcation values. For non-bifurcating one-dimensional SPDEs, see also [2].

5.4 Noise-induced passage through an unstable periodic orbit Let us now turn our attention to non-reversible diffusions, where new phenomena arise. We will focus on the simplest situation of interest, namely an SDE (5.1.4) for which the underlying deterministic dynamics (obtained by setting " D 0 in (5.1.4)) has an unstable periodic orbit. Let us remark in passing that such a dynamics cannot

122

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Figure 5.4.1. Deterministic dynamics in R2 , exhibiting an unstable periodic orbit (broken black curve) with a stable periodic orbit (solid black curve) in its interior. We show the orbits (grey curves) of the dynamics for three different initial conditions: (i) inside the stable periodic orbit, (ii) between the stable and the unstable periodic orbit, but close to the unstable periodic orbit, and (iii) outside the unstable periodic orbit.

be of gradient type. Assume from now on that d D 2 and that the domain D is such that @D is this unstable periodic orbit for the deterministic dynamics. Theorem 5.1.3 still applies and implies in particular that, for initial conditions x0 2 D , the expected first-exit time satisfies Ex0 D"  eV =2" in the small noise limit. See Figure 5.4.1 for an illustration of orbits for the deterministic dynamics in the case D contains a stable periodic orbit … instead of a stable equilibrium point x ? . Theorem 5.1.3 also generalises to this case. In either case, the quasipotential V .x0 ;
/  V or V .…;
/  V , respectively, is constant on @D . This means that at the level of large deviations we do not see any preferred exit points on @D , cf. Theorem 5.1.3(c).
Actually, the solution xt" t of the SDE (5.1.4) in this case behaves quite differently from the reversible case discussed so far. Recall that in the reversible case, exit locations concentrate near minima of the quasipotential as " ! 0. In contrast, if @D is an unstable periodic orbit, the distribution of the exit location x"" on @D does not D converge as " ! 0. Instead, the density of x"" is translated along @D proportionally D to jlog "j. This surprising phenomenon was first discovered by Day [17, 18], who named it “cycling”. In the same spirit, Maier and Stein [30] found that in a stationary regime, obtained by reinjecting the particle upon exit from D , the rate of escape ˚ " d P x 2 6 D has a jlog "j-periodic prefactor, see also [25]. t dt Passage through an unstable periodic orbit plays an important role in many applications. For instance, it determines the distribution of noise-induced phase slips in synchronised oscillators [34]. The first-exit distribution also determines the residencetime distribution in stochastic resonance [24, 31, 7], and in neuroscience, the spiking mechanism for the Morris–Lecar model may involve passage through an unstable periodic orbit [35, 38, 39, 20]. A detailed analysis provides insight into the properties of the density of the firstpassage time. For the sake of brevity, we state an informal version here, valid under non-degeneracy assumptions excluding symmetries of the system.

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 123

Theorem 5.4.1 ([6, 7, 11, informal version]). There exists an explicit parametrisation  of @D s.t. the first-passage time distribution is given by P where

n . " / D

T

o 2 Œt; t C  D  ftrans .t/C."/QT

jlog "j 2T


t t0 ;

(5.4.1)

 T is the period and  the Lyapunov exponent of the unstable periodic orbit;   is a “natural” parametrisation of the boundary in the sense that

(i)  0 .t/ is an explicitly known, model-dependent, strictly positive, T -periodic function, and

(ii) .t/ satisfies .t C T / D .t/ C T ;

 QT is a universal T -periodic function;

 0 D 1 e H=2" is the principal eigenvalue of the transition kernel of the Markov chain .Xn /n which is given by a random Poincaré map, recording the successive positions of the diffusion .xt" /t whenever .xt" /t has completed the next rotation around the periodic orbit. The value of H is close to I. 1 /, i.e., the large-deviation rate function I , evaluated at an optimal transition path 1 which connects the stable to the unstable periodic orbit.  ftrans describes the influence of a deterministic start at a fixed time on the interior stable periodic orbit and grows from 0 to 1 in time t of order jlog "j;

 C."/ is the normalising constant which is of order e

H=2"

.

The universal profile y 7! QT .y/ is periodic with period T and given by the periodicised Gumbel distribution QT .y/ D where

1 X

A.T .n

y// ;

(5.4.2)

nD 1

1 2x o e 2 is the density of a type-1 Gumbel distribution. Figure 5.4.2 shows this universal profile for different choices of T . Observe that this profile determines the concentration of first-passage times within each period. The larger T , the more pronounced are the peaks, while for smaller values of T , the peaks overlap more. Thus for larger values of T , the first-passage time is more concentrated within each period, showing that there is one preferred time window for exit per period. Figure 5.4.3 shows the resulting first-passage density over 16 periods, where we have fixed H while varying T and ". Note the effect of the transitional phase which suppresses the first peak(s) due to the time needed until exit becomes likely. For n A.x/ D exp 2x

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1.0

0.8

0.6

0.4

0.2

1

2

3

4

Figure 5.4.2. The cycling profile y 7! QT .y/, cf. (5.4.2), plotted over four periods. The parameter values are T D 1:0, T D 1:5, T D 2:0, T D 4:0 and T D 20:0. The darker shades correspond to successively larger values of T .

(a) " D 0:5, T D 1:5

(b) " D 0:5, T D 5

(c) " D 0:2, T D 1:5

(d) " D 0:5, T D 10


"j t t0 , cf. (5.4.1), shown Figure 5.4.3. The first-passage density t 7! ftrans .t/C."/QT jlog 2T for 16 periods. Note that the graphs have been rescaled in such a way that the maximum is at the same height for all panels. We fixed H D 1. Panels (a),(c) and (d) show the effect of varying T while keeping " D 0:5 fixed, while Panels (a) and (c) illustrate the “cycling” effect by varying " for fixed T .

Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 125

larger times, first-passage becomes more unlikely again because the probability density has already lost mass due to exit having occurred earlier. Comparing the panels (a) and (c), we can see the peaks moving according to the cycling phenomenon. The proof of Theorem 5.4.1 is based on combining large-deviation results with properties of random Poincaré maps described by continuous-space discrete-time Markov chains. Spectral-gap estimates for the kernels of these Markov chains allow to estimate the first-passage times relevant for the proof, see [11].

5.5 Mixed-mode oscillations Estimates on the kernels of the Markov chains which are derived from random Poincaré maps also proved extremely useful in quantifying the effect of noise on mixedmode oscillations (MMOs). We speak of MMOs, if a model exhibits a pattern of alternating large- and small-amplitude oscillations (SAOs), see [19] for a review. MMOs can occur in a variety of systems. Here we focus on MMOs generated in slow–fast systems with one fast and two slow variables, containing a folded-node singularity and an S-shaped critical manifold governing the global return mechanism. In such systems, MMOs are associated with the existence of canard solutions near the folded-node singularity, i.e., solutions which track a repelling slow manifold for a long time. The effect of noise on the dynamics near the folded-node, which is governing the SAOs, has been quantified in [12], combining general results for deterministic slow–fast systems and for canard solutions [36, 40] with the geometric approach to concentration results for sample paths of randomly perturbed slow–fast systems [8]. Sample paths near a folded node stay inside a suitably chosen neighbourhood of an attracting deterministic solution with high probability. This neighbourhood can be explicitly defined with the help of the covariance matrix of the solution of the linearised SDE. The subtle interplay between noise intensity, timescale separation and a system parameter determines when and how many small oscillations become indistinguishable from noisy fluctuations. When exiting the neighbourhood of the folded node, noise can cause sample paths to jump away from canard solutions with high probability before their deterministic counterparts do. The typical escape time can be determined rather precisely. This early-jump mechanism can drastically influence the local as well as the global dynamics and change the pattern of alternating small- and large-amplitude oscillations. The fact that early escape is likely suggests that noise decreases the number of SAOs. Is that so? Or can we tune the parameters in such a way that preselected MMO patterns are achieved? These questions were addressed in [13], where an interesting phenomenon was uncovered: While larger noise intensities generally increase the size of fluctuations, noise may still increase the number of small-amplitude oscillations between consecutive large-amplitude oscillations. This counterintuitive behaviour has its origin in the interplay of the following facts:

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 Orbits with a small number of SAOs are less affected by noise than those with a large number of SAOs;  There is an unexpected saturation effect: The typical value of the stochastic return map and its spreading become independent of the number of SAO for large enough SAO numbers.  This saturation effect sets in earlier for larger noise intensities. The techniques developed in this work will certainly prove helpful in the analysis of the effect of noise on a variety of related systems in which a local analysis is not sufficient.

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Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory 127 [15] A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein, Metastability in reversible diffusion processes. I.Sharp asymptotics for capacities and exit times, J. Eur. Math. Soc. (JEMS) 6 (2004), 399–424. [16] A. Bovier, V. Gayrard, and M. Klein, Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues, J. Eur. Math. Soc. (JEMS) 7 (2005), 69–99. [17] M. Day, Conditional exits for small noise diffusions with characteristic boundary, Ann. Probab. 20 (1992), 1385–1419. [18] M. Day, Cycling and skewing of exit measures for planar systems, Stoch. Stoch. Rep. 48 (1994), 227–247. [19] M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, , H.M. Osinga, and M. Wechselberger, Mixed-mode oscillations with multiple time scales, SIAM Review 54 (2012), 211– 288. [20] S. Ditlevsen and P. Greenwood, The Morris–Lecar neuron model embeds a leaky integrateand-fire model, J. Math. Biol. 67 (2013), 239–259. [21] H. Eyring, The activated complex in chemical reactions, Journal of Chemical Physics 3 (1935), 107–115. [22] W.G. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A 15 (1982), 3025–3055. [23] M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, 3rd ed., Springer, Berlin (2012). [24] L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Stochastic resonance, Rev. Modern Phys. 70 (1998), 223–287. [25] S. Getfert and P. Reimann, Suppression of thermally activated escape by heating, Physical Review E 80 (2009), 030101 R. [26] B. Helffer, M. Klein, and F. Nier, Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp. 26 (2004), 41–85. [27] B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker–Planck Operators and Witten Laplacians, Springer, Berlin, 2005. [28] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, 2. ed., Graduate Texts in Mathematics, Vol. 113, Springer, New York, 1991. [29] H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940), 284–304. [30] R.S. Maier and D.L. Stein, Oscillatory behavior of the rate of escape through an unstable limit cycle, Phys. Rev. Lett. 77 (1996), 4860–4863. [31] R.S. Maier and D.L. Stein, Noise-activated escape from a sloshing potential well, Phys. Rev. Lett. 86 (2001), 3942–3945. [32] R.S. Maier and D.L. Stein, Droplet nucleation and domain wall motion in a bounded interval, Phys. Rev. Lett. 87 (2001), 270601-1. [33] R.S. Maier and D.L. Stein, The effects of weak spatiotemporal noise on a bistable onedimensional system, in: Noise in complex systems and stochastic dynamics, L. SchimanskiGeier, D, Abbott, A. Neimann, and C. Van den Broeck (eds.), SPIE Proceedings Series, vol. 5114, 2003, pp. 67–78.

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[34] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization, a Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series 12, Cambridge University Press, Cambridge, 2001. [35] J. Rinzel and B. Ermentrout, Analysis of neural excitability and oscillations, in: Methods of Neural Modeling: From Synapses to Networks, (C. Koch and I. Segev, eds.), MIT Press, Cambridge, MA (1989), pp. 135–169. [36] P. Szmolyan and M. Wechselberger, Canards in R3 , Journal of Differential Equations 177 (2001), 419–453. [37] D.L. Stein, Large fluctuations, classical activation, quantum tunneling, and phase transitions, Braz. J. Phys. 35 (2005), 242–252. [38] T. Tateno and K. Pakdaman, Random dynamics of the Morris–Lecar neural model, Chaos 14 (2004), 511–530. [39] K. Tsumoto, H. Kitajima, T. Yoshinaga, K. Aihara and H. Kawakami, Bifurcations in Morris– Lecar neuron model, Neurocomputing 69 (2006), 293–316. [40] M. Wechselberger, Existence and bifurcation of canards in R3 in the case of a folded node, SIAM J. Applied Dynamical Systems 4 (2005), 101–139.

Chapter 6

Computation and stability of waves in equivariant evolution equations W.-J. Beyn and D. Otten Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs that model the combined effect of dissipation and nonlinear interaction. From an abstract viewpoint, they appear as relative equilibria of an equivariant evolution equation. In numerical computations, the freezing method takes advantage of this structure by splitting the evolution of the PDE into the dynamics on the underlying Lie group and on some reduced phase space. The approach raises a series of questions which were answered to a certain extent: linear stability implies nonlinear (asymptotic) stability, persistence of stability under discretisation, analysis and computation of spectral structures, first versus second order evolution systems, well-posedness of partial differential algebraic equations, spatial decay of wave profiles and truncation to bounded domains, analytical and numerical treatment of wave interactions, relation to connecting orbits in dynamical systems. A further numerical problem related to this topic will be discussed, namely the solution of nonlinear eigenvalue problems via a contour method.1

6.1 Equivariant evolution equations 6.1.1 Abstract setting The overall topic of the project is the numerical analysis of evolution equations which may be written in the abstract form ut D F .u/;

t > 0;

(6.1.1)

where the solution u W Œ0; T / ! X , t 7! u.t/ is defined on a real interval Œ0; T /, with 0 < T 6 1, and has values in a Banach space X with derivative ut . The map F W Z  X ! X is a vector field defined on a dense subspace Z of X . The additional structure is described in terms of a Lie group G of dimension n D dim.G/ < 1 which acts on X via a homomorphism into the general linear group GL.X / of homeomorphisms on X : a W G ! GL.X /;

7! a. /:

(6.1.2)

For the images we use the synonymous notation a.g/u D a.g; u/, g 2 G, u 2 X . 1 Projects

A9, B3

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We assume that Equation (6.1.1) is equivariant with respect to this group action, i.e., the vector field F has the following property F .a. /u/ D a. /F .u/;

8 2 G and 8u 2 Z;

(6.1.3)

where we have assumed a. /Z  Z for all 2 G. In Sections 6.2.2 and 6.3, we will deal with several classes of PDEs which fit into this general setting. All of them are formulated for functions on a Euclidean space Rd where the action is caused by the special Euclidean group SE.d / acting via rotations and translations on their arguments or on their values. Remark 6.1.1. For some applications, even this framework is not sufficient. For example, travelling fronts which have finite but non-zero limits at infinity do not lie in any of the usual Lesbesgue or Sobolev spaces, but in an affine space. To cover such cases, but also more general PDEs on manifolds, one can generalise the whole approach to Banach manifolds X , where F is a vector field defined on a submanifold Z of X mapping into the tangent bundle TX , and a takes values in the space of diffeomorphisms Diff.X; X /. Equivariance (6.1.3) is then expressed as F .a. ; u// D du Œa. ; u/F .u/, where du Œa. ; u/ W Tu X ! Ta. ;u/ X denotes the tangent map. For the sake of simplicity, we will not pursue this generalisation here (see [45]). For this article, it is sufficient to work with the simple notion of a strong solution of a Cauchy problem instead of dealing with mild solutions in time and weak solutions in space. We refer to Chapters 2, 4 and 8 for related concepts of a weak solution as they appear in the analysis of many PDEs, both local and nonlocal. Definition 6.1.2. A function u 2 C 1 .Œ0; T /; X / \ C.Œ0; T /; Z/ satisfying ut D F .u/;

t 2 Œ0; T /;

u.0/ D u0 2 Z;

(6.1.4)

is called a strong solution of the Cauchy problem (6.1.4). In the following, we will always assume that a strong solution of (6.1.4) exists locally, i.e., on some interval Œ0; T / with T > 0, and that it is unique. For applications to PDEs, it is typical that the group action is differentiable only for smooth functions. Therefore, we impose the following condition. Assumption 6.1.3. For any u 2 X the mapping a.
/u W G ! X;

7! a. /u

is continuous, and for every u 2 Z it is continuously differentiable with derivative d Œa. /u W T G ! X;

 7! d Œa. /u:

(6.1.5)

When D 1, the tangent space T1 G may be identified with the Lie algebra g of G, and we have d Œa.1/u W g ! X .

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6.1.2 Relative equilibria Relative equilibria are special solutions of (6.1.1) which lie in the group orbit of a single element. Definition 6.1.4. A pair v? 2 Z; ? 2 C 1 .Œ0; 1/; G/ is called a relative equilibrium of (6.1.1) if ? .0/ D 1 and if u? .t/ D a. ? .t//v? , t > 0 is a strong solution of (6.1.4) with u0 D v? . Sometimes (see e.g. [18]), the whole group orbit OG .v? / D fa.g/v? W g 2 Gg is called a relative equilibrium. However, we include the path t ! ? .t/ on the group as part of our definition since it will be relevant for both the stability analysis and the numerical computations. The following theorem shows that the path may always be written as ? .t/ D exp.t? / for some ? 2 g. Recall that exp W g ! G is the exponential map and that ? .t/ D exp.t? / with t 2 R is the unique solution of the Cauchy problem

?;t .t/ D dL ? .t / .1/? ; ? .0/ D 1; (6.1.6)

where L g D ı g; g 2 G, denotes the multiplication by from the left. The vector field on the right-hand side of (6.1.6) is often simply written as ? .t/? , but in analogy to (6.1.5) we keep the slightly clumsier notation dL ? .t / .1/? for clarity. Theorem 6.1.5. Under assumption 6.1.3, for every relative equilibrium v? 2 Z with

? 2 C 1 .Œ0; 1/; G/, there exists ? 2 g such that 0 D F .v? /

d Œa.1/v??

a. ? .t//v? D a.exp.t? //v? :

(6.1.7) (6.1.8)

Conversely, if v? 2 Z, ? 2 g solve (6.1.7), then v? and ? .t/ D exp.t? / with t > 0 are a relative equilibrium. Given v? and ? .
/, the uniqueness of ? will follow from (6.1.8) in the first part of the theorem if the stabiliser H.v? / of v? is simple, i.e. H.v? / D f 2 G W a. /v? D v? g D f1g:

(6.1.9)

But even then, Equation (6.1.7) does not determine the pair .v? ; ? / uniquely, since relative equilibria always come in families. More precisely, Definition 6.1.4 and the equivariance (6.1.3) show that any relative equilibrium v? ; ? of (6.1.4) is accompanied by a family .w.g/; .g;
//; g 2 G of relative equilibria given by w.g/ D a.g/v? ;

.g; t/ D g ı ? .t/ ı g

1

;

g 2 G;

(6.1.10)

see [17] for related results. This will be important for the stability analysis in Section 6.4.

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6.1.3 Wave solutions of PDEs Two important classes of semi-linear evolution equations, which fit into the above setting and to which our results apply, are the following: ut D Auxx C f .u; ux /; u.x; t/ 2 Rm ; x 2 R; t > 0; u.
; 0/ D u0 ; (6.1.11) ut D Au C f .u/; u.x; t/ 2 Rm ; x 2 Rd ; t > 0; u.
; 0/ D u0 :

(6.1.12)

In both cases, A 2 Rm;m is assumed to have spectrum  .A/ with Re. .A// > 0. Note that Re. .A// > 0 leads to parabolic systems while  .A/  iR occurs for Hamiltonian PDEs. Intermediate cases with  .A/  .f0g [ fRe z > 0g/ generally belong to hyperbolic or parabolic-hyperbolic mixed systems. The nonlinearities f W R2m ! Rm in (6.1.11) resp. f W Rm ! Rm in (6.1.12) are assumed to be sufficiently smooth and to satisfy f .0; 0/ D 0 resp. f .0/ D 0. For (6.1.11), the Lie group is .G; ı/ D .R; C/ acting on X D L2 .R; Rm / by the shift Œa. /u.x/ D u.x /, x 2 R, u 2 X . With F .u/ D Auxx C f .u; ux / for u 2 Z D H 2 .R; Rm /, equivariance is easily verified and F .u/ 2 X follows from the Sobolev embedding H 1 .R; Rm /  L1 .R; Rm / and f .0; 0/ D 0. For the derivative, we find d Œa.1/v D vx ;  2 g D R; v 2 H 1 .R; Rm /: Relative equilibria then turn out to be travelling waves, u? .x; t/ D v? .x

? t/;

x 2 R; t > 0;

(6.1.13)

where the pair .v? ; ? / solves the second order system from (6.1.7) 0 D Av?; C ? v?; C f .v? ; v?; /;

v./ 2 Rm ;  2 R:

In fact, our simplified abstract approach only covers pulse solutions (defined by v? ./; v?; ./ ! 0 as  ! ˙1), whereas fronts need the setting of manifolds; see Remark 6.1.1. It is interesting to note that our numerical approach also applies to travelling waves in evolution equations with nonlocal diffusion. Their existence and their qualitative properties are analysed in Chapter 4.6. In the multi-dimensional case (6.1.12), the phase space is X D L2 .Rd ; Rm /, and we aim at equivariance w.r.t. the special Euclidean group G D SE.d / D SO.d /ËRd . It is convenient to represent SE.d / in GL.Rd C1 / as    Q b SE.d / D W Q 2 Rd;d ; Q> Q D Id ; det.Q/ D 1; b 2 Rd ; (6.1.14) 0 1 where the group operation is matrix multiplication. We represent the Lie algebra se.d / D so.d /  Rd accordingly,    S c (6.1.15) se.d / D W S 2 Rd;d ; S > D S; c 2 Rd : 0 0 The action on functions u 2 X is defined by Œa. /u.x/ D u.Q> .x

b//;

x 2 Rd ; D



 Q b 2 SE.d /: 0 1

133

Computation and stability of waves 1 The derivative exists for functions u 2 HEucl .Rd ; Rm /, where n o k HEucl .Rd ; Rm / D u 2 H k .Rd ; Rm / W LS u 2 L2 .Rd ; Rm / 8S 2 so.d /

for k > 1 and

LS u.x/ WD ux .x/S x D

d X

Dj u.x/Sj;k xk ;

j;kD1

x 2 Rd :

The derivative of the group action is then given by
d Œc.1/v .x/ D

vx .x/.S x C c/;

d

x2R ;

 D



S 0

 c 2 se.d /: 0

Note that the first order operator LS has unbounded coefficients and that the norm in k HEucl is given by kuk2H k

Eucl

D kuk2H k C supfkLS uk2L2 W S 2 so.d /; jS j D 1g:

2 Setting Z D HEucl .Rd ; Rm / one finds F .u/ D Au C f .u/ 2 X for u 2 Z in dimension d D 2, since H 2 .R2 ; Rm /  L1 .R2 ; Rm / by Sobolev embedding. But for d > 3 one has to impose growth conditions on f to ensure this. Equivariance follows from the equivariance of the Laplacian under Euclidean transformations. Special types of relative equilibria are waves rotating about a centre x? 2 Rd :

u? .x; t/ D v? .exp. tS? /.x

When substituting  D exp. tS? /.x

x? //;

v? 2 Z; S? 2 so.d /:

(6.1.16)

x? / the system (6.1.7) reads

0 D Av? C v?; S?  C f .v? /;

 2 Rd :

Several examples of travelling and rotating waves will be dealt with in Section 6.3.

6.2 The freezing method 6.2.1 The abstract approach The idea of the freezing method, set out in [51, 13], is to separate the strong solutions of the Cauchy problem (6.1.4) into a motion on the group G and on a reduced phase space, just as for the relative equilibria in Definition 6.1.4, u.t/ D a. .t//v.t/; t > 0: (6.2.1) 1 Let 2 C .Œ0; T /; G/; .0/ D 1, let u be a strong solution of (6.1.4) and define .t/ WD .dL .t / .1// 1 t .t/ 2 g in the Lie algebra g of G. Then, ; v solve the system vt .t/ D F .v.t//

d Œa.1/v.t/.t/;

t .t/ D dL .t / .1/.t/;

v.0/ D u0 ;

.0/ D 1:

(6.2.2) (6.2.3)

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Conversely, one can show that a strong solution u 2 C 1 .Œ0; T /; X / \ C.Œ0; T /; Z/,  2 C.Œ0; T /; g/, 2 C 1 .Œ0; T /; G/ of (6.2.2),(6.2.3) leads to a strong solution of (6.1.4) via (6.2.1). According to Theorem 6.1.5, a relative equilibrium v? ; ? of (6.1.1) is a steady state of the first equation, (6.2.2). Following [51], we call Equation (6.2.3) the reconstruction equation. Due to the extra variables 2 G resp.  2 g, the system (6.2.2), (6.2.3) is not yet well posed, but needs n D dim.G/ additional algebraic constraints (called phase conditions) which we write as .v; / D 0:

(6.2.4)

Here, W X  g ! g? (the dual of g) is a smooth map typically derived as a necessary condition from a minimisation principle. For example, if .X; h
;
i/ is a Hilbert space, one can require the distance infg2G kv a.g/vk O to the group orbit of a template function vO 2 X (such as vO D u0 ) to be minimal at g D 1. For vO 2 Z, this leads to the fixed phase condition fix .v; /

D hd Œa.1/v; O v

vi O D 0;

8 2 g:

(6.2.5)

An alternative is to minimise kvt k2 D kF .v/ d Œa.1/vk2 with respect to  at each time instance, resulting in the orthogonality condition orth .v; /

D hd Œa.1/v; F .v/

d Œa.1/v i D 0;

8 2 g:

(6.2.6)

This condition requires the group orbit of v.t/ to be tangent to its time derivative. Altogether, Equations (6.2.2) and (6.2.4) constitute a partial differential algebraic equation (PDAE) for the functions v and . The reconstruction equation (6.2.3) decouples from the PDAE and may be solved in a post-processing step. Condition (6.2.6) has a unique solution  if d Œa.1/v W g ! X is one to one and then leads to a PDAE of (differentiation) index 1. Condition (6.2.5) leads to an index 2 problem, but can be reduced to index 1 by differentiating with respect to t and then inserting (6.2.2). 6.2.2 Application to evolution equations In this section, we take a closer look at the PDAEs that arise from the freezing method when applied to the two equations (6.1.11) and (6.1.12). In Section 6.3, we will provide a series of numerical examples and also discuss the influence of both spatial and temporal discretisation errors. In the following, we restrict to the fixed phase condition (6.2.5) which is particularly well-suited near a relative equilibrium and which admits rather general stability results; see Section 6.4. On the other hand, the orthogonal phase condition needs no a priori information and hence can be applied far away from any relative equilibrium. However, its stability properties are questionable and have only been investigated in a special case [14]. For the one-dimensional system (6.1.11) with shift equivariance, the freezing ansatz simply reads u.x; t/ D v.x

.t/; t/;

x 2 R; t > 0;

.t/ D t .t/;

(6.2.7)

Computation and stability of waves

135

and the corresponding PDAE is given by (cf. [57]) vt D Av C v C f .v; v /; v.
; 0/ D u0 ; 0 D hvO  ; v

t D ;

vi O L2 .R;Rm / ;

(6.2.8)

.0/ D 0;

for the unknown quantities .v; ; /. For initial data u0 close to a wave, we expect v.
; t/ ! v? , .t/ ! ? as t ! 1. Travelling waves in parabolic systems and their stability are analysed in [34, 53, 60, 57], and numerical applications of the freezing method for this case appear in [9]. Next, consider the parabolic system (6.1.12) in several space dimensions. With the special Euclidean group (6.1.14) and its Lie algebra (6.1.15), the freezing system (6.2.2),(6.2.3) takes the form vt D Av C v .S  C c/ C f .v/; v.
; t0 / D u0 ;

0 D hj vO i

Q b Q b D 0 1 t 0 1

i vO  ; v j
S c ; 0 0

vi O L2 ;

0 D hvO  ; v vi O L2 ; l   Q.t0 / b.t0 / D 1d C1 ; 0 1

(6.2.9)




b , and indices 1 6 l 6 d , for the unknown quantities v;  D S0 c0 ; D Q 0 1 1 6 i < j 6 d . Since S.t/ is skew-symmetric, it is sufficient to work with Sij , i D 1; : : : ; d 1, j D i C 1; : : : ; d and c 2 Rd when solving the reconstruction equation. Numerical methods for differential equations on Lie groups may be found are close to a stable rotating wave (6.1.16), we expect in [33]. If the initial data

v.
; t/ ! v? and S0 0c D .t/ ! ? D S0? c0? 2 se.d / as t ! 1. Rotating waves in parabolic systems are treated in [24, 25], their nonlinear stability (for d D 2) in [6], and numerical examples in [41]. Essential steps for extending nonlinear stability to higher space dimensions are taken in [7, 8], which is based on previous work [41, 42, 43, 44]. 6.2.3 Dynamic decomposition of multi-waves Consider a simplified parabolic system (6.1.11) in one space dimension ut D Auxx C f .u/; u.
; 0/ D u0 ;

(6.2.10)

under the assumptions of Section 6.1.3. Suppose this system admits several travelling waves .v?;j ; ?;j /, j D 1; : : : ; N , with different speeds ?;j and limit behaviour lim!˙1 v?;j ./ D vj˙ for j D 1; : : : ; N . If the limits fit together, i.e. if vjC D vj C1 ;

j D 1; : : : ; N

1;

then one often observes N -waves (or multi-waves) of (6.2.10) which look like concatenations of the waves v?;j .x ?;j t/; j D 1; : : : ; N , see for example Figure 6.3.6(c) for such a concatenation of two fronts from Example 6.3.1 travelling at

136

W.-J. Beyn, D. Otten

different speeds to the left. Strong interaction occurs when two or several fronts move towards each other, while all other cases are called weak interactions. Many more interaction phenomena of this type may be found in [61] and the references therein. In [9, 55], we extend the freezing method in order to handle such interactions. More precisely, we generalise (6.2.7) to u.x; t/ D

N X

vj .x

j .t/; t/;

(6.2.11)

j D1

where the values of j W R ! R denote the time-dependent position of the j -th profile vj W R ! Rm . In order for the linear superposition to make sense, we think of vj ./ limiting to 0 as  ! 1 (except for j D 1), i.e. we expect vj to satisfy ( 0; j D 1; ˙ lim vj .; t/ D vj wj ; wj D vj ; j > 2: !˙1 Figures 6.3.6(a), 6.3.6(b) show the shifted profiles v1 ; v2 for the 2-wave in Figure 6.3.6(c). The main idea is to combine (6.2.11) with a dynamic partition of unity '.x j .t//
Qj 1 .t/; : : : ; N .t/; x D PN ;

k .t// kD1 '.x

j D 1; : : : ; N;

where ' 2 C 1 .R; .0; 1/ is a mollifier function, for example '.x/ D sech.ˇx/ for some ˇ > 0. Using (6.2.11) in (6.2.10) and abbreviating vk .?/ D vk .
k .t/; t/, one finds N h i X vj;t .?/ vj; .?/ j;t D ut D Auxx C f .u/ j D1

D

N h X

j D1

Avj; .?/ C f .vj .?/ C wj /

N n X  C Qj . ;
/ f vk .?/ kD1

N X

kD1

oi f .vk .?/ C wk / :

  Equating the terms inside brackets
on both sides, substituting  D x j .t/ and adding phase conditions, and initial conditions leads to the following decompose and freeze system (see [9, 55]) vj;t .; t/ D Avj; .; t/ C vj; .; t/j .t/ C f .vj .; t// C PN

'./

kD1

 0 D vj .
; t/

j;t D j ,

'.kj /

N h X  f vk .kj ; t/

vO j ; vO j;



kD1

L2

,

vj .
; 0/ D

N X

kD1

i f .vk .kj ; t/ C wk / ;

vj0 ;

j .0/ D j0 :

(6.2.12)

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Computation and stability of waves

This is an N -dimensional PDAE to be solved for .vj ; j ; j /, j D 1; : : : ; N , where kj D 

k .t/ C j .t/;

' 2 C 1 .R; .0; 1/;

u0 D

N X

vj0 .


j0 /:

j D1

A particular difficulty of this system is that the right-hand side contains non-local terms vk .kj ; t/ that need special treatment when discretised on bounded intervals Œx ; xC ; see Section 6.3.5. We also mention that the stability of this approach for weak interaction is analysed in [9, 55] and that a generalisation of the decompose and freeze method to the abstract framework of Section 6.2.1 is proposed and applied in [9, 12, 41]; compare Section 6.3.5.

6.3 Parabolic, hyperbolic, and Hamiltonian systems 6.3.1 Travelling and rotating waves in parabolic systems Our first numerical example deals with a scalar parabolic equation of type (6.2.10) related to the classical Nagumo equation with a cubic nonlinearity. Example 6.3.1 (Quintic Nagumo equation). In the scalar case m D 1 with A D 1, f .u; ux / D

5 Y

i D1

.u

bi /; bi 2 R;

0 D b1 < b2 < b3 < b4 < b5 D 1: (6.3.1)

Equation (6.3.1) is called the quintic Nagumo equation (QNE), [9]. Figure 6.3.1(a) shows the time evolution for a travelling front of the QNE for , spatial domain Œ 100; 100, initial data parameters b2 D 25 , b3 D 21 , b4 D 17 20 u0 .x/ D tanh.x/C1 and time range Œ0; 1500. At time t  1300, the front leaves the 2 computational domain. Figures 6.3.1(b) and 6.3.1(d) show the time evolution of the front profile and the velocity obtained by solving the freezing system (6.2.8) with homogeneous Neumann boundary conditions, f from (6.3.1), parameters bj and spatial

(a)

(b)

(c)

(d)

Figure 6.3.1. QN-front: space-time of u (a), of v (b), profile v as a function of space (c), velocity  as a function of time (d).

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W.-J. Beyn, D. Otten

domain as before, initial data v0 D u0 , and the template vO D u0 on the time range Œ0; 3000. The front quickly stabilises at the shape v? shown in Figure 6.3.1(c), and the velocity quickly reaches its asymptotic value ?  0:07 as shown in 6.3.1(d). For the numerical solution of (6.1.11) resp. (6.2.8) we used an FEM space discretisation with Lagrange C 0 -elements and maximal element size 4x D 0:3, the BDF method for time discretisation with maximum order 2, time step-size 4t D 0:3, relative tolerances 10 2 and 10 3 , and absolute tolerances 10 3 and 10 4 , combined with Newton’s method for nonlinear equations. The next example is a two-dimensional system of type (6.1.12) obtained by writing a scalar complex equation as a real system. Example 6.3.2 (Quintic-cubic Ginzburg–Landau equation). Consider the PDE zt D ˛z C g.z/; z D z.x; t/ 2 C;

g.z/ D .ı C ˇjzj2 C jzj4 /z;

˛; ˇ; 2 C; ı 2 R;

known as the quintic-cubic Ginzburg–Landau equation (short: QCGL). Figure 6.3.2(a) shows the time evolution for the real part of a spinning soliton (cross-section at x2 D 0) of the QCGL for parameters ˛ D 12 C 21 i, ˇ D 52 C i, 1 i, ı D 21 , spatial domain B20 .0/ D fx 2 R2 W jxj 6 20g, initial data

D 1 10
1 u0 .x/ D .Rez0 ; Imz0 /> for z0 .x/ D x5 exp 49 jxj2 and time range Œ0; 150. At time t D 150, we take the solution data and switch on the freezing system (6.2.9). Figures 6.3.2(a) and 6.3.2(b) show the time evolution of the real part of the soliton profile and the velocities obtained by solving (6.2.9) with homogeneous Neumann boundary conditions, parameters ˛; ˇ; ; ı and spatial domain as before, initial data v0 D u.
; 150/, template function vO D u.
; 150/ and time range Œ150; 400. Approx-
imations of the real part of the soliton profile v? and the velocities ? D S0? c0?

0 1:027 and c  0:003 , are shown in Figures 6.3.3(a) and with S?  ? 1:027 0 0:017 6.3.2(b). For the numerical solution of (6.1.12) resp. (6.2.9) we used FEM for space discretisation with Lagrange C 0 -elements and maximal element size x D 0:25, the BDF method for time discretisation with maximum order 2, time step size t D 0:1

(a)

(b)

Figure 6.3.2. Spinning soliton of QCGL: space-time along x2 D 0 (a), translational velocities .1/ .t/, .2/ .t/ (two lines at the bottom) and angular velocity .3/ .t/ (topline) (b).

139

Computation and stability of waves

(a)

(b)

(c)

(d)

Figure 6.3.3. Rotating waves: spinning soliton for d D 2 (a) and d D 3 (b), spiral wave for d D 2 (c), untwisted scroll wave for d D 3 (d).

resp. t D 0:2 , relative tolerance 10 4 resp. 10 2 , and absolute tolerance 10 resp. 10 7, and Newton’s method for nonlinear systems.

5

The spinning solitons of the QCGL from Example 6.3.2 are a special kind of a localised rotating wave for d D 2; see Figure 6.3.3(a). Their extension to d D 3 dimensions is displayed in Figure 6.3.3(b), and non-localised rotating waves, such as spiral waves, are shown in Figure 6.3.3(c). Finally, we show a so-called scroll wave in Figure 6.3.3(c). These types of waves occur in various applications, for instance in the QCGL [19, 40], the -!-system [37], the Barkley model [2], and the FitzHugh–Nagumo system [26]. Their treatment via the freezing method is discussed in [41, 9, 6]. 6.3.2 Hyperbolic systems The following hyperbolic system in one space dimension may be viewed as a special case of (6.1.11) with A D 0: ut D Eux C f .u/;

u.
; 0/ D u0 :

(6.3.2)

For (6.3.2) to be well-posed, we assume E 2 Rm;m to be real diagonalisable and f W Rm ! Rm to be sufficiently smooth. As in Section 6.1.3, travelling waves of (6.3.2) are solutions of the form (6.1.13), the underlying Lie group .G; ı/ is the additive group .R; C/ acting on functions via translations. The freezing system (6.2.8) for pursuing profiles and velocities, for the unknown quantities .v; ; /, now reads as follows: vt D Ev C v C f .v/; v.
; 0/ D v0 ; 0 D hvO  ; v

t D ;

vi O L2 .R;Rm / ;

.0/ D 0:

The main difference to the parabolic case (6.2.8) is due to the fact that the unknown function .t/ of this PDAE now appears in the principal part of the spatial operator. This creates serious difficulties, both for the numerical and for the theoretical analysis. These have been successfully treated in [45, 46], and a series of numerical examples appears in [45, 46, 9]. Moreover, with a slightly generalised notion of equivariance

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(see [51, 45]), the freezing approach has found interesting applications to detecting similarity solutions in Burgers’ equation; see [51, 9] for the one-dimensional and [49, 50] for the multi-dimensional case. Finally, we refer to [47, 48] where the stability of travelling waves and the freezing approach is analysed for mixed parabolic-hyperbolic systems of the partitioned form       A11 0 g.u/ f .u/ ut D uxx C C 1 ; u.
; 0/ D u0 ; (6.3.3) 0 0 B22 u2 x f2 .u/


with u D uu12 , a positive diagonalisable matrix A11 and a real diagonalisable matrix B22 . This covers the famous Hodgkin–Huxley model for propagation of pulses in nerve axons; cf. [9, Ch.3.1].

6.3.3 Nonlinear wave equations Another area of application are systems of nonlinear wave equations in one space dimension M ut t D Auxx C fQ.u; ux ; ut /;

u.
; 0/ D u0 ;

ut .
; 0/ D v0 ;

(6.3.4)

where M 2 Rm;m is invertible, A 2 Rm;m , fQ W R3m ! Rm is smooth and u0 , v0 W R ! Rm denote the initial data. Further, we assume M 1 A to be positive diagonalisable, which implies local well-posedness of (6.3.4). In case m D 1, travelling waves (6.1.13) for Equation (6.3.4) and their global stability have been treated in [29, 28]. The freezing ansatz (6.2.7) now requires to solve the following second order PDAE (cf. [10, 11]), M vt t D .A

21 M /v C 21 M vt C 2 M v C fQ.v; v ; vt

0 D hvO  ; v

1;t D 2 ;

v.
; 0/ D u0 ;

1 v /

vi O L2 .R;Rm / ;

t D 1 ;

vt .
; 0/ D v0 C 01 u0; ;

1 .0/ D 01 ;

(6.3.5)

.0/ D 0

for the unknown quantities .v; 1 ; 2 ; /. Travelling waves .v? ; ? / appear as steady states of (6.3.5) (with 1 D ? ; 2 D 0) and satisfy the equation 2? M /v?; C f .v? ; v?; ; ? v?; /:

0 D .A

Differentiating the algebraic constraint in (6.3.5) w.r.t. time at t D 0 and inserting the initial conditions leads to a first consistency condition for 01 , 01 hu0; ; vO  iL2 C hv0 ; vO  iL2 D 0;

(6.3.6)

and differentiating twice at t D 0 gives a consistency condition for 2 .0/ D 02 : 0 D h.M

CM

1

A C .01 /2 Im /u0; C 201 v0; 1

f .u0 ; u0; ; v0 /; vO  iL2 C 02 hu0; ; vO  iL2 :

(6.3.7)

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Computation and stability of waves

(a)

(b)

(c)

(d)

Figure 6.3.4. QNWE-front: space-time of u (a), v (b), profile v as a function of space (c), velocity 1 .t/ (top) and acceleration 2 .t/ (bottom) as functions of time (d).

The local stability of the PDAE system (6.3.5) is analysed in [10], while a generalisation to several space dimensions and a numerical example appear in [11]. It is interesting to note that the system (6.3.4) may be written as a first order system (6.3.2) of dimension 3m. Taking a positive square root N D .M 1 A/1=2 and introducing the variables U1 D u, U2 D ut C N ux , U3 D ut N ux C cu (with c 2 R arbitrary) leads to a system (6.3.2) with 1 0 1 0 cU1 C U3 N 0 0 0 A ; f .U / D @ g.U / A E D @0 N g.U / C cU2 0 0 N (6.3.8) g.U / D M

1

fQ.U1 ; 12 N

1

.U2

U3 C cU1 /;

1 .U2 2

C U3

cU1 //:

Though we prefer to solve numerically the second order system (6.3.5), the first order system (with a suitable choice of the constant c) is useful for applying the stability results from [46]; see [10] and Section 6.4. Example 6.3.3 (Quintic Nagumo wave equation). Taking the quintic nonlinearity f D fQ from (6.3.1) with the wave equation (6.3.4), we obtain the quintic Nagumo wave equation (QNWE); see [11]. Figure 6.3.4(a) shows the time evolution for a travelling front of the QNWE for 17 , spatial domain Œ 50; 50, initial parameters M D 21 , b2 D 25 , b3 D 12 , b4 D 20 1 x data u0 .x/ D 2 .1 C tanh. 2 //, v0 .x/ D 0 and time range Œ0; 800. At time t  600, the front leaves our computational domain. Figures 6.3.4(b) and 6.3.4(d) show the time evolution of the front profile and the velocity obtained by solving (6.3.5) with homogeneous Neumann boundary conditions, parameters M , bj , spatial domain and initial data as before, template vO D u0 and time range Œ0; 1000. An approximation of the front profile v? (with v D 0, vC D 1) and the approach towards the limit velocity ?  0:07 are shown in Figures 6.3.4(c) and 6.3.4(d). The data for the numerical solution of (6.3.4) resp. (6.3.5) are the same as in Example 6.3.1, except for the step sizes x D 0:1 and t D 0:2. 6.3.4 Hamiltonian PDEs So far, we mainly considered waves in dissipative PDEs which are detected during simulation via the freezing method due to their asymptotic

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Figure 6.3.5. Solitary wave of the NLS with spike-like initial perturbation: direct numerical simulation (left) vs. solution of the freezing system (right).

stability. This changes fundamentally for PDEs with Hamiltonian structure which typically allow several or even infinitely many conserved quantities. They fit into the general class of evolution problems described in Section 6.1.1 but require quite different techniques for establishing existence and uniqueness of wave solutions [23] as well as their stability ([30, 31]). As a model example, consider the cubic nonlinear Schrödinger equation (NLS, see [16, 23, 36, 56]), iut D

uxx

juj2 u;

u.
; 0/ D u0 ;

(6.3.9)

which may be subsumed under (6.1.1) with X D H 1 .RI C/ , Z D H 3 .RI C/. Equivariance holds with respect to the action a. /v D e

i 1

v.


2 /;

D . 1 ; 2 / 2 G

of the two-dimensional Lie group G D S 1  R. With  D .1 ; 2 / 2 R2 , the freezing system (6.2.2) is given by ivt D

v

jvj2 v

1 v C i2 v ;

v.
; 0/ D u0 ;

(6.3.10)

and the fixed phase condition (6.2.5) with some vO 2 X reads ˝ ˛ ˝ ˛ 0 D iv; O v 0 D vO  ; v 0 ; (6.3.11) R where hu; vi0 D Re R u.x/v.x/dx. N There is a well-known two-parameter family of solitary wave solutions given by u? .x; t/ D ei1 t v? .x 2 tI 1 ; 2 /; p ! 2ei2 =2 v? .I 1 ; 2 / D ; ! 2 D 1 cosh.!/

22 ; 4

(6.3.12)

see for example [22, Ch.II.3]. For the following numerical computations, we choose parameter values 2 D 0:3, ! D 1. Discretisation in time is done via a split-step Fourier method with step size t D 10 3 . The spatial grid is formed by 2K D 256

143

Computation and stability of waves

  28:56. A equidistant points on the interval Œx ; xC  with xC D x D 0:11 spike-like perturbation at x D 11 is added to the initial data. Figure 6.3.5 shows the solution for both the original system and the freezing system. Clearly, the freezing system prevents the wave from rotating and travelling, while the interference patterns caused by the initial perturbation are essentially preserved. A theoretical result supporting these observations will be described in Section 6.4.4, and a detailed presentation can be found in the PhD thesis [20].

6.3.5 Multi-waves For a numerical experiment of decomposing and freezing multiwaves, we take up Example 6.3.1 of the QNE. Example 6.3.4 (QNE). Figure 6.3.6(c) shows the time evolution of the superposiP tion 2j D1 vj .x j .t/; t/, which can be considered as an approximation of a travelling 2-front u of the original QNE (6.2.10) with f from (6.3.1). The quantities .vj ; j / are the solutions of (6.2.12) and provide approximations of .v?;j ; ?;j /. Figure 6.3.6(c) shows that the lower front v1 (travelling at speed 1 ) is faster than the upper front v2 (travelling at speed 2 ), i.e., we may expect ?;1 < ?;2 < 0. Figure 6.3.6(a) and 6.3.6(b) (resp. 6.3.6(d)) show the time evolution of the single front profiles v1 and v2 (resp. the velocities 1 ; 2 ) obtained by solving (6.2.12) with 1 homogeneous Neumann boundary conditions, f from (6.3.1), parameters b2 D 32 , 2 73 b3 D 5 , b4 D 100 , spatial domain Œ 200; N D 2, initial data   200, multi-waves
0 1 v2 v2   0 v1 ./ D 2 tanh. 5 / C 1 , v2 ./ D 2 tanh. 5 / C 1 with v2 D b3 , 10 D 20 D

 0, templates vO j D vj0 , bump function './ D sech. 20 / and time range Œ0; 3000. Approximations of the single front profiles v?;j (with v1 D 0, v1C D a4 D v2 , v2C D 1) and velocities ?;1  0:159, ?;1  0:021 are shown in Figure 6.3.6(a), 6.3.6(b) and 6.3.6(d). For the numerical solution of (6.2.12), we used the FEM for space discretisation with Lagrange C 0 -elements and maximal element size x D 0:4, the BDF method for time discretisation with maximum order 2, intermediate time steps, time step-size t D 0:8, and the Newton method for solving nonlinear equations.

(a)

(b)

(c)

(d)

Figure 6.3.6. 2-front of QNE: profile v1 (a), profile v2 (b), superposition (c), and velocities 1 .t/ (top) and 2 .t/ (bottom) (d).

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W.-J. Beyn, D. Otten

(a)

(b)

(c)

(d)

Figure 6.3.7. Multiwaves: 3-front of QNE (a), pulse front of QCGL (b), 3-soliton of QCGL (c) and position of centres (d).

Travelling 2-fronts as in Example 6.3.4 are a special class of multi-waves. The decompose and freeze method (DFM) easily extends to larger numbers of fronts, e.g. 3-fronts (see Figure 6.3.7(a)), and can be used to analyse wave interaction processes, for example repulsion and collision of waves. Moreover, the DFM extends to general multi-structures, for instance to a superposition of a phase-rotating pulse and a travelling phase-rotating front (see Figure 6.3.7(b)), and to higher space dimensions, see the three spinning multi-solitons in Figure 6.3.7(c) with the interactions represented by the traces of their centres in Figure 6.3.7(d). For the DFM, we refer to [9, 55, 12]. Extensions of the DFM to rotating multi-solitons including numerical experiments can be found in [9, 41].

6.4 Stability of relative equilibria The issue of stability is fundamental to all wave phenomena considered here. Since relative equilibria come in families due to the group action (see Section 6.1.2), the classical notions of (Lyapunov)-stability and asymptotic (Lyapunov)-stability are replaced by the notions of orbital stability and stability with asymptotic phase. 6.4.1 Notions of stability and the co-moving frame equation In order to have some flexibility for the application to PDEs, the following definition uses two norms k:k1 and k:k2 which need not agree with the norms in the Banach spaces Z and X . Moreover, depending on the type of PDE, a solution concept different from the strong solution in Definition 6.1.2 may be necessary. Definition 6.4.1. A relative equilibrium .v? ; ? / of (6.1.4) is called orbitally stable with respect to norms k:k1 and k:k2 if, for any " > 0, there exists ı > 0 such that the Cauchy problem (6.1.4) has a unique strong solution u for u0 2 Z with ku0 v? k1 6 ı, and the solution satisfies inf 2G ku.t/

a. /v?k2 6 ";

8t > 0:

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Computation and stability of waves

It is called stable with asymptotic phase if for any " > 0 there exists ı > 0 such that for all initial values u0 2 Z with ku0 v? k1 6 ı the Cauchy problem (6.1.4) has a unique strong solution u, and, for some 1 D 1 .u0 / 2 G, the solution satisfies the two conditions ( 6 "; 8t > 0; ku.t/ a. 1 ı ? .t//v? k2 ! 0; as t ! 1: Stability in general requires to investigate the solution of (6.1.4) for initial data u0 D v? C v0 where v0 is a small perturbation of the wave profile. For this, we transform into a co-moving frame via u.t/ D a. ? .t//v.t/;

t > 0;

which by contrast to the general ansatz (6.2.1) assumes the group orbit ? to be known. One obtains a special case of (6.2.2), the co-moving frame equation vt .t/ D F .v.t//

d Œa.1/v.t/?;

v.0/ D v? C v0 :

(6.4.1)

Linearising about v? in a formal sense leads to consider the linear operator

Lw D DF .v? /w

d Œa.1/w? ;

w 2 Z:

(6.4.2)

If the topology on Z is strong enough, then DF is in fact the Fréchet derivative of F , and this point of view is sufficient for our applications to semi-linear PDEs in Section 6.3. The general procedure then is to deduce nonlinear stability in the sense of Definition 6.4.1 from spectral properties of the operator L. One says that the principle of linearised stability holds if such a conclusion is valid. A minimal requirement is that the spectrum lies in the left half-plane, i.e.  . L/  C

D f 2 C W Re./ 6 0g:

However, the special properties of the PDEs considered here usually require more: (P1) determine eigenvalues on the imaginary axis caused by the group action, (P2) analyse the essential spectrum ess .L/   .L/ which arises from the loss of compactness for differential operators on unbounded domains, (P3) compute isolated eigenvalues of the point spectrum pt .L/   .L/ different from those in (P1), either by a theoretical or by a numerical tool. Let us finally note that a proof of nonlinear stability becomes particularly delicate if there is no spectral gap between the eigenvalues from (P1) and the remaining spectrum. This occurs if the spectrum touches the imaginary axis (wave trains, spiral waves, see [21, 54]) or lies on the imaginary axis (Hamiltonian case).

146

W.-J. Beyn, D. Otten

6.4.2 Spectral structures Hardly anything can be said about problems (P2), (P3) above within the abstract framework of Equations (6.1.4) and (6.4.1). However, the eigenvalues caused by symmetry have some general structure. For this purpose, recall the Lie bracket Œ
;
 W g  g ! g (see e.g. [27, Ch. 8]) which turns g D T1 G into a Lie algebra. The abstract definition of the bracket is in terms of the adjoint representation Ad.g/ W g ! g of g 2 G given by Ad.g/ D dh Œg ı h ı g

1

jhD1 ;

Œ;  D dg ŒAd.g/jgD1./;

 2 g;

;  2 g:

It is reasonable to look for eigenfunctions of L of type w D d Œa.1/v?;  2 gC , where gC denotes the complexified Lie algebra and d Œa.1/v?  denotes the complexified operator. Theorem 6.4.2. Let v? 2 Z, ? .t/ D exp.t? /, t > 0 be a relative equilibrium of (6.1.1) such that d Œa.1/v? maps g into Z. Then, w D d Œa.1/v?,  2 gC solves the (complexified) eigenvalue problem .I

L/w D 0

(6.4.3)

if and only if  satisfies d Œa.1/v?.

Œ; ? / D 0:

In particular, if the stabiliser H.v? / is trivial , see (6.1.9), then independent eigenvectors j j D 1; : : : ; k of Œ
; ?  W g ! g lead to independent eigenfunctions wj D d Œa.1/v?j of (6.4.3), j D 1; : : : ; k. Proof. For the family of relative equilibria (6.1.10), we have by the chain rule  d  F .a. .g; t//a.g/v?/ D a. .g; t//.a.g/v?/ dt   D d a.g ı ? .t/ ı g 1 /.a.g/v?/ dh .g ı h ı g

1

/jhD ? .t / ?0 .t/;

which upon evaluation at t D 0 yields

F .a.g/v?/ D d Œa.1/.a.g/v?/ Ad.g/? : We differentiate with respect to g 2 G and apply this to  2 Tg G, DF .a.g/v?/d Œa.g/v? D d Œa.1/.d Œa.g/v?/Ad.g/?

C d Œa.1/.a.g/v?/dg ŒAd.g/? ;

(6.4.4) (6.4.5)

which upon evaluation at g D 1,  2 g gives DF .v? /d Œa.1/v? D d Œa.1/.d Œa.1/v?/? C d Œa.1/v?Œ; ? :

147

Computation and stability of waves

Therefore, the eigenvalue problem (6.4.3) with w D d Œa.1/v? is equivalent to 0 D w

DF .v? /w C d Œa.1/w?

D d Œa.1/v?

D d Œa.1/v?.

(6.4.6)

DF .v? /d Œa.1/v? C d Œa.1/.d Œa.1/v?/? (6.4.7) Œ; ? /;

(6.4.8) 

which proves our assertion.

Theorem 6.4.2 shows that the geometric multiplicity of the eigenvalue  D 0 is at least the dimension of the centraliser of ? given by g0 .? / WD f 2 g W Œ; ?  D 0g: If the group G is represented as a subgroup of the matrix group GL.RN / for some N 2 N, the Lie bracket agrees with the commutator. It is not difficult to see that the spectrum of the linear map  7! Œ; ?  always satisfies  .Œ
; ? /  f1

2 W 1 ; 2 2  .? /g D  .? /  .? /: (6.4.9)
The special elements ? D S0? c0? from se.d / (see (6.1.15)) occur with rotating waves (6.1.16) and satisfy  .? /  iR as well as  .? / D  .? /. Let 1 ; : : : ; d be the eigenvalues of the skew-symmetric matrix S? . Then, one finds  .Œ
; ? / D f 2 C W  2  .S? / or  D j C k for some j < kg;

(6.4.10)

see [15, 8] for the computation of eigenvalues and corresponding eigenvectors. 6.4.3 Stability with asymptotic phase We discuss sufficient conditions for the stability with asymptotic phase in case of our two model equations, (6.1.11) and (6.1.12) (see [9, 34, 35, 53, 58]). For a travelling wave .v? ; ? /, the linearised operator L from (6.4.2) reads

Lw D Aw C .? Im C D2 f .v? ; v?; //w C D1 f .v? ; v?; /w D Aw C B.
/w C C.
/w:

(6.4.11)

We consider the case of a front lim v? ./ D v˙ ;

!˙1

lim v?; ./ D 0;

!˙1

(6.4.12)

which is covered by our abstract approach only in case v˙ D 0; see Remark 6.1.1. However, note that L W H 2 .R; Rm / ! L2 .R; Rm / is well defined in the general case (6.4.12), and that it has the eigenvalue 0 with eigenfunction w D v?; , compare Theorem 6.4.2 and (6.4.9) with 0 2  .? /. The essential spectrum of L is determined by the constant coefficient operators

L˙ D A@2 C B˙ @ C C˙ ;

C˙ D D1 f .v˙ ; 0/;

B˙ D ? Im C D2 f .v˙ ; 0/: (6.4.13)

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W.-J. Beyn, D. Otten

Bounded solutions of .I L˙ /w D 0 are of the form w./ D ei! , ! 2 R, which leads to the definition of the dispersion set ˚ disp.L/ D  2 C W  2  . ! 2 A C i!B˙ C C˙ / for some sign ˙ and ! 2 R : (6.4.14) By Weyl’s theorem on invariance of the essential spectrum under relatively compact perturbations (see [34, 35]), one finds disp.L/  ess .L/ and, moreover, that the connected component U of C n disp.L/ containing a positive real semi-axis satisfies U  ..L/ [ pt .L//. Therefore, the issues (P2) and (P3) from Section 6.4.1 are resolved by requiring, for some ˇ > 0, the spectral conditions Re disp.L/ 6   Re pt .L/ n f0g 6

ˇR

as R ! 1 for j˛j 6 2;

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Computation and stability of waves

and stability of the linearisation at infinity in the sense of Re hDf .0/w; wi 6

ˇjwj2

for all w 2 Cm

and some ˇ > 0:

(6.4.18)

This assumption guarantees that the essential spectrum of the linear operator L W 2 HEucl .R2 ; Rm / ! L2 .R2 ; Rm / defined by

Lv D v C LS? v C Df .v? /v

(6.4.19)

lies in the open left half-plane. As for the abstract result (6.4.10), one finds that L has eigenvalues 0, ˙i? with corresponding eigenfunctions LS? v? and D1 v? ˙ iD2 v? . The appropriate assumption on the point spectrum of L then is to require that, for some ˇ > 0 (which agrees w.l.o.g with ˇ from (6.4.18)), the eigenvalues 0; ˙i? are simple and the only ones of L with Re >

ˇ:

Theorem 6.4.5. Let f 2 C 4 .Rm ; Rm / and let the rotating wave .v? ; S? / satisfy the spectral assumptions above. Then, the rotating wave is asymptotically stable with 2 asymptotic phase for Equation (6.1.12) with initial data u0 2 HEucl .R2 ; Rm /, for strong solutions in the function class C 1 .Œ0; 1/; L2 .R2 ; Rm // \ C.Œ0; 1/; H 2 .R2 ; Rm //; and with respect to the norms k:k1 D k:kH 2 , k:k2 D k:kH 2 . Eucl

Let us comment on the assumptions of this theorem and possible extensions. In [7, Cor. 4.3], it is shown that the derivatives D ˛ v? , 1 6 j˛j 6 2 of the solution decay even exponentially as R ! 1 if (6.4.18) holds and if supjj>R jv? ./j falls below a certain computable threshold. Moreover, according to [8, Thm. 2.8], the op2 erator I L W HEucl .R2 ; Rm / ! L2 .R2 ; Rm / is Fredholm of index 0 for values Re./ > ˇ. Hence, the eigenvalues 0, ˙i? are isolated and of finite multiplicity. These results generalise to arbitrary space dimensions d > 3 if the nonlinearity and the solution v? are sufficiently smooth. Then, it can also be shown that the eigenfunctions which belong to eigenvalues on the imaginary axis and which are induced by symmetry, decay exponentially in space. This suggests that the nonlinear stability in Theorem 6.4.5 genereralises to space dimensions d > 3 , but details have not yet been worked out. 6.4.4 Lyapunov stability of the freezing method The numerical experiments in Section 6.3 confirm for various types of PDEs that the abstract freezing system (6.2.2), (6.2.5) has a Lyapunov-stable equilibrium whenever the original equation, (6.1.1), has a relative equilibrium which is stable with asymptotic phase. Moreover, one expects this property to persist under numerical approximations, such as truncation to a bounded domain with suitable boundary conditions as well as discretisations of space and time. In this section, we discuss a few instances where corresponding analytical results are available. The following result for travelling waves is taken from [57, Thm. 1.13].

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Theorem 6.4.6. Let the assumptions of Theorem 6.4.3 hold and let the template function vO in (6.2.8) satisfy vO 2 v? C H 2 .R; Rm /;

hvO  ; v?

vi O L2 D 0;

hvO  ; v?; iL2 ¤ 0:

Then, the travelling wave .v? ; ? / is asymptotically stable for (6.2.8). More precisely, there exist constants ı, C , ˛ > 0 such that (6.2.8) has a unique solution O L2 D 0. Existence and uniqueness .v; / if ku0 v? kH 1 6 ı and hvO  ; u0 vi holds for solutions with regularity  2 C Œ0; 1/, v 2 C.Œ0; 1/; H 1.R; Rm //, vt , f .v; v / 2 C.Œ0; 1/, L2 .R; Rm //, and v.t/ 2 H 2 .R; Rm / for t > 0. Furthermore, the following estimate is valid: kv.t/

v? kH 1 C j.t/

? j 6 Ce

˛t

ku0

v? kH 1 ;

t > 0:

The papers [58, 59] transfer these properties to a spatially discretised sytem (time is left continuous) on bounded intervals J D Œx ; xC  with general linear boundary conditions P .v.x /

v / C Q v .x / C PC .v.xC /

vC / C QC v .xC / D 0; (6.4.20)

where P˙ , Q˙ 2 R2m;m and v˙ are given by (6.4.12). An essential condition for stability is [59, Hypothesis 2.5]     

YCu ./ Y s ./ Q PC QC ¤ 0 (6.4.21) det P YCu ./ƒuC ./ Y s ./ƒs ./

for all  2 C satisfying Re > ˇ and jj 6 C for some large constant C . Here, the m;m matrices Y˙s;u ./ 2 Rm;m are invertible and, together with ƒs;u , solve ˙ ./ 2 R the quadratic eigenvalue problem (cf. (6.4.13)) AYƒ2 C B˙ Yƒ C .C˙

Im /Y D 0

(6.4.22)

such that Re  .ƒs˙.// < 0 < Re  .ƒu˙.//. Recall the definition of B˙ , C˙ in (6.4.13). Condition (6.4.15) on the dispersion set (6.4.14) ensures that (6.4.22) has m stable and m unstable eigenvalues. A counterexample in [59, Ch. 5.2] shows that violation of (6.4.21) creates instabilities of the numerical solution even if all conditions of Theorem 6.4.6 are satisfied. We proceed with two stability results recently obtained for the freezing formulation of the semilinear wave equation (6.3.5) and of the NLS (6.3.9). The assumptions on (6.3.4) are as follows: fQ 2 C 3 .R3m ; Rm /; M is invertible;

.v? ; ? / v?;

M

1

A is diagonalisable with positive eigenvalues,

2 Cb2 .R; Rm / 3 m

 R is a travelling wave of (6.3.4) with 2 H .R; R /; lim .v? ; v?; /./ D .v˙ ; 0/; fQ.v˙ ; 0; 0/ D 0; !˙1

A

2? M is invertible:

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The spectral assumptions concern the quadratic operator polynomial obtained from linearising the comoving frame equation in the first line of (6.3.5),

P .; @ / D M 2

.D3 fQ.?/ C 2? M @ /

C .? D3 fQ.?/

.A

2? M /@2

D2 fQ.?//@ ;

D1 fQ.?/

.?/ D .v? ; v?; ; ? v?; /: (6.4.23) From this, we obtain the matrix polynomials P˙ .; !/ by replacing the argument .?/ by its limit .v˙ ; 0; 0/ as  ! ˙1, and the operator @ by its Fourier symbol i!. Then, the dispersion set is defined as disp.P / D f 2 C W det.P˙ .; !// D 0 W for some sign ˙ and ! 2 Rg: The conditions analogous to (6.4.15), (6.4.16) are then Re disp.P / 6   Re pt .P .
; @ // n f0g 6

ˇ < 0;

ˇ < 0; and the eigenvalue 0 is simple:

Theorem 6.4.7. Let the assumptions above be satisfied and let the template function vO in (6.3.5) fulfil vO 2 v? C H 1 .R; Rm /;

hvO

v? ; vO  iL2 D 0;

hv?; ; vO  iL2 ¤ 0:

Then, the pair .v? ; ? / is asymptotically stable for the PDAE (6.3.5). More precisely, for all 0 <  < ˇ, there exist , C > 0 such that, for all u0 2 v? C H 3 .R; Rm /, v0 2 H 2 .R; Rm / and 01 2 R which satisfy ku0

v? kH 3 C kv0 C ? v?; kH 2 6 

as well as the consistency condition (6.3.6), the system (6.3.5) has a unique solution .v; 1 ; 2 / with 1 2 C 1 Œ0; 1/, 2 2 C Œ0; 1/ and regularity v v? 2 C 2 .Œ0; 1/; L2 .R; Rm //\C 1 .Œ0; 1/; H 1.R; Rm //\C.Œ0; 1/; H 2.R; Rm //: The following estimate holds for the solution: kv.
; t/

v? kH 2 C kvt .
; t/kH 1 C j1 .t/ 6 Ce

t

.ku0

? j

v? kH 3 C kv0 C ? v?; kH 2 /: (6.4.24)

Note that the second consistency condition (6.3.7) does not appear in the theorem but is used in the proof to make the acceleration 2 continuous at t D 0. The proof of the theorem builds on a careful reduction to the first order system (6.3.8) and on an application of the stability theorem from [46]. The theory for first order systems is also the reason for measuring the convergence (6.4.24) in a weaker norm than the initial values.

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Finally, we state a recent result on Lyapunov-stability of the freezing method for the nonlinear Schrödinger equation (6.3.10), (6.3.11). It is a special case of a general stability result from the thesis [20, Ch. 2] which applies to Hamiltonian PDEs that are equivariant w.r.t. the action of a Lie group. The assumptions are taken from the abstract framework of [31] which is a seminal paper on the stability of solitary waves. The following result is concerned with the waves (6.3.12) for fixed values ?;1 , ?;2 subject to 4?;1 > 2?;2 . Theorem 6.4.8. Let vO 2 H 3 .R; C/ be a template function such that 

hiv; O v? i0 D 0; hiv; O iv? i0 hvO x ; iv? i0

hvO x ; v? i0 D 0;  hiv; O v?;x i0 is invertible. hvO x ; v?;x i0

Then, the solitary wave .v? ; ?;1 ; ?;2 / from (6.3.12) is Lyapunov-stable for the system (6.3.10), (6.3.11). More precisely, for every " > 0, there exists ı > 0 such that the system (6.3.10), (6.3.11) with ku0 v? kH 1 6 ı has a unique (weak) solution .v; 1 ; 2 / with 1 2 C 1 Œ0; 1/, 2 2 C Œ0; 1/ and regularity v 2 C.Œ0; 1/; H 1.R; C// \ C 1 .Œ0; 1/; H

1

.R; C//;

t > 0:

The solution satisfies kv.
; t/

v? kH 1 C j1 .t/

?;1 j C j2 .t/

?;2 j 6 ";

t > 0:

For the notion of weak solution employed here, we refer to [20, Ch. 1.2]. The proof of Theorem 6.4.8 is mainly based on Lyapunov function techniques which are quite different from the semigroup and Laplace transform approaches used in the proofs of Theorems 6.4.5–6.4.7. We also emphasise that [20] contains applications to other PDEs with Hamiltonian structure, for example the nonlinear Klein Gordon and the Korteweg–de Vries equation, and that spatial discretisations are also studied.

6.5 Nonlinear eigenvalue problems In the context of this work, nonlinear eigenvalue problems arise when computing isolated eigenvalues of differential operators obtained by linearising about a relative equilibrium. We refer to (6.4.2) for the abstract linearisation and to (6.4.11), (6.4.19), (6.4.23) for some examples of operators. There are several sources of nonlinearity in the eigenparameter; see [32] for a recent survey. Quadratic terms arise from second order equations in time (6.4.23), exponential terms occur in the stability analysis of delay equations (see [39]), and nonlinear integral operators appear in the boundary element method for linear elliptic eigenvalue problems. Here, another source of nonlinearity is of interest, namely the use of projection boundary conditions when

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solving linear eigenvalue problems for operators such as (6.4.11) on a bounded interval J D Œx ; xC . In the following, we summarise two of the major results from [5] on this problem. Contour methods have been developed over the last years ([1, 4, 32]) and have become rather popular since no a-priori knowledge about the location of eigenvalues is assumed. The paper [5] generalises the contour method from [4] to holomorphic eigenvalue problems

L./v D 0;

v 2 X;

 2   C;

(6.5.1)

where L./ W X ! Y are Fredholm operators of index 0 between Banach spaces X; Y which depend holomorphically on  in some subdomain  of C. The algorithm determines all eigenvalues of (6.5.1) in the interior 0 D int.€/ of some given closed contour € in . It is assumed that € itself lies in the resolvent set .L/ D f 2 C W N.L.// D f0gg where N denotes the null space of an operator. One chooses linearly independent elements vk 2 Y , k D 1; : : : ; `, and functionals wj 2 X ? , j D 1; : : : ; p, and computes the matrices   p;` ;  2 €; (6.5.2) E./ D hwj ; L./ 1vk ikD1;:::;` j D1;:::;p 2 C E0 D

1 2i

Z

E./d;

€

E1 D

1 2i

Z

E./d:

(6.5.3)

€

The following result from [5, Thm. 2.4] holds for the case of simple eigenvalues defined by the conditions  .L/ \ int.€/ D f1 ; : : : ; ~ g; N.L.j // D spanfxj g; N.L.j /? / D spanfyj g; 0

hyj ; L .j /xj i ¤ 0;

j D 1; : : : ; ~:

j D 1; : : : ; ~;

Theorem 6.5.1. Let the above assumptions hold and assume the following nondegeneracy condition:     kD1;:::;` (6.5.4) D ~ D rank hy ; v i rank hwj ; xk ikD1;:::;~ j k j D1;:::;~ : j D1;:::;p

Then, rank.E0 / D ~ holds. Further, let

E0 D V0 †0 W0? ; V0 2 Cp;~ ; V0? V0 D I~ ; W0 2 C`;~ ; W0? W0 D I~

(6.5.5)

be the (shortened) singular value decomposition of E0 with †0 D diag.0 ; : : : ; ~ /, 1 > 2 > : : : > ~ > 0. Then, all eigenvalues of the matrix EL D V0? E1 W0 †0 1 2 C~;~ are simple and coincide with 1 ; : : : ; ~ .

(6.5.6)

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First note that (6.5.4) implies p; ` > ~, i.e., the number of test functions and test functionals should exceed the number of eigenvalues inside the contour. In fact, in applications we expect to have p  `  ~. The key to the proof is the theorem of Keldysh (see [38, Thm. 1.6.5]) which describes the coefficients of the meromorphic expansion of L./ 1 near its singularities in terms of (generalised) eigenvectors. We mention that Theorem 6.5.1 generalises to eigenvalues of arbitrary geometric and algebraic multiplicity. With the proper definition of generalised eigenvectors of (6.5.1), it turns out that the Jordan normal form of the matrix EL in (6.5.6) inherits the exact multiplicity structure of the nonlinear eigenvalue problem; see [5, Thm. 2.8]. For the overall algorithm, one approximates the integrals in (6.5.3) by a quadrature rule (for analytical contours €, the trapezoidal sum is sufficient since it leads to exponential convergence [4]) and solves linear systems L./uk D vk with k D 1; : : : ` at the quadrature nodes  2 €. Note that these solutions can be used for both integrals in (6.5.3). The (shortened) singular value decomposition (6.5.5) involves a rank decision revealing the number ~ of eigenvalues inside the contour. Finally, solving the linear (!) eigenvalue problem for the matrix EL 2 C~;~ is usually cheap if ~ is small. Let us note that the algorithm also provides good approximations of the eigenfunctions associated to j , j D 1; : : : ; ~; see [4] and [5, Sec. 2.2]. There is even an extension of the contour method to cases where the nondegeneracy condition (6.5.4) is violated. Then, one computes some higher order moments Z 1  E./d;  D 0; 1; 2; : : : ; (6.5.7) E D 2i € and determines the eigenvalues from a suitable block Hankel matrix (see [4] for the extended algorithm and for the number of additional integrals needed). Numerical examples with more details on the algorithm may be found in [4], and applications to the travelling waves considered here appear in [5, Sec. 6]. Another favourable feature of the method is that the errors occurring in the intermediate steps (6.5.2), (6.5.3), (6.5.5), (6.5.6) are well controllable. We demonstrate this for the operator L./ D I L with the differential operator L taken from (6.4.11). The evaluation of the matrix E./ from (6.5.2) requires to solve inhomogeneous equations

L./u D v 2 L2 .R; Rm /;

 2 €  .L/;

(6.5.8)

on a bounded interval J D Œx ; xC  with linear (but possibly -dependent) boundary conditions (cf. (6.4.20))

BJ ./u WD P ./.u.x /

v / C Q ./u .x / C PC ./.u.xC/

vC /

C QC ./u ./ D 0:

Such -dependent boundary matrices P˙ ; Q˙ 2 C.; R2m;m / occur with the socalled projection boundary conditions [3] and lead to fast convergence towards the solution of (6.5.8) as x˙ ! ˙1. The matrices are determined in such a way (see [5,

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Computation and stability of waves

Sec. 4]) that  
P ./ Q ./

Y s ./ s Y ./ƒs ./




PC ./ QC ./



YCu ./ u YC ./ƒuC ./



D I2m

holds for the matrices Y˙s;u ./; ƒs;u ˙ ./ determined from (6.4.22). Condition (6.4.21) is then trivially satisfied. With these preparations, [5, Cor. 4.1] reads as follows: Theorem 6.5.2. Let the assumptions of Theorem 6.4.3 hold except for the condition (6.4.16) on the point spectrum. Let €  fz 2 C W Rez > ˇg with ˇ from (6.4.15) be a closed contour which lies in the resolvent set of the operator pencil

L./ D I

L D I

.A@2 C B.
/@ C C.
//

with L from (6.4.11). Further, let vk 2 L1 .R; Rm /, k D 1; : : : ; `, be linearly independent functions with compact support and let wj , j D 1; : : : ; p, be linearly independent functionals on L1 .R; Rm / defined by Z wO j .x/> u.x/dx; wO j 2 L1 .R; Rm /; j D 1; : : : ; p: hwj ; ui D R

Then, for J D Œx ; xC  sufficiently large, the linear boundary value problem with projection boundary conditions

L./uk;J D vkjJ in J;

BJ ./u D 0

has a unique solution uk;J .
; / 2 H 2 .J; Rm / for all k D 1; : : : ; ` and  2 €. Moreover, for every 0 < ˛ < ˇ, there exists a constant C > 0 such that the matrices E;J D



1 2i

Z

€

kD1;:::;`  hwO j jJ ; uk;J .
; /iL2 .J / d ; j D1;:::;p

 D 0; 1;

(6.5.9)

satisfy the estimate jE

E;J j 6 C exp. 2˛ min.jx j; xC //;

 D 0; 1:

(6.5.10)

Note that the integrals (6.5.9) are the quantities approximating the integrals (6.5.7) over the unbounded domain. With the estimates (6.5.10) at hand, it is not difficult to show that the singular values obtained in (6.5.5) and finally the eigenvalues of EL in (6.5.6) inherit the exponential error estimate; see [5, Sec. 4]. Let us finally note that the computation of isolated eigenvalues for the linearised operator becomes rather challenging for waves in two and more space dimensions. We consider the contour method to be a true competitor to classical methods for computing eigenvalues of linearisations at such profiles.

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References [1] J. Asakura, T. Sakai, H. Tadano, T. Ikegami and K. Kimura, A numerical method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 1 (2009), 52–55. [2] D. Barkley, Euclidean symmetry and the dynamics of rotating spiral waves, Phys. Rev. Lett. 72 (1994), 164–167. [3] W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems. IMA J. Numer. Anal. 10 (1990), 379–405. [4] W.-J. Beyn, An integral method for solving nonlinear eigenvalue problems, Linear Algebra Appl. 436 (2012), 3839–3863. [5] W.-J. Beyn, Y. Latushkin and J. Rottmann-Matthes, Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals, Integral Equations Operator Theory 78 (2014), 155–211; arXiv:1210.3952. [6] W.-J. Beyn and J. Lorenz, Nonlinear stability of rotating patterns. Dyn. Partial Differ. Equ. 5 (2008), 349–400. [7] W.-J. Beyn and D. Otten, Spatial decay of rotating waves in reaction diffusion systems, Dyn. Partial Differ. Equ. 13 (2016), 191–240; arXiv:1602.03393. [8] W.-J. Beyn and D. Otten, Spectral analysis of localized rotating waves in parabolic systems, Philos. Trans. A 376 (2018) no. 20170196. [9] W.-J. Beyn, D. Otten and J. Rottmann-Matthes, Stability and computation of dynamic patterns in PDEs. In Current Challenges in Stability Issues for Numerical Differential Equations (L. Dieci, and N. Guglielmi, eds.), Springer, Cham, 2014, 89–172. [10] W.-J. Beyn, D. Otten and J. Rottmann-Matthes, Computation and stability of traveling waves in second order equations, SIAM J. Numer. Anal. 56(3) (2018), 1786–1817. [11] W.-J. Beyn, D. Otten and J. Rottmann-Matthes, Freezing traveling and rotating waves in second order evolution equations. In Patterns of Dynamics (P. Gurevich, J. Hell, B. Sandstede, A. Scheel, eds.), Springer Proc. Math. Stat. 205, 2017, 215–241. [12] W.-J. Beyn, S. Selle and V. Thümmler, Freezing multipulses and multifronts, SIAM J. Appl. Dynam. Syst. 7 (2008), 577–608. [13] W.-J. Beyn and V. Thümmler, Freezing solutions of equivariant evolution equations, SIAM J. Appl. Dynam. Syst. 3 (2004), 85–116. [14] W.-J. Beyn and V. Thümmler, Phase conditions, symmetries, and PDE continuation. In Numerical Continuation Methods for Dynamical Systems (B. Krauskopf, H. Osinga, and J. Galan-Vioque, eds.), Springer, Dordrecht, 2007, 301–330. [15] A. M. Bloch and A. Iserles, Commutators of skew-symmetric matrices, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005), 793–801. [16] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol.10, AMS, Providence, RI, 2003. [17] A.R. Champneys and B. Sandstede, Numerical computation of coherent structures. In Numerical Continuation Methods for Dynamical Systems (B. Krauskopf, H. Osinga, and J. GalanVioque, eds.), Springer, Dordrecht, 2007, 331–358. [18] P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems, World Scientific, River Edge, 2000. [19] L.-C. Crasovan, B.A. Malomed and D. Mihalache, Spinning solitons in cubic-quintic nonlinear media, Pramana-J. Phys. 57 (2001), 1041–1059.

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Chapter 7

Initial value problems for nonlinear dispersive equations at critical regularity S. Herr Global regularity results for nonlinear dispersive equations hinge on a thorough understanding of the Cauchy problem in spaces of functions of low regularity. This is most challenging in scale invariant regimes as solutions interact strongly on multiple frequency-scales. Here, some recent progress on the critical well-posedness theory will be reviewed, with a focus on nonlinear Schrödinger and Dirac equations.1

7.1 Introduction This is a survey on certain aspects of nonlinear dispersive equations in critical Sobolev spaces. First, we illustrate key concepts and paradigms and explain the connection to harmonic analysis. Second, we report on recent progress on the well-posedness theory at the critical level of regularity and its impact on questions concerning the longtime behaviour of solutions with a certain bias towards the results of Project B8. The focus will be on nonlinear Schrödinger equations on compact manifolds in Section 7.2 and the cubic Dirac equation, the Dirac–Klein–Gordon system, and the Zakharov system in Section 7.3. 7.1.1 Scaling and conserved quantities Let us start with an example. Let p > 1 and consider the nonlinear Schrödinger equation i @t u C u D ˙jujp

1

u;

(7.1.1)

for sufficiently smooth u W I  Rd ! C. The case C is called focussing, the case is called defocusing. Often, we will study the initial value problem, where we prescribe initial data by requiring u.0;
/ D f for some given f W Rd ! C. We will be interested in a well-posedness theory for this and other initial value problems.

1 Project

B8

160

S. Herr

Roughly speaking, a well-posedness theory in the Hadamard sense comprises 1. Existence of solutions for all initial data in a given space (locally in time or globally in time), 2. Uniqueness of solutions (in a certain space), 3. Continuous dependence on the initial data (with respect to appropriate topologies), and we will be more precise about this in the examples to follow. If we rescale solutions u according to u .t; x/ WD ˛ u.2 t; x/ then one easily checks that u W 

2

for  > 0;

I  Rd ! C and

i @t u Cu D ˛C2 .i @t Cu/.2t; x/ and ju jp We conclude that

i @t u C u D ˙ju jp

1

1

u D p˛ jujp

1

u.2 t; x/:

u

iff ˛ D p 1 . Now, suppose that initial data u.0;
/ D f 2 HP s .Rd / is given, where the (semi-)norm in HP s .Rd / is defined by 2

kf kHP s .Rd / D

Z

Rd

b./j2 d jj2s jf

 12

:

With the above choice of ˛ we compute that ku .t;
/kHP s .Rd / D  p

2

1 Cs

d 2

ku.2 t;
/kHP s .Rd / ;

2 so that the HP s .Rd /-norm is scale-invariant iff s D sc WD d2 , and sc is called p 1 the critical regularity. For reasons discussed below, we will be interested in a well-posedness theory for initial data in the Sobolev space H s .Rd /, which is the space of tempered distributions f with

kf kH s .Rd / D

Z

Rd

b ./j2d  hi jf 2s

 12

1

< C1; hi D .1 C jj2 / 2 :

The scaling argument above suggests that a well-posedness theory for initial data in the critical space H sc .Rd / is key for the analysis of the longtime behaviour. The regime s > sc is called subcritical and the regime s < sc is called supercritical. We do not expect a well-posedness theory in H s .Rd / in the supercritical case s < sc , because small times of existence for small solutions can be rescaled to large times of existence for large solutions and this is known to be false in many problems.

Nonlinear Dispersive equations at critical regularity

161

Suppose that we are given a smooth and decaying solution u of (7.1.1). Then, one can easily check that Z d d ju.t; x/j2 dx D 0 and e.u.t// D 0 dt Rd dt for the energy e.u.t// D

1 2

Z

Rd

jru.t; x/j2 dx ˙

1 pC1

Z

Rd

ju.t; x/jpC1 dx;

so that both ku.t/kL2 .Rd / and e.u.t// are constant with respect to t. Observe that sc D 1 iff p D 1 C d 4 2 if d > 2. Further, the sharp Sobolev embedding implies ku.t/kLpC1 .Rd / . kru.t/kL2 .Rd / in this case. Here and in the following we will use the notation A . B if there exists a harmless (e.g. only depending on d and p) constant c > 0, such that A 6 cB. If we can choose c smaller than 1 (depending on other previously fixed parameters), we write A  B. We also write A  B if A . B and B . A. From the above consideration we conclude that e.u.t//  kru.t/k2L2 .Rd / , pro-

vided that kru.t/kL2 .Rd / is sufficiently small. In other words, when p D 1 C d 4 2 , the critical space corresponds to the energy space. Hence, this problem is called energy-critical. Notice that, for small initial data f 2 H 1 .Rd /, in this setting a local existence result with a time of existence depending only on kf kH 1 .Rd / yields global existence owing to the a-priori bound on ku.t/kH 1 .Rd / coming from conservation of the energy and the L2 .Rd /-norm. To sum up, if d > 2, the well-posedness problem for p D 1 C d 4 2 in H 1 .Rd / is both challenging, because it is scaling-critical, and rewarding, because it allows to exploit the energy conservation to establish global existence (at least for small data). We remark that (7.1.1) has a Hamiltonian structure and there are interesting special solutions, see chapter 6.3.4. For energy-critical Schrödinger equations on Rd , even for initial data of arbitrary size, many of the fundamental questions concerning well-posedness, scattering and blow-up have been solved within the last decade, see e.g. the surveys by Kenig [39, 40] and the references therein. It is obvious that Lp -estimates for solutions are crucial to derive such results. We will discuss these estimates, often called Strichartz estimates in this context, in Subsection 7.1.2. In Section 7.2 we will address corresponding problems for energycritical Schrödinger equations in a slightly different context, namely on compact manifolds. Further, we will consider analogous issues for certain systems on Rd in Section 7.3. 7.1.2 Dispersive equations and Fourier restriction Consider the linear Schrödinger equation i @t u C u D 0:

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For sufficiently smooth and decaying solutions, taking the spatial Fourier transform Z d b u.t; / WD .2/ 2 e ix
 u.t; x/ dx Rd

yields the ordinary differential equation i @t b u.t; /

jj2b u.t; / D 0; hence b u.t; / D e

i jj2 t

b u.0; /; for every  2 Rd :

The Fourier inversion theorem implies the representation formula Z d 2 b./ d e i t jj eix
 f u.t; x/ D .2/ 2 Rd

D .kt  f /.x/; where kt .y/ D

1

.4 i t/

d 2

e

2

i jyj 4t

:

Indeed, if (say) f 2 Cc2 .Rd /, one can easily check that u is a solution in the classical sense. More generally, let us assume that  W Rd ! R is measurable and that it does not grow faster than some polynomial. It is the symbol of the Fourier multiplier . i r/, i.e. b./:  . i r/f ./ D ./f

1

For, say, f 2 S .Rd /, the function Z d b./ d DW ei t . ei t ./ eix
 f u.t; x/ D .2/ 2 Rd

i r/

f .x/

solves

i @t u C . i r/u D 0: Important examples are ./ D jj2 ./ D ˙ jj ./ D ˙ hi

(Schrödinger equation); (half-wave equations); (half-Klein–Gordon equations):

Notice that, for any s 2 R, we have ku.t/kH s .Rd / D ku.0/kH s .Rd /

for all t 2 R;


and ei t . i r/ t 2R is a group of unitary operators on H s .Rd /. What about estimates in Lp -norms? This question is of particular interest with regard to nonlinear problems, which we will discuss later. We will now explain the connection between dispersive equations and one of the driving themes of Euclidean harmonic analysis. In the 1960’s, Elias Stein put forward the Fourier restriction problem, see e.g. Stein’s b book [53] and Tao’s survey [57] and the references therein. The Fourier transform F

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Nonlinear Dispersive equations at critical regularity

of an integrable function F W Rn ! C is continuous and bounded, hence its restricb j† to a smooth hypersurface †  Rn is well-defined. However, if F is squaretion F b j† , integrable instead, this is impossible due to Plancherel’s theorem. Let R† F D F for F 2 S .Rn /. The precise question is: For 1 < p < 2 and q > 1, does there exist C > 0, such that for all F 2 S .Rn / the estimate kR† F kLq .†/ 6 C kF kLp .Rn /

(7.1.2)

holds true? Here, we implicitly restrict to compact subsets by considering the measure  on † defined by Z Z h˛ dHn 1; for all h 2 L1 .†/; h d D †

†

where ˛ 2 Cc1 .Rn /, and Hn 1 denotes the .n Now, for G 2 C.†/, Z Z b G d R† F G d D F †

†

D .2/

n 2

Z

Rn

F .y/

Z

eiy


†

1/-dimensional Hausdorff measure.

G./ d ./ dy D

Z

Rn

F E† G dy ;

with the so-called Fourier extension operator Z n E† G.y/ D .2/ 2 eiy
 G./ d ./: †

By duality and density, the Fourier restriction estimate is equivalent to kE† GkLp0 .Rn / 6 C kGkLq0 .†/ ;

(7.1.3)

for all G 2 Cc1 .†/. If † D f.; / 2 R1Cd j  D ./g, d D n 1, we notice that Z d ei.t;x/
../;/ G../; / d ; E† G.t; x/ D .2/ 2 Rd

provided that G is compactly supported and ˛ is chosen appropriately. We have discussed above that u D E† G solves i @t u C . i r/u D 0;


with initial data given as the inverse Fourier transform of G .
/;
. To summarise, we have proved that estimates in space-time Lebesgue norms for solutions u are equivalent to estimates for the Fourier restriction operator. This is of particular importance in the case q 0 D 2, in which case Strichartz made this connection and proved

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such estimates for important phase functions . Let us sketch a conceptual proof which has been developed later. We notice that kR† k2Lp .Rn /!L2 .†/ D kE† k2L2 .†/!Lp0 .Rn / D kE† R† kLp .Rn /!Lp0 .Rn / ; and we compute n 2

Z

b ./ d ./ eiy
 F † Z Z ei.y z/
 d ./ F .z/ dz D .2/ n

E† R† F .y/ D .2/

Rn

†

D .k†  F /.y/;

where k† D .2/

By stationary phase type arguments, it can be proved that jb  .y/j . .1 C jyj/

` 2

n 2

b  .
/:

;

where ` is the number of non-vanishing principal curvatures of †  Rn , i.e. the 0 eigenvalues of the Hessian of . The Lp ! Lp -estimate (also mixed norm versions) now follow by interpolation and the Hardy–Littlewood–Sobolev theorem on fractional integration. The precise range of p depends on ` 6 n 1. Estimates of this type are commonly referred to as Strichartz estimates. The key ingredient here is the decay estimate for the Fourier transform of the surface measure, which corresponds to a dispersive estimate for solutions. For the paraboloid (Schrödinger equation) and the hyperboloid (half-Klein–Gordon equation) we have ` D d , whereas for the cone (half-wave equation) ` D d 1 due to the lack of curvature of the cone in the radial direction. Later, Bourgain, Wolff, and Tao, among others, found that one can also exploit transversality of characteristic surfaces in bilinear versions of estimates for the extension operator to go beyond the above range, see [57]. We will come back to this in Section 7.3. We conclude this section with a first look at the Schrödinger equation on a compact manifold, more precisely, on the flat square torus Td D .R=2Z/d . As above, we see that d X 2 u.t; x/ D .2/ 2 ei.t;x/
. jnj ;n/ cn n2Z d

solves i @t C u D 0 with initial data u.0/ with Fourier coefficients .cn / . This can be viewed as a Fourier extension operator for the discrete paraboloid † D f.; n/ 2 Z 1Cd j  D jnj2 g: Again, this defines a group of unitary operators in H s .Td /. However, in this setting, there can be no dispersive estimate implying supx ju.t; x/j ! 0 as t ! 1, because this would violate ku.t/kL2 .Td / D ku.0/kL2.Td / as Td has finite measure. Still, one can prove Lp -estimates in this setting. To a large extent, this theory has a number-theoretic flavour. Let us look at an example, which is due to Bourgain [11]

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Nonlinear Dispersive equations at critical regularity

if d > 2 and to Zygmund [61] if d D 1, and reveals the connection to estimates for lattice points and exponential sums. Assume that cn D 0 if jnj > N for some N > 0. We express the exponential sum for u2 as a space-time Fourier series, i.e. X 2 2 u2 .x; t/ D .2/ d ck cn k ei.t;x/
. jkj jn kj ;n/ n;k2Z d

D .2/

X

d

an;m ei.t;x/
.m;n/

.n;m/2Z d C1

P where an;m D k2Sn;m ck cn k and Sn;m D fk 2 Z d j jkj 6 N; jkj2 C jn mg. Plancherel’s Theorem and the Cauchy-Schwarz inequality yield X kuk4L4 .Td C1 / D ku2 k2L2 .Td C1/ D jan;m j2 n;m

6 6

max #Sn;m

jnj62N jmj62N 2

X

X

n2Z d m2Z;k2Sn;m

max #Sn;m

jnj62N jmj62N 2

X n

jcn j2


2

ˇ 2 ˇck j jcn

kj

kj2 D

2

Completing the squares implies that k 2 Sn;m if and only if j2k nj2 D 2m jnj2 . Let sd .r/ denote the number of integer lattice points on the .d 1/-sphere of radius r. We obtain 1 kukL4 .Td C1/ 6 max sd .r/ 4 ku.0/kL2 .Td / : 06r62N

This estimate and its ramifications are of fundamental importance in the analysis of the cubic nonlinear Schrödinger equation on Td . Recall that the asymptotics for sd .r/ are part of classical analytic number theory. More recently, Bourgain and Demeter proved the decoupling conjecture [12]. This yields another approach to Lp -estimates in the periodic setting more akin to Fourier restriction theory which does not rely on analytic number theory, but we will not go into details here.

7.2 Nonlinear Schrödinger equations on compact manifolds In generalisation of Bourgain’s results [11] on tori, consider i @t u C g u D 0

on R  M;

(7.2.1)

where .M; g/ is a smooth compact Riemannian manifold without boundary of dimension d > 2 and g the Laplace–Beltrami Operator. As described above, the dispersive estimate cannot hold. What are the dispersive properties of solutions in this setting? There are classical results stating that non-degenerate and stable trapped

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geodesics force spatial concentration of solutions, which is a geometric obstruction to dispersion, see e.g. [1, 49, 50] for more details. This may lead to instability results for nonlinear Schrödinger equations [60], and the failure of Strichartz estimates. Before we go into more detail, let us introduce some notation. Since M is compact, the spectrum  . g / of the Laplace–Beltrami Operator is discrete and we list the non-negative eigenvalues 0 D 20 < 21 < : : : < 2n ! C1. For an eigenvalue 2k , let hk W L2 .M / ! L2 .M / be the spectral projector onto the corresponding eigenspace Ek . We have the orthogonal decomposition L2 .M / D Let P D P N

k2N0 Wk 1. Further, we spectrally define H s .M / D .1 kf

D

kD0

hk :

hk for dyadic numbers N D 2; 4; : : :, and P1 D N >1 PN D Id, with the convention that we add up all

hk . Then,

1 X

i t 2 k

kD0

k2N0 WN 6kP 1

g /

s=2

L2 .M / with

hN i2s kPN f k2L2 .M / :

where hxi D .1 C jxj2 /1=2 . 7.2.1 Strichartz estimates and well-posedness Solutions uN .t/ D ei t g PN  travel at a speed proportional N and therefore do not leave a coordinate chart within a time proportional to N 1 . Burq–Gérard–Tzvetkov [14] used this fact to prove classical dispersive estimates on such time scales and by decomposition and Littlewood– Paley theory recovered Strichartz estimates on the time interval Œ0; 1, up to a loss of derivatives. To be more precise, for q2 C dr D d2 , q > 2, 2 6 r < 1, kei t g kLqt .Œ0;1;Lrx .M // . kkH  .M / ;

with  D q1 :

In some cases, such as tori and spheres, the order  of this derivative loss can be extenuated, by exploiting more specific information on the spectrum and the eigenfunctions. We are interested in nonlinear Schrödinger equations in this context, say i @t u C g u D ˙jujp

1

u

on R  M;

(7.2.2)

and ask the questions of local and global well-posedness analogous to the discussion in Subsection 7.1.1, again with a special emphasis on the energy-critical case p D 1 C d 4 2 and d D 3, also called the quintic nonlinear Schrödinger equation. It is immediate that for results in scale-invariant regimes one needs scale-invariant estimates, so that a loss of derivatives in Strichartz estimates is a significant problem. In [15, 16, 17] Burq–Gérard–Tzvetkov proved multilinear versions of Strichartz estimates on compact manifolds. Using spectral information, such as the precise

Nonlinear Dispersive equations at critical regularity

167

knowledge of the spectrum and sharp (multilinear) estimates for the eigenfunctions, some scale-invariant estimates could be recovered. In previous joint work with Tataru– Tzvetkov [35], we found some additional almost orthogonality principles and constructed critical function spaces to solve the first energy-critical global well-posedness problem on a compact manifold [35], see also [36]. In subsequent work of Ionescu– Pausader [38], this has been extended to large initial data in the defocusing case. This is done by an indirect argument which, among other ideas, relies on the global wellposedness result of Colliander–Keel–Staffilani–Takaoka–Tao [23] in the Euclidean setting. On flat tori, the spectrum of g , consisting of sums of squares of integers, is a rather delocalised subset of the real line. On the other hand, relative to the corresponding eigenvalue the eigenfunctions are quite small in L1 .M /. Spectrally, an other extreme case is the round sphere Sd . Here, the spectrum, being a shifted sequence of squares of integers, is well localised, but the eigenfunctions, the spherical harmonics, saturate the worst case L1 .M /-bound given by Weyl’s estimate. It is a special case of a Zoll manifold, where all geodesics are closed with a common minimal period. Due to results of Duistermaat–Guillemin [25] and Colin de Verdière [22], Zoll manifolds are characterised by the fact that the spectrum of g is contained in a union of uniformly bounded intervals centered at a shifted sequence of squares of integers. For this class of manifolds in d D 3, local and small data global wellposedness for the quintic nonlinear Schrödinger equation in H 1 .M / was proved in [29] and extended to large data by Pausader–Tzvetkov–Wang [46]. Given these results, one is inclined to ask: Under which spectral assumptions can one prove scale-invariant well-posedness results? As a starting point, consider general flat rectangular tori M D T3 WD R3 =.21 Z  22 Z  23 Z/ for j > 0 and products of spheres M D S  S2 for  > 0. Spectrally, these can be viewed as intermediate cases between T3 and S3 . Strunk [54, 55] proved local and small data global well-posedness for the quintic equation in H 1 .T3 / similar to the rational case [35], besides further critical results in dimension two. His proofs are based on the one-dimensional exponential sum estimate

X

1 2 2

cn ei t n p . .1 C jJ j/ 2 p k.cn /k`2 ;

n2J \Z

L .Œ0;2/

for any bounded interval J  R and fixed p > 4, which in case J D Œ1; N  is due to Bourgain [10], see also [29] and [55, Section 1.3.3] for the above version. Around the same time, Killip–Visan [41] derived scale-invariant versions of Strichartz estimates on irrational tori from [12], which allows for alternative proofs of critical well-posedness results. For the defocusing quintic nonlinear Schrödinger equation in dimension d D 3, Strunk [56, 55] extended his small data result to global wellposedness for initial data of arbitrary size in H 1 .T3 /, using ideas of [38]. Concerning M D S  S2 small data global well-posedness was proved in joint work with Strunk [34] in the case  D 1 and in [55] in the general case  > 0.

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7.2.2 A conditional well-posedness theory for energy critical Schrödinger equations In all instances discussed above, it turned out that multilinear versions of Strichartz estimates are powerful tools. More precisely, in all cases discussed above, the following trilinear estimate played a crucial role: There exists ı > 0, such that for all j 2 L2 .M / and N1 > N2 > N3 > 1 3

Y

ei t g PNj j

j D1

L2 .Œ0;2M /

. N2 N3

N

3

N1

C

3 1 ı Y kj kL2 .M / : (7.2.3) N2 j D1

We emphasise that this is a scale-invariant estimate without derivative loss in the sense that the constant does not grow with the highest frequency scale N1 , and we may replace any of the three factors by its complex conjugate. Concerning general compact manifolds, we established a conditional well-posedness theory in [34] based on the validity of (7.2.3), see also [55, Subsection 2.2.1] for a comprehensive account. Theorem 7.2.1. Suppose that .M; g/ is a smooth Riemannian compact boundaryless manifold, dim.M / D 3, and that (7.2.3) holds true. Then, the energy-critical nonlinear Schrödinger equation (7.2.2) with p D 5 is globally well-posed for small initial data  2 H 1 .M /. Sketch of proof. Rewrite (7.2.2) with u.0/ D  2 H 1 .M / as the integral equation Z t 0 i t g 4 ei.t t /g f .t 0 / dt 0 : u.t/ D e   i I .juj u/.t/; I .f /.t/ D 0

The aim is to invoke the contraction mapping principle in a suitable complete metric space X 1 .I /, which we describe next, see [35, 29, 34] and the references therein for details. For 1 6 r < 1, a step function a W R ! L2 .M / is called a Ur g -atom, if a.t/ D

K X

1Œtk

1 ;tk /

e

i t g

ak ;

kD1

K X

kD1

kak krL2 D 1;

for a partition 1 < t0 <


< tK 6 C1. The normed vector space Ur g is defined as the corresponding atomic space. The normed vector space Vr g consists of all right-continuous v W R ! L2 .M / such that kvkV r D

sup 1 N2 > N3 > 1 3

Y

PNj uj

j D1

L2 .Œ0;2M /

. N2 N3

N

3

N1

3 1 ı 0 Y kuj kY 0 : N2

C

(7.2.4)

j D1

This is a crucial step in scale-invariant problems and we sketch the proof devised in [29] and [34, Section 3]. In a first step, consider atoms uj D

Kj X

1Œt

kj D1

k j

;t

1 k j

/e

i t g

Kj X

kj ;

kj D1

kkj k2L2 D 1;

for j D 1; 2; 3. Then, due to 3

Y

PN uj

j

j D1

L2 .Œ0;2M /

.

3

2  X Y

PN ei t g k 2

j

j

j D1

k1 ;k2 ;k3

L .Œ0;2M /

 12

;

and the atomic structure of X 0 , the estimate 3

Y

PN uj

j j D1

L2 .Œ0;2M /

. N2 N3

N

3 1 ı Y kuj kX 0 C N1 N2 3

(7.2.5)

j D1

follows. It remains to extend this to Y 0 . Choosing 1 D 2 D 3 D  together with N1 D N2 D N3 D N in (7.2.3), we obtain 2

kei t g PN kL6 .Œ0;2M / . N 3 kkL2 .M / ; and by atomic decomposition this extends to 2

kPN ukL6 .Œ0;2M / . N 3 kukU 6 : g

Hölder’s inequality implies 3

Y

PN uj

j j D1

L2 .Œ0;2M /

2

. .N1 N2 N3 / 3

3 Y

j D1

kuj kU 6 : g

(7.2.6)

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S. Herr

In addition, for any fixed r > 1, by Hölder’s inequality and the Sobolev embedding we obtain the trivial estimate 3

Y

PN uj

j

1

L2 .Œ0;2M /

j D1

. jI j 2 kPN u1 kL1 L2 kPN u2 kL1 kPN u3 kL1 1

. .N2 N3 /

3 2

t

3 Y

j D1

2

x

t;x

3

t;x

(7.2.7) kuj kU r ; g

which is not scale-invariant, but the constant is independent of N1 . Now, we use an interpolation argument, based on [28, Prop. 2.20]. More precisely, we combine (7.2.5) with (7.2.6) if N2 N3 > N1 and with (7.2.7) if N2 N3 6 N1 to obtain (7.2.4) for any ı0 < ı. We refer to [34, p. 756] and [29] for more details on this crucial point. Now, we are in a position to finish the proof of Theorem 7.2.1. For I D Œ0; 2/ (say), we obtain kei t g   i I .juj4u/kX 1 .I / . kkH 1 .M / C kI .juj4 u/kX 1 .I / . kkH 1 .M / C

sup kvk

Y

1 .I /61

jhjuj4 u; viL2 .I M / j

. kkH 1 .M / C kuk5X 1 .I / ; provided that sup v2Y

1 .I /61

jhjuj4 u; viL2 .I M / j . kuk5X 1 .I / :

(7.2.8)

Due to the polynomial structure, one can prove the contraction property along the same lines and the proof of Theorem 7.2.1 can be completed by applying the contraction mapping principle and the energy conservation in the standard way. We omit the details. Estimate (7.2.8) follows from 5 X Y h PN vj ; PN v0 iL2 .I M / . kv0 kY 

0

j

j D1

1

5 Y

j D1

kvj kY 1

(7.2.9)

where  indicates summation over all dyadic N0 ; N1 ; : : : ; N5 > 1 under the constraint N1 > : : : > N5 . This is because Y 1 ,! X 1 , vj either denotes uj or uj , and due to the symmetry we are free to assume the ordering N1 > : : : > N5 . Now, we outline how to sum up the dyadic pieces, using (7.2.4). On a general manifold, there are two contributions, 5 X Y h PN vj ; PN v0 iL2 .I M / D †1 C †2 C †3 

j

j D1

0

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Nonlinear Dispersive equations at critical regularity

where the sum †1 is defined by the additional constraint N2 6 N0  N1 , †2 by N0 < N2  N1 , and †3 by N0 ; N2  N1 or N1  N0 . Concerning †1 , the Cauchy-Schwarz inequality and the estimate (7.2.4) imply

Y

Y

X



j†1 j . PNj vj 2 PNj vj 2

.

X

N2 N3 N4 N5

IN2 6N0 N1

. kv0 kY

1

L .I M /

j D0;2;4

IN2 6N0 N1

5 Y

j D1

N

5

N1

C

L .I M /

j D1;3;5

5 1  ı 0  N4 1 ı 0 Y C kPNj vj kY 0 N3 N0 N2 j D1

kvj kY 1 :

Concerning †2 , we start with an application of Cauchy–Schwarz as above, and (7.2.4) implies j†2 j .

X

N0 N3 N4 N5

IN0 0 and consider the map F W H 1 .M / ! H 1 .M /, F ./ D u.T /, where u is a solution of (7.2.1) with initial data u.0/ D . Assume that the
5 fifth order differential of F at the origin D 5 F .0/W H 1 .M / ! H 1 .M / is bounded. Then, for all N1 , PN1 1 2 L2 .M / and 2 ; 3 2 H 1 .M /

kei t g PN1 1 ei t g 2 ei t g 3 kL2 .Œ0;T M / . kPN1 1 kL2 .M / k2 kH 1 .M / k3 kH 1 .M / : Sketch of proof. Using the integral equation, we compute D 5 F .0/.h1 ; : : : ; h5 / Z T X D  12i ei.T t /g H.1/ .t 0 /H.2/ .t/H.3/ .t/H.4/ .t/H.5/ .t/ dt; 0



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S. Herr

where Hj .t/ WD ei t g hj and the sum is restricted to all permutations  2 S5 which give rise to different pairs . .2/;  .4//. Here, we used that DF .0/.h/ D eiTg h and D j F .0/ D 0 for 2 6 j 6 4. Choosing h2 D h3 D h4 D h5 yields X H.1/ H.2/ H.3/ H.4/ H.5/ D 6H1 jH2 j4 C 4H1 H23 H2 : 

From the boundedness of D 5 F .0/ and duality we infer ˇZ ˇ ˇ ˇ D 5 F .0/.h1; h2 ; : : : ; h2 /H1 .T / dx ˇ . kh1 kH 1 kh1 kH ˇ M

1

kh2 k4H 1 :

2

From Ref6jH1 j2 jH2 j4 C 4H1 H23 H2 g > 2jH1 j2 jH2 j4 we conclude that Z

0

T

Z

M

jH1 j2 jH2 j4 dx dt . kh1 kH 1 kh1 kH

1

kh2 k4H 1 :

By polarisation, we obtain the claimed inequality. We refer to [34] for more details, also to the earlier work [15, Rem. 2.12] in the context of cubic Schrödinger equations.  Let us reiterate that the validity of (7.2.3) has been verified for M D T3 [35], M D S3 (and 3d Zoll manifolds) [29], M D T3 [54], and M D S  S2 [34, 55]. 7.2.3 Related results There are other cases which can be treated similarly. For instance, the case of radial initial data in the 3d unit ball with Dirichlet boundary conditions is similar to the case M D S3 , as remarked in [46] and carried out explicitly in the Master’s thesis of Frieda Wall in 2016. Also, concerning critical local well-posedness of equation (7.2.1) for sufficiently large odd integers p, the same strategy based on multilinear estimates applies on, say, spheres and tori, which has been verified in Master’s theses of Jakob Herrenbrück (M D Sd ) and Magnus Winter (M D Td ) in 2016. Concerning the cubic nonlinear Schrödinger equation, which is energy-critical in dimension d D 4, we remark that bilinear estimates play a similar role. With Tataru and Tzvetkov, we established small data global well-posedness in H 1 .T4 / and certain product spaces in [36], see also [41] for the case of the irrational torus. Arguably, the most interesting setting is the L2 -critical cubic nonlinear Schrödinger equation in d D 2, where no critical well-posedness result is known. In [24], Colliander–Keel–Staffilani–Takaoka–Tao proved that there is a low-tohigh frequency cascade leading to growth of Sobolev norms beyond the energy regularity, which supports a conjecture known as weak turbulence. The construction is based on a certain discrete model dynamical system with a Hamiltonian structure. With Marzuola [31], we analyse certain rarefaction wave-like solutions to this system which transfer energy from low to high frequencies. We only mention briefly that the methods described above have been applied to rigorously derive defocusing nonlinear Schrödinger equations from the dynamics of

Nonlinear Dispersive equations at critical regularity

173

quantum many-body systems in the setting of periodic boundary conditions. In a joint paper with Sohinger [33], we establish uniqueness results for the Gross–Pitaevskii hierarchy in dimensions two and three on general rectangular tori, in particular solving an open problem put forward in [42]. Due to limitation of space, we are unable to describe this area of mathematical physics in more detail here and we refer the reader to [42, 33] for more details and extensive lists of references.

7.3 Nonlinear systems on Euclidean space On Euclidean space, the small data theory for nonlinear Schrödinger equations at the critical level of regularity is well-understood since the early nineties, see e.g. [21]. Concerning nonlinear Schrödinger and wave equations, the focus shifted towards the longtime behaviour of large solutions. However, concerning systems of dispersive equations exhibiting more complex nonlinear interactions, many problems regarding critical well-posedness remain open. We will describe some of the recent progress for systems involving nonlinear Dirac equations, the Zakharov system and related problems. 7.3.1 Dirac equations and systems Let M > 0, d D 2; 3 and D D 2 if d D 2 and D D 4 if d D 3. The Pauli matrices are defined as       0 1 0 i 1 0 1 2 3  D ;  D ;  D 1 0 i 0 0 1 and the Dirac matrices  2 CDD in the case d D 2 are given by 0 D  3 ,

1 D i  2 and 2 D i  1 , and in the case d D 3 by     0 j I2 0 j 0 ; j D 1; 2; 3: ;

D

D 0 I2 j 0 They satisfy the anti-commutation relation

  C   D 2g  ID ;

.g  / D diag.1; 1; : : : ; 1/:

Using the summation convention, the cubic Dirac equation for the spinor CD is given by . i  @ C M / D . N / ;

W R1Cd ! (7.3.1)

where N D and denotes the Hermitian adjoint. This equation is the socalled Soler model in quantum field theory for self-interacting Dirac fermions (e.g. electrons) [52]. There are other relevant cubic models such as the Thirring model and all results will hold for this, too, see [13] for explanations. The equation is Lorentz covariant. We consider the associated Cauchy problem by prescribing initial data .0/ in Sobolev spaces. Concerning scaling, it turns out that the equation is d 1 H 2 .Rd /-critical. 0

Ž

Ž

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S. Herr

In joint work with Bejenaru [4, 5] we proved the critical global well-posedness and scattering results in d D 2; 3 for small initial data in the massive case M > 0, and Bournaveas–Candy [13] established this in the massless case M D 0. We summarise these developments as follows. Theorem 7.3.1. Let M > 0 and d D 2; 3. The cubic Dirac equation is globally d 1 well-posed for small initial data in H 2 .Rd / and these solutions scatter to free solutions as t ! ˙1. Let us briefly mention selected previous results. Local well-posedness was obtained in H s .R3 / for s > 1 (subcritical range) by Escobedo–Vega in [26], global well-posedness and scattering was proved by Machihara–Nakanishi–Ozawa [43] for small initial data in H s .R3 /, s > 1 as well as for small initial data in H 1 .R3 / with some regularity in the angular variable in [44]. In d D 2, local well-posedness in H s .R3 /, s > 21 was obtained by Pecher [47]. We refer to [4, 5] for more references. The idea of proof is the following: First, the system is reduced to a system of half-Klein–Gordon equations with null structure in the nonlinearity. Second, global in time nonlinear estimates are derived which allow to invoke the contraction mapping principle. The key ingredient here are certain endpoint Strichartz and energy estimates in adapted systems of coordinates. We now describe the ideas, focusing on the easier case d D 3. By rescaling, it suffices to consider M D 1. Multiplying equation (7.3.1) from the left by the matrix 0 DW ˇ leads to i.@t C ˛
r C iˇ/

D . N /ˇ

where ˛ j D 0 j and ˛
r D ˛ j @j . Here, the operator ˛
r C iˇ is defined via the symbol ˛
 C ˇ. Due to .˛
 C ˇ/2 D .jj2 C 1/I , the matrix ˛
 C ˇ has the eigenvalues ˙hi. Let …˙ ./ denote the projections onto the eigenspaces and …˙ the Fourier multiplier operator. This leads to the equivalent system .i @t C hri/ .i @t

hri/

C

D

D

…C .. N /ˇ /; … .. N /ˇ /

(7.3.2)

D …C C … . ˙ D …˙ , Next, we will discuss the issue of Strichartz estimates for half-Klein–Gordon equations. Let U.t/ D ei t hri . The endpoint estimate for

. kkHP 1 .R3 / kUkL2 .R;L1 3 x .R // t

fails to hold, as for the wave equation, even if b  is localised to some dyadic annulus Aj D f 2 R3 j jj  2j g. Let .j /j 2N0 be a smooth partition of unity subordinate to .Aj /j 2N0 , where A0 is the full unit ball, and Pj the associated Fourier localisation operator. In oder to deal with cubic nonlinearities we construct a useful replacement. As discussed in Section 7.1, we need to understand the decay properties of the kernel Z e˙i.t;x/
.hi;/ 2j ./ d D 2j H .t; x/; Kj .t; x/ D R3

1

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Nonlinear Dispersive equations at critical regularity

of .Pj U / Pj U , where H is the hyperboloid. H has 3 non-vanishing principal curvatures, but for  2 Aj the radial curvature is proportional to 2 2j only. This implies 3 jKj .t; x/j . 23j .1 C 2j jtj/ 1 and jKj .t; x/j . 24j .1 C 2j jtj/ 2 . For any  > 0, . 22j.1C/ and this yields kKj kL1 L1 x t

kUkL2 .R;L1 .R3 // . kkHP 1C .R3 / ; t

x

which is not scale invariant. This can be improved substantially by introducing new coordinates, similar to the ideas of Tataru in the context of the wave maps problem [58]. Let Kj be a covering of S2 by approximately 22j maximally separated caps  of radius 2 j and center !./ and corresponding cone € , and let . /2Kj be a subordinate smooth angular partition of unity, and P  D  b . Consider the angularly localised kernel Z ei.t;x/
.hi;/ 2j ./ ./ d: Kj; .t; x/ D

b

R3

Let  D j D h2 j i, ! D !./ and ‚;! D hi 1 .; !/, ‚? D hi 1 . 1; !/, ;! and t D ‚;
.t; x/; x1 D ‚?
.t; x/ and .t ; x / the associated orthogonal ; coordinates. For any fixed N 2 N, we obtain for 2 2j j.t; x/j  jt j jKj; .t; x/j .N 2j .1 C 2j jt j/

N

which implies kKj; kL1

. 1, from which we obtain the desired Strichartz type estimate: For all j > 0,  2 Kj and  2 L2 .R3 / s.th. supp.b /  Aj we have X kUP kL2 L1 . kkHP 1 : 1 t Lx

t

2Kj

x

In addition, there are corresponding energy estimates in these coordinates: If b  has angular support in another cap  0 and 2 j  ˛ D dist.;  0 /, then kUkL1 L2x . ˛ t



1

kkL2

In the case d D 2, this is much more difficult, as L2t L1 x is the forbidden endpoint for the Schrödinger equation. The strategy is to fix a finite time horizon T > 0 and prove estimates in T -dependent norms, uniformly in T , similar to the Schrödinger maps problem [9]. We omit the details and refer to [5] instead. Now, we come back to the nonlinear system (7.3.2). The nonlinearity exhibits a so-called null structure. It damps down products of waves with parallel frequencies, similar to other nonlinear wave equations arising e.g. in general relativity and geometric wave equations. Using the anti-commutativity relations of the Dirac matrices, one can show that …˙ ./… ./ D O .†.; // C O .hi

1

C hi

1

/

176

S. Herr

One then constructs function spaces N ˙ , S ˙ such that k…˙ kS ˙ . k…˙ .0/kH 1 .R3 / C k.i @t ˙ hri/…˙ kN ˙ k…˙ .h…˙

1 ; ˇ…˙

2 iˇ…˙

3 /kN ˙

.

3 Y

k…˙

mD1

m kS ˙

The function spaces have several components and a multiscale structure. S ˙  P 1 3 2 Cb .R; H .R // is such that k kS ˙ D j >0 22j kPj k2 ˙ . Let us take a glimpse

at its structure if d D 3: One property is that X n 2 2j kP k2L2 L1 C kP k2L2 t

2Kj

Sj

1 t Lx

x

o

. k k2 ˙ Sj

and, if the modulation of is small enough and ` 6 j 10, o X n . k k2 ˙ : 2 2` kP 0 k2L1 L2 kP 0 k2L1 L2 C sup t

 0 2K `

x

t

2K` dist.; 0 /2 `

Sj

x

Let us describe some aspects of the proof, in particular a crucial bilinear estimate. Suppose that supp. m .t//  fjj  2jm g, 2j1 6 2j2 . Then,

1

kh…˙

1 ; ˇ…˙

2 ikL2 .RR3 /

. 2 j1 k

1 kS ˙ k 2 kS ˙ : j1

(7.3.3)

j2

To prove this, we decompose the spinors on the Fourier side into caps of radius 2 for 2`1  2`2 . 2j1 . Due to orthogonality, it suffices to prove X kh…˙ P1 1 ; ˇ…˙ P2 2 ikL2 . 2j1 `1 k 1 kS ˙ k 2 kS ˙ j1

.1 ;2 /2D`1 ;`2

`j

,

j2

where D`1 ;`2 is the set f.1 ; 2 / 2 K`1  K`2 j dist.1 ; 2 /  2

`1

or . 2

`1

if `1  j1 g:

Consider 1 ; 2 of low modulation and 2`1  2`2  2j1 only. The null-structure yields, ignoring terms of lower order, X kh…˙ P1 1 ; ˇ…˙ P2 2 ikL2 .1 ;2 /2D`1 ;`2

. 2

`1

X

X

.1 ;2 /2D`1 ;`2 2Kj1

kP1 \

kP2 1 kL2 L1 t x

2 kL1 L2 x t



which implies (7.3.3) in this case. Given the bilinear estimate (7.3.3), it requires more work to establish the contraction property, but we do not go into further detail here.

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Nonlinear Dispersive equations at critical regularity

Instead, let us briefly discuss some related problems. First of all, the model (7.3.1) is also interesting in d D 1 and critical global well-posedness has been established in [18]. Very recently, Candy–Lindblad [20] proved a modified scattering result in this case. The Dirac equation can be coupled to other equations, which is relevant in physics and gives rise to mathematical challenges. For instance, the Dirac–Klein–Gordon system is a basic model of proton-proton interactions (one proton is scattered in a meson field produced by a second proton) or neutron-neutron interaction. It is i  @

CM

D  ;  C m  D N :

(7.3.4)

2

We will study the Cauchy problem with initial condition . ; ; @t /jt D0 D .

0 ; 0 ; 1 /:

(7.3.5)

The masses M; m > 0 play an important role, as they determine the structure of the set of resonances. Fix the space dimension d D 3. The scale-invariant space is 1

L2 .R3 I C4 /  HP 2 .R3 I R/  HP

1 2

.R3 I R/:

In a joint work with Bejenaru [6], we obtain a result in the full subcritical range. Theorem 7.3.2. Assume that  > 0 and 2M > m > 0. Then, (7.3.4) is globally well-posed for small initial data 0

2 H  .R3 I C4 /;

1

.0 ; 1 / 2 H 2 C .R3 I R/  H

1 2 C

.R3 I R/

and these solutions scatter to free solutions for t ! ˙1. The condition 2M > m > 0 guarantees that there are no nontrivial resonances. The proof is based on Fourier-localised Strichartz estimates. Further, we exploit the null-structure of this system, which is akin to the one described above for the cubic Dirac equation, in conjunction with a careful analysis of the set of resonances. In joint work with Candy [19], we work in scale-invariant subspaces of the critical space, where we impose additional angular regularity via the vector fields ij D xi @j

xj @i :

In [19] we prove the following. 7 . Then, Theorem 7.3.3. Let 2M > m > 0 and  > 0, or m > 2M > 0 and  > 30 the system (7.3.4) is globally well-posed, provided that the initial data satisfy

hi . ;  ;  / 1  1; 1 0 0 1 2 2 2 L H

H

and these solutions scatter to free solutions as t ! ˙1.

178

S. Herr

Here, we allow for resonances in the regime m > 2M > 0. These lead to slow oscillations in the Duhamel integral, hence to weaker decay. However, we observe that in resonant interactions the characteristic surfaces intersect transversally. The key idea now is to use a novel version [19] of the bilinear Fourier restriction theory [57], which allows to prove that for any 32 < p 6 2, kU1
U2 kLp .R1C3 / .

   p4 ˛

2

k1 kL2 .R3 / k2 kL2 .R3 /

c b b provided that supp  j  fjj  g, †.supp 1 ; supp 2 /  ˛. For certain p, such estimates also follow from Hölder’s inequality and standard Strichartz estimates (only relying on curvature), but the above goes beyond this in the sense that one can choose smaller p. In fact, in [19] we prove such estimates for more general surfaces under appropriate transversality, curvature and regularity assumptions. Furthermore, we prove that these estimates extend to V 2 -perturbations of free solutions, which is important for an application to nonlinear systems. The problem (7.3.4) is also of interest in other spatial dimension. In particular, in dimension d D 1, its local well-posedness theory is quite well-understood. Recently, Selberg–Tesfahun [51] proved that solutions are real analytic. One of the most challenging open problems seems to be the longtime behaviour of solutions in dimension d D 2, as the decay of free solutions marks the border of short-range and long-range scattering. Finally, let us mention a couple of related results obtained in our group. A more complex system is the Yang–Mills equation. Here, one needs to fix a gauge. With refined Fourier-analytic methods Tesfahun [59] recently proved a local well-posedness result in the Lorenz gauge below the energy regularity. Simpler models are semi-relativistic Hartree equations p . i @t C m2 /u D .V  juj2 /u with Coulomb or Yukawa potential V .x/ D e jxj jxj 1 and mass m > 0, where  denotes spatial convolution. Formally, they can be obtained by ignoring the vector structure in (7.3.4) and the second order time derivatives in the Klein–Gordon equation. With Lenzmann [30], we obtain sharp results on local well-posedness for radial and for non-radial initial data in dimension d D 3. With Tesfahun [37] we extended this to small data global well-posedness and scattering for the Yukawa potential ( > 0), while it has been shown by Pusateri [48] that there is modified scattering for the Coulomb potential  D 0. We would like to refer to [30, 37] and the references therein for a more complete account on this equation. 7.3.2 Waves in plasmas In this subsection, we will review recent results on the Zakharov system i @t u C u D nu @2t n

n D juj2

(7.3.6)

179

Nonlinear Dispersive equations at critical regularity

with an initial condition .u; n; @t n/jt D0 2 H s .Rd /  H  .Rd /  H 

1

.Rd /:

This system is a simplified model for Langmuir oscillations in a plasma, where u W R1Cd ! C denotes the (scalar) electric field envelope and n W R1Cd ! R the ion density fluctuation. Due to the coupling of a Schrödinger and a wave equation, there is no scaling heuristic to determine critical regimes. In previous joint work, we established well-posedness results in d D 2 [7] and in d D 3 [3] in low regularity spaces which were optimal in some sense. Recently, normal form methods have been successfully applied in dimension d D 3 by Guo and Nakanishi to prove global well-posedness and scattering in the energy space. From a scaling point of view, the space dimension d D 4 is the most interesting. In a joint paper with Bejenaru–Guo– Nakanishi [2], we proved small data global well-posedness and scattering results in dimension d D 4. Further, we extended the region of local well-posedness significantly. In the Master’s theses of Anne Rubel and of Melissa Meinert in 2016 this could be extended to other dimensions and was partly checked for optimality by showing unboundedness of the flow map in certain regimes. A recent joint paper with Schratz [32] goes in a different direction. We construct a numerical time-discretisation scheme for the Zakharov system and prove its convergence (with rates) without imposing a so-called CFL-condition, i.e. without assuming that the solutions have compact Fourier-support for all times. The proof of convergence is challenging in this setting because there is a derivative in the nonlinearity. A model to describe the unidirectional propagation of waves in a magnetised plasma is the Zakharov–Kuznetsov equation @t u C @3x u C @x y u D @x .u2 / ; .x; y/ 2 R  Rd with initial condition

1

;

(7.3.7)

u.0/ D  2 H s .Rd /

for d D 2; 3. It is a generalisation of the Korteweg-de Vries equation to higher dimensions. In a joint work with Grünrock [27], we proved a local well-posedness result if d D 2, and shortly after, Molinet–Pilod proved further results in d D 2 and global well-posedness in some Besov space in d D 3 in independent work [45]. However, this simple model will provide further challenges for the future, mostly due to the fact the the set of resonances is quite complex.

References [1] V.M. Babiˇc, Eigenfunctions which are concentrated in the neighborhood of a closed geodesic. Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 9 (1968), 15–63. [2] I. Bejenaru, Z. Guo, S. Herr and K. Nakanishi, Well-posedness and scattering for the Zakharov system in four dimensions. Anal. PDE 8 (2015), 2029–2055.

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[3] I. Bejenaru and S. Herr, Convolutions of singular measures and applications to the Zakharov system. J. Funct. Anal. 261 (2011), 478–506. [4] I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in H 1 .R3 /. Commun. Math. Phys. 335 (2015), 43–82. 1

[5] I. Bejenaru and S. Herr, The cubic Dirac equation: small initial data in H 2 .R2 /. Commun. Math. Phys. 343 (2016), 515–562. [6] I. Bejenaru and S. Herr, On global well-posedness and scattering for the massive Dirac–Klein– Gordon system. J. Eur. Math. Soc., 19 (2017), 2445–2467. [7] I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with L2 Schrödinger data. Nonlinearity 22 (2009), 1063–1089. [8] I. Bejenaru, S. Herr and D. Tataru, A convolution estimate for two-dimensional hypersurfaces. Rev. Mat. Iberoam. 26 (2010), 707–728. [9] I. Bejenaru, A. Ionescu, C. Kenig and D. Tataru, Global Schrödinger maps in dimensions d > 2: small data in the critical Sobolev spaces. Ann. of Math. 173(2) (2011), 1443–1506. [10] J. Bourgain, On ƒ.p/-subsets of squares. Israel J. Math. 67 (1989), 291–311. [11] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107–156. [12] J. Bourgain and C. Demeter, The proof of the l 2 decoupling conjecture. Ann. of Math. 182(2) (2015), 351–389. [13] N. Bournaveas and T. Candy, Global well-posedness for the massless cubic Dirac equation. Int. Math. Res. Not. (IMRN) (2016), 6735–6828. [14] N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), 569–605. [15] N. Burq, P. Gérard and N. Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations. Ann. Sci. École Norm. Sup. 38(4) (2005), 255–301. [16] N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. Invent. Math. 159 (2005), 187–223. [17] N. Burq, P. Gérard and N. Tzvetkov, Global solutions for the nonlinear Schrödinger equation on three-dimensional compact manifolds. In Mathematical Aspects of Nonlinear Dispersive Equations (J. v. Bourgain, C.E. Kenig, and S. Klainerman, eds.), Ann. of Math. Stud. 163, Princeton Univ. Press, Princeton, NJ, 2007, 111–129. [18] T. Candy, Global existence for an L2 critical nonlinear Dirac equation in one dimension. Adv. Diff. Equations 16 (2011), 643–666. [19] T. Candy and S. Herr, Transference of bilinear restriction estimates to quadratic variation norms and the Dirac–Klein–Gordon system. Anal. PDE 11 (2018), 1171–1240. [20] T. Candy and H. Lindblad, Long range scattering for cubic Dirac equation on R1C1 . Differ. Integral Equ., 31 (2018), 507–518. [21] T. Cazenave, Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics 10. AMS, Providence, RI, 2003. [22] Y. Colin de Verdière, Sur le spectre des opérateurs elliptiques a bicaractéristiques toutes périodiques. Comment. Math. Helv. 54 (1979), 508–522.

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[23] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3 . Ann. of Math. 167(2) (2008), 767–865. [24] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math. 181 (2010), 39–113. [25] J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29 (1975), 39–79. [26] M. Escobedo and L. Vega, A semilinear Dirac equation in H s .R3 / for s > 1. SIAM J. Math. Anal. 28 (1997), 338–362. [27] A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov–Kuznetsov equation. Discr. Cont. Dynam. Syst. 34 (2014), 2061–2068. [28] M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space. Ann. Inst. Henri Poincaré (C) Anal. Non Linéaire 26 (2009), 917–941. Erratum ibid. 27 (2010), 971–972. [29] S. Herr, The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds. Amer. J. Math. 135 (2013), 1271–1290. [30] S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity. Nonlinear Anal. 97 (2014), 125–137. [31] S. Herr and J.L. Marzuola, On discrete rarefaction waves in an NLS toy model for weak turbulence. Indiana Univ. Math. J. 65 (2016), 753–777. [32] S. Herr and K. Schratz, Trigonometric time integrators for the Zakharov system. IMA J. Numer. Anal. 37 (2017), 2042–2066. [33] S. Herr and V. Sohinger, The Gross–Pitaevskii hierarchy on general rectangular tori. Arch. Ration. Mech. Anal. 220 (2016), 1119–1158. [34] S. Herr and N. Strunk, The energy-critical nonlinear Schrödinger equation on a product of spheres, Math. Res. Lett. 22 (2015), 741–761. [35] S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in H 1 .T3 /. Duke Math. J. 159 (2011), 329–349. [36] S. Herr, D. Tataru and N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications. J. Reine Angew. Math. (Crelle) 690 (2014), 65–78. [37] S. Herr and A. Tesfahun, Small data scattering for semi-relativistic equations with Hartree type nonlinearity. J. Diff. Eqs. 259 (2015), 5510–5532. [38] A.D. Ionescu and B. Pausader, The energy-critical defocusing NLS on T3 , Duke Math. J. 161 (2012), 1581–1612. [39] C.E. Kenig, Recent developments on the global behavior to critical nonlinear dispersive equations. In Proceedings of the International Congress of Mathematicians. Volume I, Hindustan Book Agency, New Delhi, 2010, 326–338. [40] C.E. Kenig, Critical non-linear dispersive equations: global existence, scattering, blow-up and universal profiles. Jpn. J. Math. 6 (2011), 121–141. [41] R. Killip and M. Visan, Scale invariant Strichartz estimates on tori and applications. Math. Res. Lett. 23 (2016), 445–472.

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[42] K. Kirkpatrick, B. Schlein and G. Staffilani, Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems: the periodic case. Amer. J. Math. 133 (2011), 91–130. [43] S. Machihara, K. Nakanishi and T. Ozawa, Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation. Rev. Mat. Iberoam. 19 (2003), 179–194. [44] S. Machihara, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal. 219 (2005), 1–20. [45] L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), 347–371. [46] B. Pausader, N. Tzvetkov and X. Wang, Global regularity for the energy-critical NLS on S3 . Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 315–338. [47] H. Pecher, Local well-posedness for the nonlinear Dirac equation in two space dimensions. Commun. Pure Appl. Anal. 13 (2014), 673–685; corrigendum ibid. 14 (2015), 737–742. [48] F. Pusateri, Modified scattering for the boson star equation. Commun. Math. Phys. 332 (2014), 1203–1234. [49] J.V. Ralston, On the construction of quasimodes associated with stable periodic orbits. Commun. Math. Phys. 51 (1976), 219–242. [50] J.V. Ralston, Approximate eigenfunctions of the Laplacian. J. Diff. Geom. 12 (1977), 87–100. [51] S. Selberg and A. Tesfahun, On the radius of spatial analyticity for the 1D Dirac–Klein– Gordon equations. J. Diff. Eqs. 259 (2015), 4732–4744. [52] M. Soler, Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy. Phys. Rev. D 1 (1970), 2766–2769. [53] E.M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. With the assistance of Timothy S. Murphy. Princeton University Press, Princeton, NJ, 1993. [54] N. Strunk, Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions. J. Evol. Eq. 14 (2014), 829–839. [55] N. Strunk, Critical Well-posedness Results for Nonlinear Schrödinger Equations on Compact Manifolds. Dissertation, Bielefeld University, 2015. [56] N. Strunk, Global Well-posedness of the Energy-Critical Defocusing NLS on rectangular tori in three dimensions. Differential Integral Eqs. 28 (2015), 1069–1084. [57] T. Tao, Some recent progress on the restriction conjecture, in: Fourier Analysis and Convexity. Appl. Numer. Harmon. Anal., (L. Brandolini, L. Colzani, A. Iosevich and G. Travaglini, eds.), Birkhäuser, MA, 2004, 217–243. [58] D. Tataru, On global existence and scattering for the wave maps equation. Amer. J. Math. 123 (2001), 37–77. [59] A. Tesfahun, Local well-posedness of Yang–Mills equations in Lorenz gauge below the energy norm. Nonlinear Differ. Equ. Appl. 22 (2015), 849–875. [60] L. Thomann, The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces. Bull. Soc. Math. France 136 (2008), 167–193. [61] A. Zygmund, On Fourier coefficients and transforms of functions of two variables. Stud. Math. 50 (1974), 189–201.

Chapter 8

Variational solutions to nonlocal problems M. Kaßmann We present recent results on nonlocal operators acting on real-valued functions that are defined on subsets of Rd . The operators under consideration exhibit a fractional order of differentiability and have attracted a lot of attention in the last twenty years. It turns out that several ideas and approaches developed for the study of partial differential operators of second order can be applied after suitable modifications. In this report, we explain similarities and differences with regard to the well-established theory for differential operators of second order. We concentrate on stationary linear symmetric operators in variational form.1

8.1 Introduction An operator L acting on functions from Rd to R is called local if, for every function u in the domain of L, the support of Lu is contained in the support of u. Examples of local operators are u 7! jruj and u 7! u. An operator L is called nonlocal if it is not local. Convolution operators provide simple examples of nonlocal operators. Here, we will study a small class of nonlocal operators only. We require them to be linear and unbounded on L2 .Rd /. Furthermore, we assume the operators L under consideration to satisfy the global maximum principle, i.e., we assume Lu.x/ 6 0 to hold, whenever the function u has a global maximum in x 2 Rd . A prominent example of such operators is the fractional Laplace operator . /˛=2 for 0 < ˛ < 2. For functions u 2 Cc1 .Rd /, it can be defined via

4

. /˛=2 u./ D jj˛b u./

. 2 Rd /:

(8.1.1)

For our purposes, it is very convenient that there is a different representation of . /˛=2 as an integrodifferential operator. For u 2 Cb2 .Rd / and x 2 Rd , one has Z u.x C h/ u.x/ ˛=2 dh . / u.x/ D C˛;d lim "!0 Rd nB" jhjd C˛ Z u.y/ u.x/ dy D C˛;d p: v: xjd C˛ Rd jy C˛;d Z u.x C h/ 2u.x/ C u.x h/ D dh : 2 jhjd C˛ Rd 1 Projects

A8, A10

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The constant C˛;d can be chosen such that (8.1.1) holds true. It turns out that C˛;d  ˛.2 ˛/ for ˛ 2 .0; 2/. This representation formula has been known for a long time; [34]. We mainly will be concerned with nonlocal operators of the form Z .u.y/ u.x//k.x; y/ dy .x 2 Rd /; (8.1.2) Lu.x/ D p: v: Rd

where k W Rd  Rd ! Œ0; 1 is a symmetric function. Note that one needs to be careful when writing (8.1.2) because the principal value integral might not exist for smooth functions u if k is not chosen accordingly. It is important to note that for several choices of k, the operator L as in (8.1.2) can be realised with the help of the Fourier transform, i.e., one obtains

bLu./ D

.x; /b u./

. 2 Rd /

(8.1.3)

for u 2 Cc1 .Rd /. Here, is a space dependent multiplier, cf. [24]. Since we do not want to assume smoothness of k, and since we want to establish local regularity results, we do not make use of the representation by the Fourier transform. Pseudodifferential operators as in (8.1.3) have been studied for several decades. For the study of such operators in (bounded) domains, the recent article [20] provides several references. Our approach is based on energy forms. Note that, for real-valued functions u; v 2 Cc1 .Rd /, one has Z C˛;d Z Z .u.y/ u.x//.v.y/ v.x// ˛=2 . / uv D dy dx 2 jy xjd C˛ Rd Rd Rd DW Œu; vH ˛=2 : Note that Œu; uH ˛=2 is the seminorm in the Sobolev–Slobodeckij space H ˛=2 .Rd / with differentiability order ˛2 . This bilinear form is a very special instance of forms introduced in (3.2.1) of Chapter 3. We will not Rinvestigate the limit ˛ % 2, but we remark that Œu; uH ˛=2 converges to Œu; uH 1 D Rd jruj2 if u is sufficiently regular. The analogous result holds for subsets of Rd as shown in [5], [29] and [12]. The nonlocal operators considered here have a strong connection to Markov jump processes. We do not dwell on this connection in this chapter but note that Markov jump processes of the same type are central objects in Chapter 3 and Chapter 4. In short, the nonlocal operators arise as generators on L2 .Rd / or Cb .Rd / of the semigroups that are generated by the stochastic process. In the translation invariant k.x y/ for some function e k, the case, e.g., in the case of (8.1.2) with k.x; y/ D e corresponding jump processes are Lévy processes [39]. An important subclass is provided by ˛-stable processes, which—in the simplest case—are generated by the fractional Laplace operator as defined above. Basic definitions and contributions to the potential theory can be found in [3]; see also the references in [4]. Beside Markov jump processes, there are many topics which would fit well into the framework of this report but are not covered, e.g., stationary linear nonlocal symmetric operators in variational form. We do not mention questions related to minimal

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surfaces and the fractional perimeter. Moreover, we do not study semilinear equations, which have attracted attention recently; see the references in [40, 42, 33, 16, 32]. Last, let us mention that nonlocal variational problems arise also in the study of models in peridynamics; see [30, 13, 31].

8.2 Variational solutions to the Dirichlet problem In this section, we provide the basic setup of nonlocal variational problems of fractional order. We explain the approach by Hilbert space methods and concentrate on function spaces. The order of differentiability is given by a number ˛ 2 .0; 2/. Recall that the Sobolev–Slobodeckij space H ˛=2 .Rd / is defined as the subspace of functions v 2 L2 .Rd / such that the seminorm Z Z .v.y/ v.x//2 2 dy dx (8.2.1) ŒvH ˛=2 D ˛.2 ˛/ jx yjd C˛ Rd Rd is finite. The corresponding bilinear form is denoted by Œ
;
. The constant ˛.2 ˛/ in (8.2.1) is important only when considering the limiting behaviour as ˛ & 0 or ˛ % 2. When considering domains   Rd , the following subspaces are of interest. Definition 8.2.1. Assume   Rd is open. Let m be the measure on Rd defined by dx m.dx/ D .1Cjxj/ d C˛ . (i) The function space H˛=2 .Rd / is defined as the subspace of all v 2 H ˛=2 .Rd / such that vjRd n D 0.

(ii) The linear space V ˛=2 .jRd / consists of all v 2 L2 .Rd ; dm/ such that Z Z .v.y/ v.x//2 dy dx < 1: jx yjd C˛  Rd

The space has been used in several papers and has systematically been introduced in [18], see the references in [18], [6] and in [1]. It is not hard to see that H˛=2 .Rd / and V ˛=2 .jRd / are separable Hilbert spaces. If a function u belongs to V ˛=2 .jRd /, then the bilinear form Z Z .u.y/ u.x//.v.y/ v.x// .u; v/ 7! dy dx d d jx yjd C˛ R R is finite for every test function v 2 H˛=2 .Rd /. In a certain sense, the space H˛=2 .Rd / ˛=2 is a nonlocal resp. fractional analogue of H01 ./, whereas .jRd / corresponds R V 1 to H ./ when considering classical local energy forms  jruj2 . A particular property of V ˛=2 .jRd / is that it encodes regularity across the boundary @.

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Example 8.2.2. Assume  D B1 .0/  Rd . For ˇ 2 R, define g W Rd ! R by ( .jxj 1/ˇ ; if 1 6 jxj < 2; g.x/ D 0; elsewhere: Then, g 2 V ˛=2 .jRd / if and only if ˇ >

˛ 1 . 2

Note that the condition v 2 H˛=2 .Rd / carries some regularity information about the behaviour of u across the boundary, too. The finiteness of the term  Z Z jx yj d ˛ dy dx v 2 .x/ Rd n



implies, provided @ is sufficiently regular, Z v 2 .x/ dist.x; @/ 

˛

dx < 1:

This resembles the classical Hardy inequality for functions v 2 H01 ./. We are now in the position to formulate the concept of weak solutions for nonlocal operators. Assume that k W Rd  Rd n diag ! Œ0; 1/ is a measurable function that satisfies k.x; y/ D k.y; x/ for all x; y 2 Rd together with .2 jx

˛/ƒ 1 .2 ˛/ƒ 6 k.x; y/ 6 d C˛ yj jx yjd C˛

.x; y 2 Rd /

for some constant ƒ > 1. We define an operator u 7! Lu by Z .u.y/ u.x//k.x; y/ dy Lu.x/ D p: v: d ZR .u.y/ u.x//k.x; y/ dy D lim

(8.2.2)

(8.2.3)

"!0 Rd nB .x/ "

for all functions u W Rd ! R for which the principal value integral exists. In the case k.x; y/ D jx yj d ˛ , the integral exists for every x 2 Rd if u 2 Cb2 .Rd /. In the same informal way, we define the carré du champ operator € and the corresponding bilinear form E as follows, 1 .L.uv/ uLv v Lu/ 2Z 1 .u.y/ u.
//.v.y/ D 2 Rd Z €.u; v/.x/ dx: E .u; v/ D

€.u; v/ D

v.
//k.
; y/ dy;

Rd

Note that symmetry of k and (8.2.2) do not imply that the expression Lu.x/, as defined above, exists if u is smooth, say u 2 Cc1 .Rd /.

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Definition 8.2.3. Assume that   Rd is open. Assume f belongs to the dual space ˛=2 ˛=2 H .Rd / . We say that a function u 2 H .Rd / is a weak solution to

Lu D f in  if

E .u; '/ D .f; '/ 8' 2 H˛=2 .Rd /: Given the operator L and a function f W  ! R, the task to find a function u W Rd ! R with the properties

Lu D f u D 0

in  on Rd n 

(8.2.4)

is called Dirichlet problem with complement data zero. The definition above provides the notion of weak solutions for this problem. On the one hand, the concept of weak solutions is important because it allows to study the equation Lu D f even if Lu.x/ does not exist. On the other hand, weak solutions naturally appear when studying minimisers of variational functionals. Good early sources on the Dirichlet problem for nonlocal operators in domains include [21, 23]. Define I ˛=2 W H˛=2 .Rd / ! R by Z Z 1 ˛=2 .v.y/ v.x//2 k.x; y/ dy dx .f; v/ : I .v/ D 4 Rd Rd Proposition 8.2.4. (i) Every minimiser of I ˛=2 satisfies (8.2.4) in the weak sense. (ii) If  is bounded, then there is a unique minimiser of I ˛=2 . The proof of this result is standard. For the proof of (i), one considers the map t 7! I ˛=2 .u C t'/ for a minimiser u and some ' 2 H˛=2 .Rd /. When rewriting d ˛=2 I .u C t'/jt D0 D 0; dt one obtains E .u; '/ D .f; '/, which is a weak formulation of (8.2.4). The proof of (ii) makes use of the fact that the bilinear form E is coercive on H˛=2 .Rd /. This property ˛=2 itself follows from a nonlocal Poincaré–Friedrichs inequality. For v 2 H .Rd /, one has  Z Z Z Z 2 2 k.x; y/dy dx .v.y/ v.x/// k.x; y/ dy dx C 2 v .x/ E .v; v/ D Rd n    Z > c1 v 2 .x/ dx; 

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where c1 > 0 is a constant, which depends on  and on ˛ 2 .0; 2/. One can modify the proof such that c1 is independent of ˛ for ˛ > ˛0 > 0, but that is a bit tricky. Let us formulate the weak solution concept for the inhomogeneous Dirichlet problem, i.e., with prescribed data on the complement of the domain   Rd under consideration. Definition 8.2.5. Assume that   Rd is open. Assume f 2 H˛=2 .Rd / together with g 2 V ˛=2 .jRd /. We say that a function u 2 V ˛=2 .jRd / is a weak solution to

Lu D f

u D g

if u

in ; on Rd n 

˛=2

g 2 H .Rd / and

E .u; '/ D .f; '/ 8' 2 H˛=2 .Rd / : We refer the reader to [18] and to [15] for the setup of the Dirichlet problem and the study of related nonlocal extension and trace theorems. The issue of trace theorems is a bit special because, in the nonlocal context, “traces” are not supported on lower dimensional subsets. The “trace space” corresponding to the space V ˛=2 .jRd / is the space of all functions v 2 L2 .Rd n ; dm/ such that the quantity Z Z .v.x/ v.y//2 dx dy yj C ıx C ıy /d C˛ c c .jx is finite, where ız D dist.z; ı/ and   Rd is a bounded domain with a Lipschitzcontinuous boundary. Extension results and “trace” properties are established in [15]. Note that, in this report, we do not address the tedious question of regularity of solutions up to the boundary of the domain under consideration. The approach via Fourier analysis is explained in [20]. The boundary Harnack inequality for the fractional Laplace is established in [2, 44]. Boundary regularity has been approached for a large class of nonlocal operators using the concept of viscosity solutions; see [35, 38, 36, 37]. There are many results on boundary regularity available for classes such as generators of subordinate Brownian motion or unimodal Lévy processes. Let us remark that parabolic equations, i.e., equations of the form @t u

Lu D f;

can be dealt with in a similar way. In this report, however, we concentrate on stationary equations. Hölder regularity results for variational solutions to parabolic nonlocal problems have been established in [7] and [17], where the first article also treats nonlinear problems and the second one provides a robust result, which is truly local. For corresponding results for fully nonlinear equations and viscosity solutions, see [9] and the references in [9] and [28].

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189

8.3 Ellipticity and coercivity of nonlocal operators In the previous section, we have introduced the operator u 7 ! Lu, given by Z .u.y/ u.x//k.x; y/ dy; (8.3.1) Lu.x/ D p: v: Rd

where k was assumed to be symmetric and to satisfy (8.2.2). We have chosen this framework because it is quite simple but still wide enough for several interesting phenomena to occur. In this section, we comment on possible modifications of this setup. Condition (8.2.2) can be understood as imposing “bounded uniformly elliptic coefficients”. For given ˛ 2 .0; 2/, the condition implies .u 2 Cc1 .Rd // ;

E .u; u/  Œu; u2H ˛=2.Rd /

(8.3.2)

i.e., the ratio of the two quantities is bounded from above and below by some positive constants independent of u. The operator L as given by (8.3.1) is special because, for fixed x, the integral is taken with respect to a measure k.x; y/dy, which has a density with respect to Lebesgue measure. There are several applications of nonlocal operators, where one studies more general measures. Thus, it is desirable to consider measures .x; dy/ instead of k.x; y/dy. We consider only the symmetric case. Recall that a family of measures .x; dy/ is called symmetric if for all measurable sets A; B  Rd Z Z Z Z .x; dy/ dx: .x; dy/ dx D A

B

B

A

Let us formulate a stronger condition than (8.3.2) that takes into account all spatial scales. Given .x; dy/, a set M  Rd and a function v 2 L2 .M /, we set Z Z  .v.y/ v.x//2 .x; dy/dx : EM .v; v/ D M

M

The following condition turns out to be important: There is a constant A > 1 such that for every ball B  Rd and every v 2 L2 .B/ Z Z .v.y/ v.x//2   (8.3.3) dy dx 6 A EB .v; v/ : A 1 EB .v; v/ 6 d C˛ jy xj B B Property (8.3.3) ensures several important regularity results for solutions to nonlocal equations such as Hölder regularity estimates. In this section, we will study the question whether and how (8.2.2) can be relaxed without losing the property (8.3.3). Since the quantities of (8.3.3) are defined with the help of double integrals, there is ample space for generalisations. Here is a first example.

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ˇ P Example 8.3.1. Assume V D fx 2 Rd ˇ jxd j2 > idD11 jxi j2 g. Let k be symmetric with ƒ 1 .2 ˛/ 1 .x jx yjd C˛ V

y/ 6 k.x; y/ 6

ƒ.2 ˛/ jx yjd C˛

.x; y 2 Rd a:e:/

for some ƒ > 1; ˛ 2 .0; 2/. Set .x; dy/ D k.x; y/dy. Then, (8.3.3) holds true. One can prove the assertion in the example as follows. For every pair .x; y/ for which k.x; y/ equals zero, there is a point z 2 Rd with jz xj  jz yj such that both k.x; z/ and k.z; y/ are comparable with jx yj d ˛ . Thus, one can prove (8.3.3) in this case using a simple chaining argument and the Fubini theorem. It is possible to relax the bound in (8.2.2) significantly further, still keeping (8.3.3). For example, the cone V could be assumed to depend on the respective pair .x; y/. Then, the problem becomes related to the study of the Boltzmann equation; compare with [22]. The connection of kinetic equations and regularity questions for nonlocal operators seems to be very promising. Many interesting cases of operators Z .u.y/ u.x// .x; dy/; (8.3.4) L u.x/ D p: v: Rd

d satisfy R .x; dy/  .dy fxg/, where  is a Borel measure on R n f0g such that min.1; jhj2 / .dh/ is finite. Here, the symbol  denotes comparability after integrating arbitrary non-negative measurable functions. Note that, for every operator u 7! L u of the form Z  .u.x C h/ u.x// .dh/; L u.x/ D p: v: Rd

there is a corresponding Lévy jump process. The quantity .A/ describes the number of jumps of this process from x into the set x C A within a unit time interval; see [39]. An interesting example is given by .dh/ D m.dh/ with m.dh/ D

d X i D1

jhi j

1 ˛

dhi

Y

j ¤i

ıf0g .dhj / :

(8.3.5)

Note that m only charges those sets which have an intersection with one of the axes in Rd . If L is as in (8.3.4) with .x; dy/ D m.dy fxg/ and m as in (8.3.5), the corresponding energy form E again can be shown to satisfy the comparability condition (8.3.3), cf. Theorem 8.3.2. The weaker condition (8.3.2) can be verified by looking at the Fourier transform. For the above choice of , for functions u 2 Cc1 .Rd / and  2 Rd ,

b

.Lm u/./ D c˛

d X i D1

 u./ ji j˛
b

(8.3.6)

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191

P for some appropriate constant c˛ > 0. Since the multiplier diD1 ji j˛ is comparable with jj˛ , the comparability result (8.3.2) for the quadratic forms follows. The approach by the Fourier transform is not helpful if one wants to verify (8.3.3). Let us mention that the general case .x; dy/  .dy fxg/ with  a Borel measure on R Rd n f0g satisfying min.1; jhj2 / .dh/ < 1 is challenging and several interesting questions are open until now. We have seen two examples of families ..x; dy//x2Rd that are not absolutely continuous with a density satisfying (8.2.2), but they still lead to (8.3.3). It is a challenging task to find a natural condition on , which is as general as possible, still allowing for (8.3.3). Let us explain a recent result in this direction. We recall that a measure  on Rd is called an ˛-stable measure for ˛ 2 .0; 2/, if Z 1 Z 1E .r /r 1 ˛ dr .d / .E/ D ˛.2 ˛/ Sd

1

0

for every Borel set E  Rd , where  is some finite measure on S d 1 ; see [39] for details. The measure is called nondegenerate if the span of supp ./ equals the whole space Rd . The following theorem provides satisfactory sufficient conditions for (8.3.3). It is a special case of one of the main results in [14]. Theorem 8.3.2. Let  D ..x; dy//x2Rd be a symmetric family of measures on Rd and ˛ 2 .0; 2/. Let  be a nondegenerate ˛-stable measure. Assume that, for some ƒ > 1, Z Z Z 1 ƒ f .x; x C z/ .dz/ 6 f .x; y/ .x; dy/ 6 ƒ f .x; x C z/ .dz/;

(8.3.7)

holds true for every x 2 Rd and every measurable non-negative function f . Then (8.3.3) holds true.

8.4 (Weak) Harnack inequalities, and Hölder regularity In this section, we study local regularity results for nonlocal operators. We explain analogies and differences between the cases of local and nonlocal operators. We assume L to be given as in (8.2.3), with condition (8.2.2) in place. One key local regularity result in the study of partial differential equations is the Harnack inequality. Assume  and 0 are open bounded subsets of Rd with 0 b . Then there is a positive constant c1 such that, for every function u W  ! R which is positive and harmonic in , the following holds, u.x/ 6 c1 u.y/

.x; y 2 0 / :

(8.4.1)

The Harnack inequality holds not only for harmonic functions, i.e., for the Laplace operator, but for a large class of elliptic differential operators. An important example

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is the operator u 7! @x@ .aij .
/ @x@ u/, where the functions aij W Rd ! R are bounded i j and the matrix .aij / is uniformly positive definite. For purposes of regularity theory, a weaker statement than (8.4.1) is often sufficient. The weak Harnack inequality asserts that supersolutions u to an elliptic equation in  satisfy inf00 u > c0 

Z

0

juj

p0

1=p0

;

(8.4.2)

where p0 and c0 are positive constants and 00 is another domain whose closure is contained in 0 . An important consequence of the Harnack inequality resp. its weaker version (8.4.2) are regularity estimates for u in Hölder spaces, cf. [19]. Inequalities (8.4.1) and (8.4.2) cannot be expected to hold for solutions to nonlocal equations without further changes. Let u W Rd ! R satisfy . /˛=2 u D 0 in B2 D B2 .0/  Rd for some ˛ 2 .0; 2/. The additional assumption u > 0 in B2 is not sufficient for (8.4.1) or (8.4.2) to hold, because negative values of u in regions outside of B2 influence the values of u inside of B2 . It is not complicated to construct an example of a bounded function u 2 C.Rd / that solves . /˛=2 u D 0 in B2 , is positive in B2 n f0g, and satisfies u.0/ D 0, cf. Theorem 3.3.1 in [6] or [26]. Let us formulate some recent results that make clear how one can deal with the influence of nonlocal terms. In the case of the fractional Laplace operator, the Harnack inequality can be formulated as follows: Proposition 8.4.1. There is a constant c > 1 such that, for 0 < ˛ 6 2 and u 2 C.Rd / with . /˛=2 u D 0 in B2 and u > 0 in B2 , the following inequality holds, Z  u .z/  u.x/ 6 c u.y/ C ˛.2 ˛/ dz .x; y 2 B1 /: (8.4.3) d C˛ Rd jzj Here, u D min.u; 0/. The proof of this result is straightforward and uses the representation of the Poisson kernel for . /˛=2 ; see [14]. Note that the influence of the nonlocal term can nicely be seen in (8.4.3). This influence vanishes for ˛ ! 2 , as it should. Next, let us look at the general case, i.e., we define L as in (8.2.3) and assume (8.2.2) to hold. If one aims at a more general result, one can assume L to be as in (8.3.4) with condition (8.3.3) in place. Given a function f 2 L2 ./ on a bounded open subset   Rd , we say that u 2 V ˛=2 .jRd / is a weak subsolution to Lu D f in  if, for every ' 2 H˛=2 .Rd / with ' > 0, one has “ E .u; '/ WD .u.y/ u.x//.'.y/ '.x//k.x; y/ dy dx 6 .f; '/: In this case, we say that Lu 6 f in  holds in the weak sense. The notion of supersolution and solution is defined accordingly. The nest result states that the weak Harnack inequality holds true for functions u that satisfy Lu > f in the weak sense. It is one of the main results of [14].

193

Variational solutions to nonlocal problems

Theorem 8.4.2. Assume 0 < ˛0 6 2 and ˛0 6 ˛ < 2. Let p0 > 0. Assume further f 2 L2 .Rd / and u 2 V ˛=2 .B2 jRd /, u > 0 in B2 and E .u; '/ > .f; '/ for every .Rd /, ' > 0. Then, ' 2 HB˛=2 2 inf .u/ > c1

B1 2

Z

p0

u.x/ dx

B1

1=p0

c2

Z

Rd nB2

u .z/ jzjd C˛

kf kL2 .B

15 16

/

;

with positive constants c1 ; c2 , which are independent of ˛; u and f . There are two important features of this result. The result is truly local in the sense that Lu > f is required to hold only in a ball, not in the whole space. Moreover, the constants c1 ; c2 do not depend on ˛. At least informally, one can consider the limit ˛ % 2 and recover the weak Harnack inequality for local differential operators of second order with bounded measurable coefficients. The proof of the weak Harnack estimate is very similar to the one in the classical case developed by Moser. One uses test functions of the form ' D u pk
k , where .pk / is a sequence of numbers tending to 1 and .k / is a sequence of localising or cutoff functions with supp.k / & B 1 . The critical case pk D 1 can be dealt with using a modification of the John– 2 Nirenberg lemma or using ideas of Bombieri–Giusti. Note that [10] establishes two similar results: (a) the weak Harnack inequality for nonlocal operators of the form (8.2.3) under the condition (8.2.2), and (b) a Harnack inequality with a tail term in the same context. Both results are established in the more general nonlinear context of p fractional Laplace operators. The proof in [10] is based on the method of De Giorgi and Stampacchia rather than on the method of Moser. The main contribution of [10] is the concise study of the nonlocal influence on the resulting inequalities such as the Caccioppoli inequality. The most important implication of Theorem 8.4.2 are regularity estimates in Hölder spaces. Let us formulate this result in a simple case. Theorem 8.4.3. Assume 0 < ˛0 6 ˛ < 2. Assume that u 2 V ˛=2 .B2 jRd / is a weak solution to Lu D 0 in B2 . Then, the following Hölder estimate holds for almost every x; y 2 B1 , ju.x/

u.y/j 6 ckuk1 jx

yjˇ ;

(8.4.4)

where c > 1 and ˇ 2 .0; 1/ are constants which depend only on d; ˛0 and ƒ from (8.2.2). In particular, they are independent of ˛. Theorem 8.4.3 is proved in [25]. The nonlinear case of operators, which are comparable to the p-Laplace operator, is covered in [11]. Note that, in the latter article, the concept of solutions is slightly different and not all estimates are robust. The main idea of the proof in both articles is to deduce the regularity estimate from the weak Harnack inequality. Note that (8.4.4) is robust in the sense that the constants stay bounded for ˛ % 2. Several extensions of this result have been established.

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Theorem 1.3 of [14] proves (8.4.4) for L as in (8.3.4) and .x; dy/  .dy fxg/ for some nondegenerate stable measure . Let us briefly mention related results for nonlocal operators that are in non-divergence form. Linear examples of such operators u 7! Au are provided by Z u.x C h/ 2u.x/ C u.x h/ Au.x/ D a.x; h/ dh; jhjd C˛ Rd where a W Rd Rd ! .0; 1/ is bounded between two positive constants and satisfies a.x; h/ D a.x; h/ for all x; h. Regularity conditions on a are not imposed. Note that, unlike L as in (8.2.3), the operator A is not symmetric in L2 .Rd / in general. The condition a.x; h/ D a.x C h; h/ would imply that A can be represented as in (8.2.3) with k.x; y/ D 21 a.x; y x/. However, assuming this condition in addition to a.x; h/ D a.x; h/ is too restrictive. First regularity results for non-divergence form operators in the spirit of Theorem 8.4.3 are proved in [43]. The proof is rather short and uses comparison functions. A robust version of this result for a large class of linear and nonlinear operators in non-divergence form is established in [8]. Anisotropic cases have been studied in [27]. The reader is referred to [41] for ultimate results and references in this direction.

References [1] N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, preprint, arXiv:1710.11567. [2] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), 43–80. [3] K. Bogdan and T. Byczkowski, Potential theory for the ˛-stable Schrödinger operator on bounded Lipschitz domains, Studia Math. 133 (1999), 53–92. [4] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondraˇcek, Potential Analysis of Stable Processes and its Extensions, (P. Graczyk and A. Stos, eds.), LNM 1980 Springer, Berlin, 2009, 194. [5] J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for W s;p when s " 1 and applications, J. Anal. Math. 87 (2002), 77–101. [6] C. Bucur and E. Valdinoci, Nonlocal diffusion and applications. In Lecture Notes of the Unione Matematica Italiana 20, Springer, Cham, 2016. [7] L. Caffarelli, C.H. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Amer. Math. Soc. 24 (2011), 849–869. [8] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Commun. Pure Appl. Math. 62 (2009), 597–638. [9] H.A. Chang-Lara and D. Kriventsov, Further time regularity for nonlocal, fully nonlinear parabolic equations, Commun. Pure Appl. Math. 70 (2017), 950–977. [10] A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), 1807–1836.

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[11] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), 1279–1299. [12] J. Dávila, On an open question about functions of bounded variation, Calc. Var. Part. Diff. Equations 15 (2002), 519–527. [13] Q. Du, M. Gunzburger, R.B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54 (2012), 667–696. [14] B. Dyda and M. Kassmann, Regularity estimates for elliptic nonlocal operators, Analysis of PDEs, to appear, arXiv:1509.08320. [15] B. Dyda and M. Kassmann, Function spaces and extension results for nonlocal Dirichlet problems, J. Funct. Anal. (2018), in press. [16] M.M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half-space, Commun. Contemp. Math. 18 (2016), 1550012 (25 pp.). [17] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Commun. Partial Diff. Eq. 38 (2013), 1539–1573. [18] M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z. 279 (2015), 779–809. [19] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. [20] G. Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528. [21] W. Hoh and N. Jacob, On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Funct. Anal. 137 (1996), 19–48. [22] C. Imbert and L. Silvestre, The weak Harnack inequality for the Boltzmann equation without cut-off, to appar in J. Europ. Math. Soc., arXiv:1608.07571 [23] N. Jacob, Various approaches to the Dirichlet problem for non-local operators satisfying the positive maximum principle, in: Potential theory—ICPT 94 (Kouty, 1994), (J. Král, J. Lukeš, I. Netuka and J. Vselý, eds.), de Gruyter, Berlin (1996), pp. 367–375. [24] N. Jacob and R.L. Schilling, Lévy-type processes and pseudodifferential operators. In Lévy Processes (O.E. Barndorff-Nielsen, S.I. Resnick, and T. Mikosch eds.), Birkhäuser Boston, MA, 2001, 139–168. [25] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Diff. Eq. 34 (2009), 1–21. [26] M. Kassmann, The classical Harnack inequality fails for nonlocal operators Sonderforschungsbereich 611 preprint no. 360, University of Bonn, 2007. [27] M. Kassmann, M. Rang and R.W. Schwab, Integro-differential equations with nonlinear directional dependence, Indiana Univ. Math. J. 63 (2014), 1467–1498. [28] M. Kassmann and R.W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.) 5 (2014), 183–212. [29] V. Maz’ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 (2002), 230–238. [30] T. Mengesha and Q. Du, The bond-based peridynamic system with Dirichlet-type volume constraint, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), 161–186.

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[31] T. Mengesha and Q. Du, On the variational limit of a class of nonlocal functionals related to peridynamics, Nonlinearity 28 (2015), 3999–4035. [32] G. Molica Bisci and J. Mawhin, A Brezis–Nirenberg type result for a nonlocal fractional operator, J. London Math. Soc. 95 (2017), 73–93. [33] G. Molica Bisci, V.D. Radulescu and R. Servadei, Variational Methods for nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016. [34] M. Riesz, Intégrales de Riemann–Liouville et potentiels, Acta Litt. Sci. Szeged 9 (1938), 1–42. [35] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. 101 (2014), 275–302. [36] X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations, Duke Math. J. 165 (2016), 2079–2154. [37] X. Ros-Oton and J. Serra, Boundary regularity estimates for nonlocal elliptic equations in C 1 and C 1;˛ domains, Ann. Mat. Pura Appl. 196 (2017), 1637–1668. [38] X. Ros-Oton and E. Valdinoci, Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains, Adv. Math. 288 (2016), 732–790. [39] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 2013. [40] O. Savin and E. Valdinoci, Density estimates for a nonlocal variational model via the Sobolev inequality, SIAM J. Math. Anal. 43 (2011), 2675–2687. [41] R.W. Schwab and L. Silvestre, Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE 9 (2016), 727–772. [42] R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898. [43] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 (2006), 1155–1174. [44] R. Song and J.-M. Wu, Boundary Harnack principle for symmetric stable processes, J. Funct. Anal. 168 (1999), 403–427.

Chapter 9

Spectral and arithmetic structures in aperiodic order M. Baake, F. Gähler, C. Huck and P. Zeiner Systems with aperiodic order can display a variety of arithmetic, combinatorial and spectral phenomena, some of which are reviewed and discussed here. At the same time, the underlying compact tiling spaces can be compared via their topological and spectral invariants. The latter are explicitly computable for substitution systems and provide an important tool for their classification.1

9.1 Introduction This contribution covers selected results from the theory of aperiodic order, with special emphasis on spectral structures. Also covered are topological aspects of tiling spaces with mixed spectrum and combinatorial problems around lattice enumeration problems. A running theme will be that and how systems without any periodicity extend the notions and results that are known from systems of classic crystallography, including perfect (periodic) crystals. The chapter is organised as follows. We begin, in Section 9.2, with the diffraction spectra of weak model sets, under a rather natural condition on their density, which extends the classic theory of regular model sets. Our focus is on systems with pure point spectrum. Particular interest in such systems emerges from the connection with dynamical systems of number-theoretic origin. In Section 9.3, we go one step in another direction, by considering a decorated version of the silver mean chain. This is the first example of a tiling with incommensurate tile lengths and mixed singular spectrum. It demonstrates that this structure, hitherto only known from constant-length substitutions, is more general, and opens an alley to the systematic study of finite-index covers of regular model sets. Next, Section 9.4 discusses a paradigmatic example of a system with mixed spectrum (pure point and absolutely continuous) that is generated by a random inflation rule. This system was originally suggested in [30], and has turned out to be an interesting element in a versatile class of random dynamical systems. Finally, with a more algebraic and number-theoretic approach, we discuss, in Section 9.5, the structure of various lattice enumeration problems and their generalisations to systems that are needed for an extension to the theory of aperiodic order. We 1 Projects

A1, B2

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focus on similar and coincidence sublattices and their generalisations to embedded Z-modules. Of particular interest is the connection between these two problems, and its relation to group theory.

9.2 Spectral properties of weak model sets The theory of regular model sets (that is, cut and project sets with sufficiently nice windows) is well established; see [5] and references therein for general background. Prominent examples are the vertex sets of the rhombic Penrose tiling and of many other related tilings [23, 5]. One cornerstone result for this class is the pure pointedness of the diffraction measure [35, 53, 16]. Equivalently, the dynamical spectrum of the uniquely ergodic hull defined by the model set is pure point as well; compare [38, 11, 40]. The regularity of the window is vital to many of the existing proofs (such as that in [53]) and also enters the characterisation of regular model sets via dynamical systems [13]. On the other hand, systems such as the visible lattice points or the kth power-free integers have been known for quite some time to be pure point diffractive as well [17, 51]. These point sets can also be described as model sets, but the windows are no longer regular. In fact, for each of the above examples, the window has no interior and coincides with its boundary, which has positive measure. This way, many properties of regular model sets are lost. In particular, there are many invariant probability measures on the orbit closure (or hull) of the point set under the translation action of the lattice. Yet, as explicit recent progress has shown, the natural cluster (or patch) frequency measure of this hull is ergodic, and the visible points are generic for this measure [9]. It follows from an application of the general equivalence theorem [11] that the dynamical spectrum is still pure point. Since this example is one out of a large class with similar properties, it is natural to ask for a general approach that includes all of them. Such a class is provided by weak model sets, where more general windows are allowed. The name was coined by Moody [47, 48] (see also [5, Rem. 7.4]), while the class of weak model sets was first looked at by Schreiber [54] as a natural extension of the harmonious sets introduced by Meyer [44]. Starting from a general cut and project scheme .G; H; L/ (see diagram (9.2.1) below), the investigation of the diffraction properties of weak model sets turns out to be feasible under the fairly natural assumption of maximal (resp. minimal) density with respect to a given van Hove averaging sequence A D .An /n2N in the group G; see [5] for background and definitions. This assumption implies pure point diffraction and dynamical spectrum. Moreover, one can also analyse the cluster frequency measure for a given van Hove sequence A. The ergodicity of this measure has interesting consequences on the dynamical properties of the hull. In particular, weak model sets of extremal density have pure point dynamical spectrum. For proofs of the results below, we refer the reader to [10].2 2

Recently, and in parallel to our work, Keller and Richard [37] have developed an alternative view on model

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Let us start with a cut and project scheme (CPS), which is a triple .G; H; L/ as introduced by Meyer in [44]. Explicitly, it is given by a diagram G [



.L/ k

1-1

int

G H [

! !

L ?

L

!

H [

dense

int .L/ k

(9.2.1)

L?

with locally compact Abelian groups G; H and a lattice L  G  H (a discrete and co-compact subgroup of G  H ); see also [46, 47, 5] for background.3 Denote by G and H the Haar measures on G and H , respectively, which we assume to be consistently normalised to facilitate explicit calculations around densities of the involved point sets, lattices (L in particular) and their sublattices. Then, for a relatively compact set W  H with H .W / > 0, usually refered to as the window of the construction, one obtains a weak model set as

f.W /

WD fx 2 L j x ? 2 W g:

Subsequently, we shall always assume that G is  -compact and H is compactly generated. Here,  -compactness of G is needed for the existence of van Hove averaging sequences and their properties, while H compactly generated is no restriction, as can be shown constructively [56]. In general, the lower and upper densities of  D f.W / with respect to a given van Hove averaging sequence A [56, 5] obey a chain of inequalities [36], dens.L/ H .W ı / 6 dens./ 6 dens./ 6 dens.L/ H .W /:

(9.2.2)

The following result characterises, in terms of the autocorrelation of , the case when the upper bound is attained. Note that, for a CPS .G; H; L/ according to Eq. (9.2.1) and some c 2 Cc .H / which is a positive definite function on H , the weighted Dirac comb X !c WD c.x ? / ıx x2L

is a Fourier transformable, translation bounded pure point measure. This observation is useful for the autocorrelation measure of model sets as follows. Given a uniformlyP discrete point set , one considers the corresponding Dirac comb, which is ı WD x2 ıx . With respect to a given van Hove averaging sequence A D .An /n2N , one defines the autocorrelation  of  relative to A as the Eberlein convolution [5, Sec. 8.8]

 D ı ~ f ı WD lim

n!1

ın  ı

n

vol.An /

;

sets via a systematic exploitation of the torus parametrisation for such systems; compare [8, 32, 53, 13]. Their work includes the class of weak model sets and provides an independent way to derive several of our key results. 3 Note that our use of the notation A  B includes the case that A D B.

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where n D  \ An , provided the limit exists. The latter can always be achieved by going to a suitable subsequence of A. The autocorrelation is a positive definite measure, and thus Fourier transformable. In the setting of weak model sets, one has the following result. Proposition 9.2.1 ([10, Prop. 5]). Let .G; H; L/ be a CPS as in Eq. (9.2.1), with G being  -compact and H compactly generated, and let ¿ ¤ W  H be relatively compact. Next, consider the weak model set  D f.W / and assume that a van Hove averaging sequence A is given relative to which the density dens./ and the autocorrelation measure  are to be defined. Then, the following statements are equivalent. (i) The lower density of  is maximal, dens./ D dens.L/ H .W /;

(ii) The density of  exists and is maximal, dens./ D dens.L/ H .W /;

(iii) The autocorrelation of  exists and satisfies  D dens.L/ !c . W

Here, cW D 1W  1f is the covariogram function of W . W This motivates the following concept.

Definition 9.2.2. For a given CPS .G; H; L/ with  -compact G and compactly generated H , the set f.W / is called a weak model set of maximal density relative to a given van Hove averaging sequence A if the window W  H is relatively compact with H .W / > 0, if the density of f.W / relative to A exists, and if the density condition dens.f.W // D dens.L/ H .W / is satisfied. One obtains the following general result on the diffraction of weak model sets of maximal density. Theorem 9.2.3 ([10, Thm. 7]). Let  D f.W / be a weak model set of maximal density for the CPS .G; H; L/, in the setting of Proposition 9.2.1. Then, the autocorrelation  is a strongly almost periodic, pure point measure. It is Fourier transformable, b and c

 is a translation bounded, positive, pure point measure on the dual group G. It is explicitly given by c

 D

X ˇ ˇ ˇa.u/ˇ2 ı ; u

u2L0

with amplitude a.u/ D

dens./ c 1W . u? /; H .W /

b and, with L 0 being the annihilator where 1c is a bounded, continuous function on H W b is the corresponding Fourier of L  G  H in G H , the group L0 D .L 0 /  G module in additive notation.

1

There is another class of weak model sets which behaves nicely.

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Definition 9.2.4. For a given CPS .G; H; L/ as in Definition 9.2.2, a projection set f.W / is called a weak model set of minimal density relative to a given van Hove averaging sequence A if the window W  H is relatively compact with H .W ı / > 0, if the density of f.W / relative to A exists, and if the density condition dens.f.W // D dens.L/ H .W ı / is satisfied. These sets are Meyer sets, and therefore perhaps less interesting than their counterparts with maximal density. Nevertheless, Theorem 9.2.3 still holds for weak model sets of minimal density if one replaces W by W ı in the formulas. In the spirit of the chain of density inequalities in (9.2.2), one can derive the following ‘sandwich result’ for an arbitrary autocorrelation of a weak model set. Corollary 9.2.5 ([10, Cor. 10]). Let  be a weak model set for the CPS .G; H; L/ from above, with relatively compact window W. If is any autocorrelation of , it satisfies 0 6 dens.L/ !c ı 6 6 dens.L/ !c ; W

W

which is to be understood as an inequality of measures. We can now proceed with the general spectral theory, aiming at a result on the dynamical spectrum of the hull of weak model sets of extremal density. In order to do so, we first need to construct a suitable measure and establish its ergodicity. Let  D f.W / with compact W  H be a weak model set of maximal density, relative to a fixed van Hove averaging sequence A. The (geometric) hull of   G is the orbit closure G C in the local topology; compare [5, Sec. 5.4] for background. Note that our point set  is of finite local complexity, so that the local topology suffices (it is a special case of a Fell topology [11]). The group G acts continuously on the hull by translations. P For our further reasoning, we represent  by its Dirac comb ı WD x2 ıx , which is a translation bounded, positive pure point measure with support . Its hull is X WD fıt  ı j t 2 Gg; where the closure is taken in the vague topology. By standard arguments, X is vaguely compact, with a continuous action of G on it. Clearly, ıt  ı D ıt C , so
that the topological dynamical systems G C; G and .X ; G/ are topologically conjugate, wherefore we tacitly identify them from now on. The compactness of W together with the regularity of the Haar measure H implies the existence of a non-empty compact set K  H and a net of Œ0; 1-valued functions g˛ 2 Cc .H / such that 1K > g˛ > 1W holds for all ˛ together with lim˛ H .g˛ / D H .W /. Consider the weighted Dirac combs !g˛ D

X

x2f.K/

g˛ .x ? / ıx :

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Since each !g˛ , as well as ı , is supported in the same Meyer set f.K/, we have pointwise convergence, lim˛ !g˛ D ı . This also implies norm convergence, for the norm defined by kkK D supt 2G jj.t CK/. Note that the topology induced by k:kK does not depend on the choice of K as long as it has non-empty interior. Moreover, for each weighted comb !g˛ , there is a hull X˛ D fıt  !g˛ j t 2 Gg that is compact in the vague topology and defines a topological dynamical system .X˛ ; G/. In fact, one has more; see [39, Thm. 3.1] as well as [41]. Fact 9.2.6. Each dynamical system .X˛ ; G/ of the above type is minimal and admits precisely one G-invariant probability measure, ˛ say, and is thus strictly ergodic. Moreover, the system is topologically conjugate to its maximal equicontinuous factor, wherefore it has pure point diffraction and dynamical spectrum, and the hull possesses a natural structure as a compact Abelian group. The Dirac comb ıf.K/ clearly is a translation bounded measure. Thus, there is a compact set K 0  G with non-empty interior and a constant C > 0 such that kıf.K/ kK 0 6 C . By construction, we also have   f.K/. It follows that both our Dirac comb ı and the measures !g˛ are elements of ˇ ˚ Y WD  2 M1 .G/ ˇ kkK 0 6 C ;

which is a compact subset of the space M1 .G/ of translation bounded measures on G. In fact, for all ˛, we have the relation 0 6 ı 6 !g˛ 6 ıf.K/ 2 Y

(9.2.3)

as an inequality between pure point measures. Moreover, we also have X  Y as well as X˛  Y for all ˛. Clearly, the measures ˛ have a trivial extension to measures on Y, still called ˛ , such that supp.˛ / D X˛ . In particular, !g˛ is then generic for ˛ . We can now work within Y for approximation purposes. In order to do so, we need a smoothing operation, which is based on the linear mapping  W Cc .G/ ! C.Y/, defined by
c 7! c ./ WD   c .0/:

Note that this is the standard approach to lift continuous functions on G with compact support to continuous functions on a compact measure space such as Y. It underlies the fundamental relation between diffraction and dynamical spectra via the Dworkin argument; compare [11, 14] and references therein. One now obtains the following result. Theorem 9.2.7 ([10, Thm. 17]). The net .˛ / of ergodic, G-invariant probability measures converges in the vague topology, and the limit,  say, is an ergodic,

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G-invariant probability measure on Y. Moreover, the weak model set  D f.W / of maximal density is generic for . For our approach, we started with an individual weak model set  of maximal density, which is then pure point diffractive by Theorem 9.2.3. Now, we obtained a measure-theoretic dynamical system .X ; G; / with an ergodic measure  as constructed above. Note that, relative to the (fixed) van Hove sequence A, it is the cluster frequency measure. Moreover, our weak model set of maximal density is generic for this measure  by Theorem 9.2.7, hence the individual autocorrelation  of  is also the autocorrelation of the dynamical system, and its Fourier transform, c

 , is the diffraction measure both of  and of our dynamical system [11]. By the general equivalence theorem between diffraction and dynamical spectrum in the pure point case [11, 14], the following consequence is now immediate. Corollary 9.2.8. Let  be a weak model set of maximal density, relative to a fixed van Hove averaging sequence A, for a CPS .G; H; L/ as above. Then,  is pure point diffractive, and the dynamical system .X ; G; / with the measure  from Theorem 9.2.7 has pure point dynamical spectrum. Note that regular model sets are special cases of weak model sets of maximal density, and thus covered as well. Repeating the above analysis for weak model sets of minimal density, one also obtains the following analogous result [10]. Corollary 9.2.9. Let  be a weak model set of minimal density, relative to a fixed van Hove averaging sequence A, for a CPS .G; H; L/ as above. Then, the autocorrelation of  relative to A exists, and  is pure point diffractive. Moreover, the dynamical system .X ; G; /, where  is the cluster frequency measure relative to A, has pure point dynamical spectrum. Let us turn our attention to concrete systems with mixed spectrum, where we begin with certain covers of regular model sets.

9.3 A decorated quasiperiodic tiling with mixed spectrum There are many inflation tilings with both a pure point and a continuous part in their dynamical or diffraction spectrum. The most prominent examples are the Thue– Morse and Rudin–Shapiro tilings (compare [3, 5]), but many more are known [25, 26, 4, 3, 6, 22], including also higher-dimensional ones. Most of these examples are generated by constant-length substitutions, and are thus lattice based. In such cases, the symbolic and the geometric picture are equivalent, as letters can be represented by prototiles of the same length (usually chosen as 1). Here and below, we speak of

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a substitution whenever we work on the symbolic level, and of an inflation when we deal with the corresponding tilings on the line. What we describe below is a procedure to construct a mixed spectrum, almost 2-1 extension of any pure point inflation tiling, and illustrate it by the well-known silver mean tiling. This means that we construct a tiling dynamical system with mixed spectrum that is based on a quasiperiodic tiling, and thus extends the constant-length case considerably. A more detailed account of this procedure has been given in [2]. The starting point is the observation that many of the examples with mixed spectrum possess an inflation rule with a particular symmetry [3]. There exists an involution P on the set of prototiles that has no fixed points and commutes with the inflation. The prototiles come in geometrically equal pairs that are only distinguished by their labels, and P simultaneously exchanges the two tiles of each such pair. If we label the tiles in a pair by ai ; aN i , where i labels the set of pairs, then P just exchanges the bar status of a tile: P .ai / D aN i , P .aN i / D ai , wherefore we call P a bar swap symmetry. Wiping out all bars defines a factor map which is 2-1 almost everywhere. Provided the maximal equicontinuous factors (MEF) of both the corresponding barred and unbarred tiling spaces are the same, this implies that its spectrum is mixed [21]. To be specific, we focus on the one-dimensional case. Let  be a primitive inflation admitting a bar swap symmetry P as above, so that  ı P D P ı  . The hull Y generated by  consists of the closure of all translation orbits of tilings generated by  . We thus have a topological dynamical system .Y; R/ with respect to the translation action. From here, we will construct a globally 2-1 factor map ' from .Y; R/ to a factor dynamical system .Y0 ; R/, which identifies tilings related by a bar swap P . We then have '.!/ D ' P .!// for all ! 2 Y, and the map ' commutes with the translation action by construction. One might be tempted to take a factor map which just wipes out all bars. This, however, would not generally lead to a map which is 2-1 everywhere. Rather, we first rewrite the inflation  in terms of collared tiles. The latter consist of pairs of tiles t1 t2 , where t1 is the actual tile and t2 is a (one-sided) collar of t1 . This is seen as a label attached to t1 that specifies the type of the right neigbour tile of t1 . Obviously, there is an induced inflation rule e  on the collared tiles. Its hull e Y is in fact mutually locally derivable (MLD) to Y, and the two dynamical systems are conjugate; compare  has a bar swap symmetry, which simulta[5, Sec. 5.2]. We now observe that also e neously swaps the bar status of a tile and its collar. We can now consider the factor map ' that is induced by identifying collared tiles related by the bar swap P . Clearly,
' also identifies pairs of global tilings related by P , so that '.e ! / D ' P .e ! / for all e ! 2 e Y. Since Y ' e Y, our procedure induces a unique mapping from Y to 0 e Y WD '.Y/, which we simply call ' again. Let ! 0 2 Y0 now be a tiling in the image of ', given by a bi-infinite sequence of tiles ti , where i 2 Z. The collared tile t0 t1 has exactly two possible preimages—let us choose one of them. The neighbouring collared tile, t1 t2 , also has two preimages, but as we have already chosen a preimage of the collared tile t0 t1 , and thus of t1 , there is only one choice left. Continuing like this, we see that, once we have chosen

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a preimage of the collared tile t0 t1 , the lifts of all other tiles to the right are fixed, and analogously to the left as well. Consequently, ! 0 has precisely two preimages, and the mapping ' W Y ! Y0 is globally 2-1. As we have already remarked, wiping out all bars from the tiles of the original inflation  also induces a factor map, but it need not be globally 2-1. The image Y00 of this factor map must also be a factor of Y0 , so that we actually have a sequence of factor maps Y

'

! Y0

2 -1

'0

! Y00 ;

1-1 a.e.

where Y0 is obtained by identifying collared tiles related by a bar swap, whereas Y00 is obtained by identifying original tiles related by a bar swap. The second map, ' 0 , is 1-1 a.e., because the composition of ' and ' 0 is a.e. 2-1. Almost all tilings in Y00 consist of a single, infinite order supertile, and these have exactly two preimages in Y, which differ by a bar swap. Only tilings consisting of two adjacent infinite order supertiles may have more than two preimages, but these are of measure zero. Note that Y0 and Y00 may coincide, but they are different in general. Each of the translation dynamical systems has a maximal equicontinuous factor (MEF), which is the topological counterpart of the Kronecker factor from measurable dynamics, so that the above sequence of factor maps leads to Y ? ? y

YMEF

'

!

2-1

Y0 ? ?0 y

! Y0MEF

'0

!

1-1 a.e.

Y00 ? ? 00 y

(9.3.1)

Y00MEF

It is well known (compare [21]) that a tiling dynamical system has pure point dynamical spectrum if and only if the factor map to the underlying MEF is 1-1 almost everywhere. Let us now assume that Y0 , and thus also Y00 by standard results [12], has pure point spectrum, so that both  0 and  00 are 1-1 a.e. As ' 0 is 1-1 a.e., Y0MEF and Y00MEF are equal. There still remain two possibilities as follows. Fact 9.3.1. The map in the commuting diagram (9.3.1) is a group homomorphism, and can thus be either 1-1 or 2-1. If is 1-1, this implies that  is 2-1 a.e., so that Y must have mixed spectrum, whereas, if is 2-1, the mapping  is 1-1 a.e., and Y has pure point spectrum. In order to compare YMEF and Y0MEF , we need to compare their respective return modules. Recall that r is a return vector of a tiling dynamical system Y if there exist two tiles t1 and t2 of the same type in some tiling ! 2 Y such that the distance of their left endpoints is r. Let RY be the Z-span of all return vectors, which is a finite index submodule of the Z-module generated by all tile lengths, TY . The (additive) pure point spectrum of Y now consists of all those k 2 R such that, for any return n vector r, one has e2i kr ! 1 as n ! 1 by [55]. This pure point spectrum can

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only be non-trivial if the inflation factor (or multiplier)  is a Pisot–Vijayaraghavan (PV) number. This property of  is satisfied for our example. The quantity of interest now is the Z-module of eventual return vectors ˝ ˛ MY D fx 2 TY j n x 2 RY ; for some n 2 Ng Z :

Clearly, Y and Y0 have the same MEF if and only if MY D MY0 . For constant-length substitutions, the module MY is known as the height lattice; compare [26]. Let us now look at a concrete example. Our goal is to construct an almost 2-1 extension of the silver mean inflation [5, Ex. 4.5] sm W

a 7! aba ;

b 7! a;

in such a way that we gain a bar swap symmetry. In line with our previous comments, we consider the geometric inflation rule here, forptwo prototiles. The scaling factor (or inflation multiplier) of  D sm is  D 1 C 2, and the natural tile lengths for a (long) and b (short) are  and 1, respectively. Here,  is a PV unit, and it is well known that  generates tilings with pure point spectrum; see [5, Chs. 7 and 9] for details. We now add a barred version of each prototile, thus enlarging the prototile set to N and look for an inflation which is primitive, commutes with the bar A D fa; b; a; N bg, swap involution P , and reduces to  under the identification of a with aN and b with N It is easy to see that the inflation b. N W

a 7! ab aN ;

b 7! aN ;

N ; aN 7! aN ba

bN 7! a:

(9.3.2)

satisfies these criteria. Moreover, we have RY D MY D TY , so that N must have mixed spectrum. We call it the twisted silver mean (TSM) inflation. It is instructive to have a closer look at Y0 . For that purpose, we rewrite the inflation in terms of collared tiles, 1 W A 7! CD BN ; B 7! CD AN ; C 7! CD AN ; D 7! AN ;

together with ˛N 7! P .1 .˛// for all ˛ 2 fA; B; C; Dg, where A D aa, B D aa, N C D ab, and D D b a. N Note that A; B; C; D are considered as single tiles with a right-collar of length 1 in the original tiles. There are three variants of the long tile, and one short tile. However, since 1 .B/ D 1 .C /, we can actually identify the two prototiles B and C , now called B, and then rename the old D as the new C , so that we arrive at N 2 W A 7! BC BN ; B 7! BC AN ; C 7! A; (9.3.3)

once again with ˛N 7! P .2 .˛//. The hulls of 1 and 2 are MLD, so that we can stick to the latter. If we wipe out the bars in 2 , we obtain the inflation for Y0 , which is an a.e. 1-1 extension of the original silver mean hull, and thus has pure point spectrum. In fact, by standard methods from [5, 16], we conclude that any tiling ! 0 2 Y0 is (a

207

Spectral and arithmetic structures in aperiodic order C C B B A A

Figure 9.3.1. Covering windows of the inflation 2 . For each pair of prototiles, such as C and C , the lift of their left endpoints give disjoint point sets in internal space with the same closure; see text for details.

translate of) a model set, and there exists a CPS

dense

R [ p

ZŒ 2  k L



1-1

RR [

L

?

int

!

1-1

! !

R [ p

dense

ZŒ 2  k

(9.3.4)

L?

such that, for each tile type i , there exists a window Wi that produces the point set i D f.x/ j x 2 L; x ? 2 Wi g of left endpoints of tiles of type i in ! 0 . Choosing a preimage ! of ! 0 under ' induces a decomposition of i into a disjoint union of two subsets, C tiles i and i , coming from the left endpoints of the unbarred resp. barred p C of !. These subsets i and i are no model sets, but as subsets of ZŒ D ZŒ 2  we can still lift them to L, project them by int to internal space, and take the closure. In this way, we can determine a covering window for each tile type of !. Of course, we cannot expect these covering windows for the different tile types to be disjoint. The result is shown in Figure 9.3.1. We see that a barred and an unbarred tile always share the same covering window. Of course, every left endpoint of a tile is either the left endpoint of a barred or an unbarred tile, but not both. Consequently, the identical covering windows emerge as the closure of disjoint point sets. This means that one cannot determine the bar status of a tile by looking at its internal space coordinate. The bars represent a sort of chemical modulation of the original silver mean tiling, where the modulation cannot be described as a function of internal space coordinates in a reasonable way. Another interesting point is that the hull Y00 of the original silver mean tiling is indeed different from Y0 as generated by the inflation (9.3.3). It is not p even MLD to 00 0 ? Y . Even though all window boundaries of Y are in ZŒ  D ZŒ 2 , there is an additional accumulation point not contained in ZŒ? . Therefore, in contrast to Y00 , the corresponding fibre over the MEF has size 2. Two tilings correspond to this point, both of which project to the same silver mean tiling. This degeneracy also shows up

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in the cohomology. Using the Anderson–Putnam method [1], it is routine to compute H 1 .Y/ D Z 3 and H 1 .Y0 / D Z 2 ,p which is in line with [27, Thm. 5.1]: The extra dimension comes from the extra ZŒ 2 -orbit of singular points. We now know that the TSM tiling dynamical system has mixed spectrum. In addition to the well-known pure point part, the spectrum must contain also a continuous component, whose nature still needs to be investigated. As the dynamical system of the TSM tiling is an almost 2-1 extension with bar swap symmetry of the silver mean dynamical system, the Hilbert space of square-integrable functions on the hull, L2 .Ytsm /, is a tensor product and can be split into an even and an odd sector under the bar swap symmetry, L2 .Ytsm / D L2 .Ysm / ˝ C2



D L2 .Ysm / ˝ C ˚ L2 .Ysm / ˝ 

(9.3.5)

DW HC ˚ H ;

where ˙ is the even/odd character of the bar swap P . The factor map induces a corresponding map on the Hilbert spaces, which sends the first summand isomorphically to L2 .Ysm /, and has the second summand as kernel. The translation group acts on this Hilbert space via a unitary representation, and leaves both sectors invariant separately. Lemma 9.3.2. The spectral measure of the translation action on Ytsm , confined to either of the two sectors HC and H , is spectrally pure, so that it has only one non-vanishing component in its Lebesgue decomposition. In particular, it is a pure point measure on HC , and a continuous one on H , where the latter is either purely singular continuous or purely absolutely continuous. Proof. This really is a consequence of the corresponding result on index-2 extensions of irrational rotations [52, 31], to which our system is measurably conjugate. For details, see [2].  Lemma 9.3.2 still leaves two possibilities for the spectral type in the odd sector. In order to discriminate between the two, we look at the asymptotic behaviour of the correlation functions ˛ˇ . Because of the purity result of Lemma 9.3.2, we can in fact take any combination of correlation functions which is odd under the bar swap. A simple such combination is the autocorrelation of a weighted model set with an odd weight function. Specifically, let  be the set of all left endpoints of tiles of a tiling !, and set w.x/ D ˙1 for x 2 , depending on whether x is the left endpoint of an unbarred or a barred tile. With R WD  \ . R; R/, the relevant autocorrelation coefficients then are X 1 w.x/w.y/: (9.3.6) tsm .z/ D lim R!1 jR j x;y2R x yDz

The nature of the spectrum in the odd sector now depends on the decay or non-decay of tsm .z/ as z ! 1.

Spectral and arithmetic structures in aperiodic order

Lemma 9.3.3. One has limn!1 tsm .zn / D 1

p

209

2, where zn D .1 C /n .

Proof. We only sketch the strategy of the proof here; compare [2]. The contributions to tsm .1 C / come from pairs of tiles where the second tile t2 is at distance 1 C  to the right of tile t1 . For any such pair, t2 is the second neighbour to the right of t1 . We can therefore determine all possible triples of three consecutive tiles in the tiling, and add up their contributions to the correlation at distance 1 C , weighted with the relative frequency of each triple. The latter can be determined by Perron–Frobenius theory [52, 5]. For the correlation at distance .1 C /n , we do the same with triples of supertiles of order n. These have the same relative frequencies as the triples of tiles. The pairs of tiles contributing to the correlation then consist of a left tile in the left supertile of the triple, and a corresponding right tile at distance .1 C /n , which is contained in one of the other two supertiles. Since the underlying silver mean tiling has pure point spectrum, the density of tiles t1 not having a corresponding tile t2 with the same geometry at distance .1 C /n asymptotically vanishes as n ! 1, so that we just have to add up the contributions of matching and anti-matching pairs of tiles, the latter ones having opposite bar status, but matching geometry. The details of the computation are given in [2].  The main result can now be formulated as follows. Theorem 9.3.4. The TSM dynamical system has two spectral components, both of which are singular. The dynamical spectrum from the even sector under the bar swap is pure point, whereas that from the odd sector is purely singular continuous. Proof. By Lemma 9.3.2, the dynamical spectrum in the even sector is pure point, whereas in the odd sectorPit is continuous and of pure spectral type. By Lemma 9.3.3, the correlation measure z2  tsm .z/ ız does not decay to zero towards infinity, so that, by the Riemann–Lebesgue lemma and the fact that   is uniformly discrete for a Pisot inflation, its Fourier transform must be singular. This implies that there must be a singular continuous component in the diffraction spectrum, and hence in the dynamical spectrum of the odd sector. As that spectrum is of pure type, it must be purely singular continuous.  Remark 9.3.5. Instead of splitting each tile type of a pure point tiling into a barred and an unbarred variant, we could have split it into k copies each, and modified the inflation such that a permutation symmetry acting on these k copies results. Such a procedure would lead to a mixed spectrum tiling, too, but a simple spectral purity result for the continuous spectrum sector, as in Lemma 9.3.2, can only be expected if k D 2. Remark 9.3.6. The set C of left endpoints of all unbarred tiles of a TSM tiling is a Meyer set (it is a relatively dense subset of a model set), which is linearly repetitive (it is a component of a primitive inflation tiling), and which has mixed spectrum

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of singular type. This shows that there are highly ordered Meyer sets with mixed spectrum. In fact, C has zero entropy, and must be distinguished from Meyer sets arising from model sets with an (a posteriori) thinning disorder of positive entropy. The latter would have a mixed spectrum with a non-trivial pure point part and an absolutely continuous component in the spectrum. The simplest example would be an uncorrelated thinning disorder of Bernoulli type, but more interesting are random inflation systems [30, 45], as discussed below. These have a highly correlated thinning disorder, but still positive entropy and an absolutely continuous component in the spectrum.

9.4 Random inflations Here, we explain some recent developments connected with the spectral theory of substitution systems in the presence of a certain type of randomness. Since the general theory is still at its beginning, we do so along a paradigmatic example. The classic Fibonacci substitution on the binary alphabet A D fa; bg is defined by % W a 7! ab; b 7! a; S where % is considered as a homomorphism of the monoid A D feg [ A [ m>2 Am into itself. Here, Am denotes the words of length m in the alphabet, e is the empty word, and concatenation of words acts as multiplication for A . Moreover, % also defines a mapping of the two-sided shift space AZ into itself, where elements x 2 AZ are written as x D .: : : ; x 2 ; x 1 ; x0 ; x1 ; : : :/. For definiteness, we equip x with a marker between x 1 and x0 , and call x 1 jx0 the core of x. By elementary arguments, one can see that there are two elements in AZ which are fixed points under %2 , namely one with core aja and one with core bja. Let w be either of them, and define XF WD fS n w j n 2 Zg where S is the left shift on AZ , defined by .S x/i WD xi C1 , and the closure is taken in the obvious product topology, which is also known as the local topology. XF is called the discrete hull of %, and is unique. In particular, it does not depend on which fixed point we use, and it can alternatively be described as the set of all two-sided sequences in AZ with the property that all finite subwords are legal, which is to say that they also occur as subwords of %m .a/ for some m 2 N. A variant of % is given by %0 W

a 7! ba;

b 7! a;

and one can check explicitly that both % and % 0 define the same hull [5, Rem. 4.5]. As a consequence, both % and % 0 map XF into itself, as does any concatenation of them. In fact, to any one-sided binary sequence, we can attach a sequence in these

Spectral and arithmetic structures in aperiodic order

211

two substitutions. The hull defined by the resulting concatenation would still be XF . The topological dynamical system defined that way, .XF ; Z/ with the continuous shift action of Z D hS i, has a number of interesting properties as follows; see [52, 5] for details. Proposition 9.4.1. The topological dynamical system .XF ; Z/ is strictly ergodic, and has pure point dynamical spectrum. Moreover, it is deterministic, which is to say that its topological entropy vanishes. There is a natural way to turn this symbolic system into a geometric one, which is analogous to what we did in the previous p
section. Here, one chooses the two prototiles 1 as intervals of length  D 2 1 C 5 for type a and 1 for type b. This maps each sequence from XF to a tiling of R by two intervals, where the marker is always sent to 0. Taking now the closure of all translates by t 2 R gives a new hull, denoted by YF , which is known as the continuous hull. This step, which can be formalised by a standard suspension argument from ergodic theory, leads to another topological dynamical system, .YF ; R/. Via standard results from suspension theory [24], one has the following consequence of Proposition 9.4.1. Corollary 9.4.2. The topological dynamical system .YF ; R/ is strictly ergodic, with pure point dynamical spectrum and vanishing topological entropy. Things change drastically if one allows to mix % and % 0 on a local level. Indeed, let us consider the (Fibonacci) random substitution [30, 45] defined by ( 8 ˆ 2

n .k/ n

where the n are uniformly bounded, smooth functions. They are defined recursively ˇ n 1 ˇ2 by nC1 .k/ D ˇp Cqe 2ik ˇ n .k/ for n > 2 together with the initial condition
cos 2 k , and the above sum converges uniformly. 2 .k/ D 1

9.5 Enumeration of lattices Due to the connection with questions and problems from classical crystallography, one is not only interested in results on the Fourier side of the coin, but also in various properties in direct (Euclidean) space, with particular reference to the actual spatial arrangement of the structures. This is justified by the fact that local arrangements of clusters (or ‘atoms’) are governed by local rules in direct space, while spectral structures are often better captured in Fourier (or reciprocal) space. In this context, we have studied various enumeration problems for sublattices of a given lattice  Rd , as well as the corresponding generalisations to (properly) embedded Z-modules as they emerge in the theory of aperiodic order. Here, a free Z-module M of rank n is called (properly) embedded in Rd when M  Rd and when the R-span of M is Rd . In particular, we have concentrated on counting similar sublattices (SSLs) and submodules (SSMs), coincidence site lattices (CSLs) and modules (CSMs), as well as well-rounded sublattices (the latter mainly for d D 2). If  Rd is a lattice and R 2 O.d /, then \ R is called a CSL if it is a sublattice of full rank—the corresponding R then being called a coincidence isometry, with coincidence index ˙.R/ D Œ W . \ R /: Equivalently, one may require that R be commensurable with in the sense explained in the paragraph preceding Def. 14.3.1. An important tool in counting the CSLs is the group of coincidence isometries, OC. /. For similar sublattices, there exists an analogous group, the group OS. / of similarity isometries, which contains OC. / as a normal subgroup, see below. The sublattices are usually counted according to their subgroup index, which gives rise to arithmetic functions. For root lattices in low dimensions, these functions are usually multiplicative. Using generating functions of Dirichlet series type, one gets further insight into their properties, in particular into the asymptotic growth rate of the number of the various kinds of sublattices via Delange’s theorem; see [57] for background. Let us mention some results on SSLs first. Recall that any similar sublattice of is of the form ˛R with 0 ¤ ˛ 2 R and some R 2 O.d /. Then, SOS. / is the collection of all rotation matrices that emerge in this way, which obviously is a subgroup of SO.d /. As similarity isometries are—roughly speaking—in one-

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to-one correspondence with similar sublattices (up to symmetry operations), similar sublattices can be counted by means of determining the group OS. /. This works particularly well in low dimensions for highly symmetric lattices such as root lattices, as they are related to special quadratic number fields or quaternion algebras; see [15] and references therein. For more general lattices, the situation is more difficult, but considerable progress in two dimensions was achieved in [18]. As an example, we state a result on the group SOS. / of similarity rotations; see [20] for further details. Theorem 9.5.1. Let be a planar lattice, written as a discrete, co-compact subgroup of C, with multiplier ring MR. / D f˛ 2 C j ˛  g. Generically, one has MR. / D Z and SOS-group SOS. ˚ w ˇ/ D f˙1g ' C 2 . ˇ 0 ¤ w 2 O , where O D MR. / is an Otherwise, one has SOS. / D jwj order in an imaginary quadratic number field K, which also satisfies MR.O / D O . Moreover, one has ˚w ˇ ˇ 0 ¤ w 2 OK ; SOS. / D SOS.O / D SOS.OK / D jwj

where OK is the maximal order of K and thus contains O . Finally, the group SOS. / is the same for all lattices in sim. /, the lattice similarity class of .

The multiplier ring is known for arbitrary planar lattices . In addition, the Dirichlet series can be given explicitly for all that are similar to a maximal order OK with class number 1; see [18] for details. The analogous problem of counting CSLs is more difficult. This is partly due to the fact that coincidence isometries are generally not in one-to-one correspondence with CSLs, which gives rise to different counting functions. This can be illustrated nicely in the case of the A4 lattice, where the Dirichlet series generating function of the coincidence isometries [7] is simpler than the corresponding generating function of the CSLs [34, 33, 61]. Here, it turns out that the counts are all multiples of 120, due to an underlying action of the icosahedral group. In particular,p we have the following result for the number of coincidence rotations, where K D Q. 5 / is a real quadratic number field and K its Dedekind zeta function. Also,  denotes Riemann’s zeta function. Theorem 9.5.2 ([7, Thm. 4]). Let 120 cArot4 .m/ be the number of coincidence rotations of the root lattice A4 of index m. Then, the Dirichlet series generating function for cArot4 .m/ is given by ‰Arot4 .s/ D D

X cArot .n/ 4

n2N

ns

1 C 51 1 52

D 1C

5 2s

s

s

D

K .s 1/ .s/.s 2/ 1 C 5 s .2s/.2s 2/

Y

p˙1.5/

C

10 3s

C

20 4s

.1 C p s /.1 C p1 s / .1 p1 s /.1 p2 s / C

30 5s

C

50 6s

C

and the spectrum of coincidence indices is N.

50 7s

C

80 8s

Y

p˙2.5/

C

90 9s

C

1Cp 1 p2 150 10s

C

s s 144 11s

C


;

Spectral and arithmetic structures in aperiodic order

215

This allowed us to determine the asymptotic behaviour of cArot4 .m/; compare [7].
Corollary 9.5.3. With Arot4 D ressD3 ‰Arot4 .s/ D totic behaviour of cArot4 .m/ is given by X

m6x

cArot4 .m/  Arot4

p 450 5 6 

.3/, the summatory asymp-

x3  0:419375 x 3; 3

as x ! 1. Analogous results hold for the number of CSLs of A4 . However, the formulas are more complicated. Still, we have been able to calculate the Euler factors of the corresponding generating function ‰A4 .s/ explicitly, but there seems to be no easy way to write ‰A4 .s/ as product of Riemann -functions; compare [33, 61, 20]. Another result establishes the link between CSMs and SSMs for arbitrary embedded Z-modules. Let scalM .R/ WD f˛ 2 R j ˛RM  M g;

(9.5.1)

which means that scalM .R/ is the set of all scaling factors ˛ such that ˛RM is commensurate to M , by which we mean that M \˛RM is a submodule of M of full rank. Then, G D fscalM .R/ j R 2 OS.M /g has a natural group structure, which allows to define a homomorphism OS.M / ! G. In particular, we have the following result, where RC WD fx 2 R j x > 0g. Theorem 9.5.4 ([61, Thm. 3.2.2]). Let M be an embedded Z-module. Then, the kernel of the homomorphism  W OS.M / ! RC =.scalM .1/ \ RC /; R 7 ! scalM .R/ \ RC ; is the group OC.M /. Thus, OC.M / is a normal subgroup of OS.M /, and the factor group OS.M /= OC.M / is Abelian. This result was first proved for lattices in [28] and later generalised to S -lattices in [29]. Here, we call a Z-module M an S -lattice, if S is a ring with identity that is also a finitely generated free Z-module and if M is the S -span of d linearly independent vectors in Rd . If M  Rd is a lattice or an S -lattice, all elements of OS.M /= OC.M / have finite order. In particular, their order is a divisor of d ; see [28, 29] for details.

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Theorem 9.5.5. Let M  Rd be a lattice or an S -lattice. Then, the factor group OS.M /= OC.M / is the direct sum of cyclic groups of prime power orders that divide d . Let us continue with a result p on the icosian ring, I, which is a maximal order in the quaternion algebra H.Q. 5 //. It is both a Z-module of rank 8, embedded in R4 , and a ZŒ -module of rank 4; see [49] or [5, Ex. 3.9] for an explicit basis and the relation to the root lattice E8 , which is the Minkowski embedding of I into R8 . In particular, I is an S -lattice in the above sense, with S D ZŒ . Since I has class number 1, all SSMs of I can be determined through the ideal theory of I and their prime factorisation; see [16] for more. The result reads as follows, where K stands for the Dedekind zeta function of a quadratic field K and I for that of the icosian ring [58]. Theorem 9.5.6 ([16, Thm. 3]). The possible indices of similar submodules of the icosian ring are precisely the squares of rational integers that can be represented by the quadratic form x 2 C xy y 2 with x; y 2 Z. The number of SSMs of a given index is a multiplicative arithmetic function, whose Dirichlet series generating function ˆI reads
2
2 K .2s/ K .2s 1/ I .s/ D ˆI .s/ D K .4s/ K .4s/ p

with K D Q. / D Q. 5 /. Again, the cases of coincidence isometries and CSMs are more complicated; see [20, 34, 61] for the known results in this direction. Let us close this section with a result on well-rounded lattices in the plane. A lattice in Euclidean d -space is called well-rounded if it contains d linearly independent vectors of minimal (positive) length. In d D 2, the well-rounded sublattices of a lattice are in close connection to its CSLs [19], though counting well-rounded sublattices is generally more difficult than counting SSLs and CSLs. However, we obtained precise results for the asymptotic growth rates for certain planar lattices. In particular, we have proved the following result [19]. Theorem 9.5.7. Let a .n/ be the number of well-rounded sublattices of index n in P the square lattice. Then, the summatory function A .x/ D a .n/ possesses n6x  the asymptotic growth behaviour

log.3/ L.1;  4/ x log.x/ 1 C c x C O x 3=4 log.x/ 3 .2/  
log.3/ log.3/ D x C O x 3=4 log.x/ x log.x/ C c 2 2

A .x/ D

217

Spectral and arithmetic structures in aperiodic order

where, with denoting the Euler–Mascheroni constant, c WD

L.1;  4 / log.3/ .2/ C .2/ 3  log.3/ 2 C 3 4 3

1 X

kD0



log.3/ 4 

1 1 log.3/ 2k C 1 4

 0:6272237 is the coefficient of .s

1/

1

L0 .1;  4 / C L.1;  4 /

log.2/ 6



k 0 and ˇ < N C 1. Moreover, for any N 2 Z and ˇ 2 R such that ˇ < N C 1, there is an affine special Kähler metric satisfying (10.2.8). (In particular, for any N 2 Z there is an affine special Kähler metric satisfying w D jzjN C1 log jzj.C C o.1//.)

Remark 10.2.6. In [17], the first formula of (10.2.8) appears in the form w D jzjN C1 log jzj eO.1/ , which follows from McOwen’s analysis of solutions of the Kazdan–Warner equation [27]. The asymptotics as stated in Theorem 10.2.5 can be

Affine special Kähler structures in real dimension two

227

obtained from [25, Prop. 3.1], which in fact provides even more refined asymptotics near the origin. By analysing the asymptotic behaviour of solutions of the Kazdan–Warner equation with singular coefficients it is possible to compute the monodromy of the flat symplectic connection. Namely, very recently we proved the following result. Theorem 10.2.7 ([5]). Let g D wjdzj2 be a special Kähler metric on B1 such that w D jzjˇ C C o.1/




or

w D


jzjN C1 log jzj C C o.1/ ;

where ˇ < N C 1 (in the second case, we put by definition ˇ D N C 1). Let Hol.r/ denote the monodromy of r along a loop that goes once around the origin. Then, the following holds:   ˇ sin ˇ  If ˇ … Z, Hol.r/ is conjugate to cos ; sin ˇ cos ˇ  If ˇ 2 2Z, Hol.r/ is trivial or conjugate to

11 01

 If ˇ 2 2Z C 1, Hol.r/ is id or conjugate to


;

1 1 0 1


.

Corollary 10.2.8. Hol.r/ is conjugate to a matrix lying in Sp.2; Z/ if and only if ˇ 2 21 Z [ 31 Z. Proof. Since Hol.r/ 2 Sp.2; R/ D SL.2; R/, the characteristic polynomial of Hol.r/ has integer coefficients if and only if tr Hol.r/ 2 Z. This implies that Hol.r/ is conjugate to a matrix lying in Sp.2; Z/ if and only if cos ˇ 2 f0; ˙ 12 ; ˙1g.  10.2.4 A link between two local descriptions Our next goal is to obtain a link between the two descriptions of special Kähler structures in terms of solutions of (10.2.7) and in terms of special holomorphic coordinates. Notice that special holomorphic coordinates always exist in a neighbourhood of a point, where the special Kähler structure is regular. However, in a neighbourhood of a singular point there may be no special holomorphic coordinates. More precisely, we have the following. Proposition 10.2.9. Let  be a disc or a punctured disc. A special Kähler structure on  admits a special holomorphic coordinate on  if and only if the triple .h; u; a/ appearing in Corollary 10.2.3 is given by .h; u; a/ D .h; log. h/; 0/ for some negative harmonic function h on . Proof. Observe first, that for any negative harmonic function h the triple .h; u; a/ D .h; log. h/; 0/ is a solution of (10.2.7). Moreover, in this case by (10.2.6) we have !11 D 0, which implies that rdx D 0. In other words, z is a special holomorphic coordinate.

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Assume now that z is a special holomorphic coordinate. Then r Re dz D 0 implies !11 D 0, which yields dh C a' e u du D 0. Since ' is not exact, we must have a D 0. This yields h D e u , i.e., h is a negative harmonic function.  For the special Kähler structure determined by a single negative harmonic function as in the above proposition, we compute   1 @h 1 0 0 2 3 g D hjdzj ; „ D : idz ; !r D 2 @z h dh dh Furthermore, by (10.2.1) the holomorphic prepotential satisfies Im

@2 F D @z 2

2h:

If  is a disc, this equality determines F up to a polynomial of degree 2, cf. [12, Prop. 1.38(c)]. If  D B1 , by [3, Thm. 3.9] there is A > 0 such that h D A log jzj C h0 , where h0 is a smooth harmonic function on B1 . Hence, we have the following result. Corollary 10.2.10. If a special Kähler structure on B1 admits a special holomorphic coordinate in a neighbourhood of the origin, there are some constants A > 0 and B such that
g D A log jzj C B C o.1/ jdzj2 as z ! 0; Moreover, ord0 „ >

1.

In particular, a special Kähler structure on B1 such that ord0 „ 6 2 does not admit a special holomorphic coordinate in a neighbourhood of the origin. 10.2.5 A relation with metrics of constant negative curvature Recall that if g and gQ D e2u g are two metrics on a two-manifold, then their curvatures K and KQ Q 2u . In particular, if g is flat, then u D Ke Q 2u . are related by u D K Ke u 2 Hence, with the help of (10.2.3) we conclude: If g D e jdzj is special Kähler, then gQ D e2u jdzj2 has a non-positive curvature. On the other hand, if gQ D e2u jdzj2 is a metric of constant negative curvature on 2u some domain   C, then p u solves u D Ke with K > 0. Hence, the triple .h; u; / with h.x; y/ D Kx and D 0 solves (10.2.5) and therefore the metric g D e u jdzj2 is special Kähler. Summarising, we obtain the following result. Proposition 10.2.11 ([17, Prop. 3.1]). If gQ D wjdzj2 is a metric of constant negative curvature K on a domain , then g D p1w jdzj2 is a special Kähler metric on the same domain . Moreover, the associated holomorphic cubic form is given by p Ki 3 dz : „ D 4

Affine special Kähler structures in real dimension two

229

10.2.6 Examples Example 10.2.12. In the setting of Proposition 10.2.9, choose h D A log jzj C B, where A > 0 and B are some constants. We require also B < 0 so that h is negative on B1 . For the corresponding special Kähler structure we have  
A Ai 3 0 0 2 : dz ; !r D g D A log jzj C B jdzj ; „ D 4z h d log jzj d log jzj Moreover, z is a special holomorphic coordinate. The dual “coordinate” w is given by w D 2.B A/iz C 2Aiz log z:

Of course, w is not a coordinate in any neighbourhood of the origin if A ¤ 0, but choosing a suitable branch of the logarithm the above expression defines a dual coordinate in a neighbourhood of any point in B1 . Similarly, on a suitable branch of the logarithm (or by going to the universal covering of B1 ), the extrinsic description of this metric is given by the holomorphic 1-form
# D wdz D 2 .B A/iz C Aiz log z dz; or the corresponding prepotential F D i.B

2



2

A/z C iA z log.z/

z2 2



D w, as in Example 10.2.1. with @F @z This special Kähler structure is related to the Ooguri–Vafa metric [29], see [7]. The monodromy of r along the circle of radius 1 centered at the origin can be computed explicitly and equals   2A Hol.r/ D 1 B : 0

1

Example 10.2.13. Apply Proposition 10.2.11 to the Poincaré metric on the punctured
2 disc gQ D jzj 2 log jzj jdzj2 to obtain that the metric g D jzj log jzj jdzj2

is special Kähler. Example 10.2.14 (Special Kähler metrics via meromorphic functions). If  is simply connected, the conformal factor of any metric of constant negative Gaussian curvature K can be written [22] in the form w D 4

jf 0 .z/j2

1 C Kjf .z/j2


2 ;

(10.2.9)

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where f is a meromorphic function on  with at most simple poles such that f 0 .z/ ¤ 0 on . Conversely, for any  (not necessarily simply connected) and any f as above, (10.2.9) determines a metric of constant negative curvature K. Hence, by Proposition 10.2.11 the metric ˇ ˇ 1 ˇ1 C Kjf .z/j2 ˇ jdzj2 g D 2 jf 0 .z/j is special Kähler for any meromorphic f as above. For example, put K D 1 and f .z/ D z n , where n > 1. Then we obtain that gD

1 1 jzj 2n

is a special Kähler metric on B1 .

n

1


jzj2n jdzj2

Example 10.2.15. By a classical result of Picard [30], for any given n > 3 pairwise distinct P points .z1 ; : : : ; zn / in C and any n real numbers .˛1 ; : : : ; ˛n / such that ˛j < 1 and ˛j > 2, there exists a unique metric of constant negative curvature gQ on C n fz1 ; : : : ; zn g satisfying gQ D jz zj j 2˛j .c C o.1//jdzj2 near zj . Hence, the corresponding special Kähler metric g has a conical singularity near zj , g D jz

zj j˛j .c C o.1//jdzj2:

Explicit examples of constant negative curvature — hence special Kähler — metrics on the three times punctured complex plane can be found in [21] and references therein.

10.3 Some global aspects of special Kähler geometry on P 1 Even though the methods of Section 10.2.3 are mainly local, some global conclusions can be also derived. The main objective for this section is to show that by allowing singular special Kähler metrics we have a lot of examples on a compact manifold and even a non-trivial moduli space. 10.3.1 A constraint from the Gauss–Bonnet formula Let g be a special Kähler metric on the complex projective line P 1 with singularities at fz1 ; : : : ; zk g. Assume that at each zj the metric g has a conical singularity of order ˇj =2 > 1, i.e.,
g D jzjˇj Cj C o.1/ jdzj2 ;

where Cj is positive. A restriction on the cone angles of special Kähler metrics as above can be obtained from the Gauss–Bonnet formula [31, (1.3)], which in this case

231

Affine special Kähler structures in real dimension two

reads 1 2

Z

P1

k 1X K D .P / C ˇj : 2 1

j D1

Here, K is the curvature of g and  is the Euler characteristic. Since K > 0, compare [12, Rem. 1.35], we obtain k X

j D1

ˇj > 2.P 1 / D

4:

(10.3.1)

10.3.2 Families of special Kähler metrics on P 1 Just like in Example 10.2.15, for any k > 3 points z1 ; : : : ; zk on P 1 and any ˛1 ; : : : ; ˛k such that ˛j < 1

and

k X

˛j > 2;

j D1

there is a unique metric gQ of constant negative curvature on P 1 with conical singularity at zj of order ˛j . Think of P 1 as C [ f1g, where each zj is distinct from 1. 2 If z is a holomorphic coordinate on C, we can write gQ D w.z; z/jdzj N with w.z; z/ N D jzj

4

.c C o.1//

as z ! 1:

Applying Proposition 10.2.11 we obtain a special Kähler metric g on C with conical singularity of order ˛j =2 at zj for each j D 1; k. Moreover,
g D jzj2 c C o.1/ jdzj2 ; as z ! 1: (10.3.2) In other words, g can be thought of as a special Kähler metric on P 1 with conical singularities at z1 ; : : : ; zk and zkC1 D 1 of order ˛1 =2; : : : ; ˛k =2; and 3, respectively. Summarising, we obtain the following result. Proposition 10.3.1. For any k > 3 points on P 1 and any ˛1 ; : : : ; ˛k such that ˛j < 1

and

k X

˛j < 2;

j D1

there is a special Kähler metric on P 1 such that g D jz


zj j˛j cj C o.1/ jdzj2

for all j D 1; k. Moreover, near 1, this metric satisfies Eq. (10.3.2), which corresponds to
g D jj 6 c C o.1/ jdj2 in a local coordinate  near 1.

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Applying a Möbius transformation, we can move .z1 ; z2 ; z3 / into any given triple of points. Hence, Proposition 10.3.1 yields a family of special Kähler metrics with singularities at k C 1 > 4 points parameterised by k C 2.k 3/ D 3k 6 real parameters. Remark 10.3.2. Restriction (10.3.1) does not apply to the special Kähler metrics constructed in Proposition 10.3.1, since such metrics always have singularities of order 3.

References [1] D.V. Alekseevsky, V. Cortés and C. Devchand, Special complex manifolds, J. Geom. Phys. 42 (2002), 85–105. [2] D.V. Alekseevsky, V. Cortés, M. Dyckmanns and T. Mohaupt, Quaternionic Kähler metrics associated with special Kähler manifolds, J. Geom. Phys. 92 (2015), 271–287. [3] Sh. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, GTM 137, 2nd ed., Springer, New York, 2001. [4] O. Baues and V. Cortés, Realisation of special Kähler manifolds as parabolic spheres, Proc. Amer. Math. Soc. 129 (2001), 2403–2407. [5] M. Callies and A. Haydys, Local models of isolated singularities for affine special Kähler structures in dimension two, Int. Math. Res. Notices, arXiv:1711.09118. [6] S. Cecotti, S. Ferrara and L. Girardello, Geometry of type II superstrings and the moduli of superconformal field theories, Int. J. Mod. Phys. A 4 (1989), 2475–2529. [7] K. Chan, The Ooguri–Vafa metric, holomorphic discs and wall-crossing, Math. Res. Lett. 17 (2010), 401–414. [8] V. Cortés, On hyper-Kähler manifolds associated to Lagrangian Kähler submanifolds of T  Cn , Trans. Amer. Math. Soc. 350 (1998), 3193–3205. [9] V. Cortés, Special Kähler manifolds: a survey, in: Proceedings of the 21st Winter School “Geometry and Physics” (Srní, 2001) Circolo Matematico die Palermo 69 (2002), 11–18. [10] B. de Wit and A. Van Proeyen, Potentials and symmetries of general gauged N D 2 supergravity-Yang-Mills models, Nuclear Phys. B 245 (1984), 89–117. [11] R. Donagi and E. Markman, Cubics, integrable systems, and Calabi–Yau threefolds. In Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) (M. Teicher, ed.), Bar-Ilan Univ., Ramat Gan, 1996, 199–221. [12] D. Freed, Special Kähler manifolds, Commun. Math. Phys. 203 (1999), 31–52. [13] D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via threedimensional field theory, Commun. Math. Phys. 299 (2010), 163–224. [14] D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013), 239–403. [15] S.J. Gates, Jr., Superspace formulation of new nonlinear sigma models, Nuclear Phys. B 238 (1984), 349–366.

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233

[16] A. Haydys, HyperKähler and quaternionic Kähler manifolds with S 1 -symmetries, J. Geom. Phys. 58 (2008), 293–306. [17] A. Haydys, Isolated singularities of affine special Kähler metrics in two dimensions, Commun. Math. Phys. 340 (2015), 1231–1237. [18] N. Hitchin, The moduli space of complex Lagrangian submanifolds, Asian J. Math. 3 (1999), 77–91. [19] N. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc., III. Ser. 55 (1987), 59–126. [20] J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2) 99 (1974), 14–47. [21] D. Kraus, O. Roth and T. Sugawa, Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky, Math. Z. 267 (2011), 851–868. [22] J. Liouville, Sur l’équation aux différences partielle 1re série 18 (1853), 71–72.

d 2 log  dudv

˙ 2a2 D 0, J. Math. Pures Appl.

[23] Zh. Lu, A note on special Kähler manifolds, Math. Ann. 313 (1999), 711–713. [24] O. Macia and A. Swann, Twist geometry of the c-map, Commun. Math. Phys. 336 (2015), 1329–1357. [25] R. Mazzeo, Y. Rubinstein and N. Sesum, Ricci flow on surfaces with conic singularities, Anal. PDE 8 (2015), 839–882. [26] R. Mazzeo, J. Swoboda, H. Weiss and F. Witt, Ends of the moduli space of Higgs bundles, Duke Math. J. 165 (2016), 2227–2271. [27] R. McOwen, Prescribed curvature and singularities of conformal metrics on Riemann surfaces, J. Math. Anal. Appl. 177 (1993), 287–298. [28] A. Neitzke, Notes on a new construction of hyperkahler metrics. In Homological mirror symmetry and tropical geometry, Springer, Cham, 2014, 351–375. [29] H. Ooguri and C. Vafa, Summing up Dirichlet instantons, Phys. Rev. Lett. 77 (1996), 3296– 3298. [30] E. Picard, De l’équation U D ke U sur une surface de Riemann fermée, J. de Math. 9 (1893), 273–291. [31] M. Troyanov, Metrics of constant curvature on a sphere with two conical singularities. In Differential Geometry (F.J. Carreras, O. Gil-Medrano, and A.M. Naveira, eds.), LNM 1410, Springer, Berlin, 1989, 296–306.

Chapter 11

Non-crossing partitions B. Baumeister, K.-U. Bux, F. Götze, D. Kielak and H. Krause Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical noncrossing partitions are associated to Coxeter and Artin groups of type An , which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments.1 The authors would like to thank Holger Kösters for carefully reviewing parts of this text and for his very helpful comments.

11.1 The poset of non-crossing partitions A partition p of a set U is a decomposition of U into pairwise disjoint subsets Bi : ] U D Bi i

The subsets Bi are called the blocks of the partition p. Another way to look at this is to consider p as an equivalence relation on U . In this perspective, the subsets Bi are the equivalence classes. Let q be another partition of the same set U . We say that q is a refinement of p if each block of q is contained in a block of p. In terms of equivalence relations, if two elements of U are q-equivalent, they are also p-equivalent. We also say that q is finer than p or that p is coarser than q; and we write q  p. Let P.U / be the set of all partitions on the underlying set U . The refinement relation  is a partial order on the set P.U /, which is therefore a poset. Moreover, it is a lattice, i.e., every non-empty finite subset P  P.U / has a least upper bound and a greatest lower bound. We remark that the partition lattice is complete, i.e., even arbitrary infinite subsets have least upper and greatest lower bounds. Remark 11.1.1. It is interesting that the definition of a complete lattice can be weakened by breaking the symmetry between upper and lower bounds. If a poset has upper bounds and greatest lower bounds, it is already a complete lattice (i.e. it also has lowest upper bounds). 1 Projects

A4, C3, C13

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B. Baumeister, K.-U. Bux, F. Götze, D. Kielak, H. Krause

v3 v4

v2

v5

v1 v6

v8 v7

Figure 11.1.1. Visualisation of the partition f f 1 g; f 2; 6; 7 g; f 3; 5 g; f 4 g; f 8 g g.

Sketch of proof. Let P be a non-empty subset of the poset. We consider the the set B C .P / of all common upper bounds for the non-empty subset P . Since the poset has upper bounds, B C .P / is non-empty. Hence it has a greatest lower bound, which turns out to be the lowest upper bound of P .  Consider the following reflexive and symmetric relations on U : xy xy

W, W,

9p 2 P W x and y are p-equivalent 8p 2 P W x and y are p-equivalent

V It is clear that  is itself an equivalence relation. It corresponds to the meet P of the partitions in P , i.e., the greatest lower bound of P .W The transitive closure of  is an equivalence relation, which corresponds to the join P of the partitions in P . Now, we restrict our consideration to finite sets. For a natural number m 2 N, let us denote by Œm the set f 1; 2; 3; : : : ; m g. We fix the natural cyclic ordering on Œm and represent its elements as the vertices v1 ; : : : ; vm of a regular m-gon inscribed in the unit circle. Let p be a partition of Œm. We say that two blocks B and B 0 of the partition p cross if their convex hulls intersect. The partition p is called non-crossing if its blocks pairwise do not cross. A non-crossing partition can thus be depicted by colouring the convex hulls of its blocks. For blocks of size one or two, we fatten up the convex hull. It is clear from the visualisation that the complements of the coloured regions also are pairwise disjoint. This gives rise to the Kreweras complement. Here, we put dual vertices w1 ; : : : ; wm within the arcs vi vi C1 . There is no natural numbering, and we choose to place w1 within the arc from v1 to v2 . Let p be a non-crossing partition. Two dual vertices lie in the same block of the complement pc if they lie within the same complementary region of the convex hulls of blocks of p. The set NC.m/ of all non-crossing partitions of Œm is partially ordered with respect to refinement. It is thus a subposet of the set of all partitions of Œm. It turns out that NC.m/ is also a lattice. This is clear from Remark 11.1.1 since greatest lower bounds are inherited from the partition lattice and upper bounds exist trivially since the trivial partition with a single block is noncrossing.

237

Non-crossing partitions

v3 v4

w3 v3 w2 v2

v4

v2

w4 v5

v1

w1

v5

v1

w5 v6

v8 v7

w8 v6

v8 w6 v7 w7

Figure 11.1.2. The partition p D f f 1 g; f 2; 6; 7 g; f 3; 5 g; f 4 g; f 8 g g and its Kreweras complement p c D f f 1; 7; 8 g; f 2; 5 g; f 3; 4 g; f 6 g g.

However, the noncrossing partition lattice is not a sublattice of the whole partition lattice: the join operation in both structures differ, i.e., the finest partition coarser than some given non-crossing partitions does not need to be non-crossing; see Remark 11.1.3 for a counterexample. The complement map NC.m/ ! NC.m/ p 7 ! pc is an anti-automorphism of the lattice NC.m/: it reverses the refinement relation and interchanges the roles of meet and join. It is, however, not an involution. In the picture, taking the Kreweras complement twice seems to get you back to the original partition. This is true; however, the indexing of the vertices shifts by one. Thus, the square of the Kreweras complement is given by cyclically rotating the element of the underlying set f 1; : : : ; m g. The bottom (finest) element ? of NC.m/ is the partition with m blocks, each of size one. The top (coarsest) element > of NC.m/ is the partition with a single block. For each non-crossing partition p, we define its rank rk.p/ in terms of its number of blocks: rk.p/ WD m #f blocks of p g

For any non-crossing partition p, all maximal chains from the bottom element ? to p have the same length, which coincides with the rank rk.p/. Let us summarise the properties and non-properties of the poset of non-crossing partitions:

Fact 11.1.2. The set NC.m/ of non-crossing partitions of an m-element is partially ordered by refinement. This poset is a lattice and self-dual with respect to the Kreweras complement, i.e., .p ^ q/c D pc _ q c .p _ q/c D pc ^ q c for any two p; q 2 NC.m/.

238

B. Baumeister, K.-U. Bux, F. Götze, D. Kielak, H. Krause

The automorphism p 7! .pc /c has order m. All maximal chains from bottom to top have length m 1. For any non-crossing partition p, there is a maximal chain from bottom to top going through p. The noncrossing partition lattice is graded and one has m

1 D rk.p/ C rk.pc /

for any p. Remark 11.1.3. For m > 4, the non-crossing partition lattice NC.m/ is not a sublattice of the partition lattice: the join operations do not coincide. A counterexample for m D 4 is p D f f 1; 3 g; f 2 g; f 4 g g and q D f f 1 g; f 2; 4 g; f 3 g g. The join of these partitions in the partition lattice is f f 1; 3 g; f 2; 4 g g whereas the join in NC.4/ is the top element. These two partitions also show that the non-crossing partition lattice NC.m/ is not semi-modular, i.e., the following inequality does not hold for all partitions p and q, rk.p/ C rk.q/ > rk.p _ q/ C rk.p ^ q/: Enumerative properties of the noncrosing partitition lattice are well understood. Kreweras counted the number of non-crossing partitions. Fact 11.1.4 (see [41, Cor. 4.2]). For any m, we have

where Cm D

1 mC1


2m m

j NC.m/ j D Cm D

.2m/Š mŠ.mC1/Š

is the mth Catalan number.

Kreweras also determined the Möbius function for the lattice of non-crossing partitions. Recall that, for a finite poset P , the Möbius function  W f .u; v/ 2 P  P j u 6 v g ! Z is defined by the following recursion: .u; u/ D 1; .u; v/ D

X

.u; w/:

u6w/ D . 1/m 1Cm

1

D . 1/m

1

.2m 2/Š .m 1/ŠmŠ

(11.1.1)

239

Non-crossing partitions

v5

v4

w4

v3

w3

w5 v6

w2

v2

v7

w6

w1

w7

w12

v1 v12

v8 v9

v10

v11

w8

w11 w9

w10

Figure 11.1.3. Two nested partitions p  q and their blockwise complement. For the dual vertices w4 and w10 , different conventions are possible to determine which dual vertex is to be used with which block of q.

Let p be a non-crossing partition, and consider a non-crossing partition q  p. Let B be a block of p. The blocks of q contained in B may be thought of as a non-crossing partition of B. Thus, we have the following: Observation 11.1.6. Let p 2 NC.m/ be a non-crossing partition, and let B1 ; : : : ; Bk be its blocks. Then the order ideal p WD f q 2 NC.m/ j q  p g is isomorphic as a poset to the cartesian product NC.B1 / 


 NC.Bk /. 0 Let B10 ; : : : ; Bm be the blocks of the Kreweras complement pc . Since the kC1 complement is an antiautomorphism of the non-crossing partition lattice, the filter p WD f q 2 NC.m/ j q  p g is isomorphic as a poset to the cartesian product 0 NC.B10 / 


 NC.Bm /. kC1 For non-crossing partitions p  q, the interval Œp; q is the filter for p within the order ideal of q. Hence, by combining the previous isomorphisms, we see that Q Œp; q is isomorphic to the product B NC.B/ where B ranges over the blocks of the “blockwise Kreweras complement” of p in q. Since the Möbius function is multiplicative with respect to cartesian products of posets, Observation 11.1.6 allows one to derive the values of .p; q/ in terms of the blockwise complement of p in q from Kreweras’ formula (11.1.1). Remark 11.1.7. To every poset .P; 6/, one associates the order complex. This is the simplicial complex .P; 6/ whose vertices are the elements of P and whose simplices are chains in P , i.e., non-empty subsets of P on which 6 is a total order. By a theorem of P. Hall, one can interpret the Möbius function as the Euler characteristic of order complexes [49, Prop. 3.8.6], .u; v/ D ...u; v///;

for u < v:

Here .u; v/ WD f w 2 P j u < w < v g is the open interval from u to v.

240

B. Baumeister, K.-U. Bux, F. Götze, D. Kielak, H. Krause

A significant implication is that the Möbius function is invariant with respect to reversing the order relation: let 6 be the Möbius function of .P; 6/ and let > be the Möbius function of the reversed poset .P; >/; then, we have 6 .u; v/ D > .v; u/:

11.2 Non-crossing partitions in free probability Classical probability spaces .; F ; P/ can be reformulated using the commutative C  -algebra A D L1 .; F ; P/ as follows. Real valued (bounded) random variables correspond to elements of RA and their expectations are given by evaluation of the linear functional '.a/ WD  a dP. The ’distribution’ of a random variable a is the 1 k distribution Rinduced R k a .A/ WD R P.a .A// and its kth moment is given by '.a / D k  a dP D R x a .dx/ D R x ak .dx/. This construction admits the following non commutative extension. Denote by .Md .C/; tr/ the space of d  d complex matrices, together with the normalised trace and the usual matrix conjugation. Consider now the algebra of random matrices R A WD Md .L1 .; F ; P// together with the linear functional '.a/ WD  tr.a/ dP. This represents a genuine non-commutative C  -probability space .A; '/, which is a unital C  -algebra over C together with a unital and tracial positive linear functional ' W A ! C, that is '.1/ D 1;

'.a a/ > 0;

'.ab/ D '.ba/;

for all a; b 2 A:

Furthermore, we shall assume that ' is faithful, that is '.a a/ D 0 is equivalent to a D 0. See the survey [50]. Many constructions in non commutative probability are parallel to those in classical probability, and this is also reflected in the notation: If a is a self-adjoint element in A, i.e. a D a, the value '.a/ is sometimes called the expectation of a, the values '.ak /, k 2 N, are calledR the moments of a, and the compactly supported probability measure a on R with x k a .dx/ D '.ak /, k 2 N, is also called the distribution of a which always exists for self-adjoint elements in a C  -probability space. If the measure a admits a density fa , the latter is also called the density of a. Similarly, given two self-adjoint elements a and b in A, the joint moments of a and b are given by the values '.w/, w being a “word” in a and b. Recall R that a compactly supported Borel measure  on R (and more generally any  with Rezx .dx/ locally analytic around z D 0) is uniquely characterised by its moments x k .dx/ since then the Fourier transform of  is a convergent power series with coefficients given by the moment sequence. In order to define a corresponding notion of independence for self-adjoint elements (like that for random variables in classical probability theory), recall that two random variables a; b 2 L1 .; F ; P/ endowed with expectation ' as above are

241

Non-crossing partitions

independent, if '.ak b l / D '.ak /'.b l / or equivalently   ' .ak '.ak //.b l '.b l // D 0

(11.2.1)

for all k; l 2 N0 . Let A1 and A2 denote unital sub-algebras in A, for instance generated by elements a and b respectively. They are called ‘free’ if the expectations of all products with factors alternating between elements from A1 and A2 vanish whenever the expectations of all factors vanish. Hence the elements a; b 2 A are called free if ' .aj1

'.aj1 //.b k1

'.b k1 //


.ajm

'.ajm //.b km

'.b km //




D 0 (11.2.2)

for all m 2 N and all j1 ; : : : ; jm ; k1 ; : : : ; km 2 N. Hence for m D 1 this rule for the evaluation of joint moments coincides with the classical rule '.ab/ D '.a/'.b/ but is apparently different for m > 1. The rules (11.2.2) as well as (11.2.1) allow to reduce by induction the evaluation of joint moments '.aj1 b k1


ajm b km / of these free or independent elements to the moments '.aj / and '.b k /, which determine the marginal distribution of a resp. b. Thus freeness may be regarded as a (noncommutative) analogue of the notion of independence in classical probability theory, allowing the development of a free probability theory. In particular (11.2.2) allows to to compute the expectation of '..a C b/n / for any n 2 N, a 2 A1 and b 2 A2 , thus determining the distribution in the sense described above of the ‘free’ sum of a and b via the moments of a and b only. Hence, this assigns to compactly supported measures ;  (with moments given by those of a; b) a free additive convolution 
, see the survey [50]. This notion may be considered as an asymptotic limit of a corresponding notion for sequences of random matrices with independent entries of increasing dimension and their limiting spectral measures, see Chapter 1. More generally, a set of unital sub-algebras Aj  A; j 2 I , indexed by a set I , is called free if for any integer k and aj 2 Aij ; j D 1; : : : ; k; ij 2 I , '.a1 : : : ak / D 0 provided that '.aj / D 0; j D 1; : : : ; k; and i1 ¤ i2 ; i2 ¤ i3 ; : : : ; ik 1 ¤ ik ;

(11.2.3)

that is, all adjacent elements in a1 : : : ak belong to different sub-algebras Aji . This notion has similar properties as classical independence. For instance, polynomials P .aj / of free self-adjoint elements aj (generating a sub-algebra) are free again. The density .x/ D p1 exp. x 2 =2/ defines the standard Gaussian distribu2 tion. Hence, the classical central limit theorem (CLT) may be stated for independent random elements ai ; i 2 N from a commutative C  -probability space .A; '/ with identical distribution such that '.ai / D 0; '.ai2 / D 1 (such variables are called standardised).

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Theorem 11.2.1 (Commutative C  -version of CLT). The moments of the normalised N p sum SN WD a1 C:::Ca satisfy N

k lim '.SN /D

N !1

Z

x k .x/ dx;

k 2 N:

(11.2.4)

Consider free random elements ai from a (non-commutative) C -probability space .A; '/, standardised via '.ai / D 0; '.ai2 / D 1 with identical distribution, that is '.ajl / depends on l only. In order to describe a corresponding free ‘central limit theorem’ for this setup we have to determine the asymptotic behaviour of moments of type '.ai1 : : : aik / subject to the assumption of freeness (11.2.3). Note that by freeness all mixed moments vanish provided an element aj occurs only once in the product. (Note that this holds as well for mixed moments of independent random variables). Thus, we only need to consider mixed moments with factors occurring at least twice. For a product ai1


aik of k factors, such that s of them, say b1 ; : : : ; bs , are different, let p D fB1 ; : : : ; Bs g denote the corresponding partition of the set f1; : : : ; kg into jpj WD s nonempty blocks Bj of the positions of bj in 1 6 j 6 k. One can show by induction that all mixed moments of free or independent elements '.ai1 ai2


aik / where 1 6 ij 6 N , can be computed via (11.2.3) resp. (11.2.1) as above also for s > 2 in terms of moments cl D '.bjl / for j D 1; : : : ; s which depend on l only by the assumption of identical distribution. Thus these mixed moments depend on the partition scheme of i1 ; : : : ; ik , say p, only and will be denoted by mp . The number of such mixed moments in a1 ; : : : ; aN corresponding to a given partition scheme depends on jpj only and is given by AN;p D N.N 1/


.N jpj C 1/ . Thus X k '.SN /D mp AN;p N k=2 : (11.2.5) p

For a partition p we have AN;p < N jpj . If all parts of p satisfy jBj j > 2 and one block is of size at least three, the corresponding contribution in (11.2.5) is of order jmp jAN;p N k=2 6 jmp jN 1=2 , that is all these terms are asymptotically negligible as N tends to infinity. k Hence, computing the asymptotic limit of '.SN / reduces to considering all mixed moments of k factors with each random element occurring precisely twice, a consek quence being that limN !1 '.SN / D 0 for k odd. Recall that NC.n/ denoted the lattice of all non-crossing partitions on the set Œn D f1; : : : ; ng. Furthermore, let NC2 .2k/ denote the subset of non-crossing partitions with blocks of size 2 only, called ’non-crossing pair partitions’ on a set of 2k elements. Now consider as an example three free standardised variables a, b, c. Then the product abc 2ab corresponds to a pair partition with a crossing, that is p D ff1; 5g; f3; 4g; f2; 6gg. Hence '.abc 2 ab/ D '.abab/'.c 2/ D 0 by freeness, that is (11.2.2). Otherwise, for a non-crossing pair partition like ca2 b 2 c, we have

Non-crossing partitions

243

'.ca2 b 2 c/ D '.cb 2 c/'.a2/ D '.cc/'.b 2/ D 1. These simple observations can be generalised by induction in the following Lemma to determine the values of joint moments mp D '.ai1 ai2


aik / for pair partitions p of free variables. Lemma 11.2.2. For any pair partition p, ( 0 if p has a crossing mp D 1 if p is non-crossing. Thus, we conclude from (11.2.5) and the previous results that 2k lim '.SN / D lim

N !1

N !1

X

p2NC2 .2k/

AN;p D j NC2 .2k/j: N k=2

Furthermore, one shows that Ck WD j NC2 .2k/j D j NC.k/j;

(11.2.6)


2k 1 where Ck D kC1 is the kth Catalan number. Among its numerous interpretak tions, it represents as well the 2k th moment of a compactly supported measure with 1 .4 x 2 /1=2 ; jxj 6 2. This is the so-called Wigner measure or density w.x/ WD 2 semi-circular distribution. See [44, Rem. 9.5]. Now the free central limit theorem for a sequence of bounded free variables N p aj ; j 2 N, which are standardised via '.aj / D 0; '.aj2 / D 1, and SN WD a1 C:::Ca N may be stated as follows. Theorem 11.2.3 (Free Central Limit Theorem). SN converges in distribution to w which serves as the Gaussian distribution in free probability, i.e. Z k lim '.SN / D x k w.x/ dx; k 2 N: (11.2.7) N !1

p This means e.g. that the rescaled sum .a1 C a2 /= 2 of two free elements a1 ; a2 of a non-commutative probability space .A; '/ which both have density w.x/ again has a Wigner distribution. In free probability an element s of .A; '/ with density w.x/ is called semi-circular and its moments are given by ( 2k
1 ; if n D 2k; n '.s / D kC1 k (11.2.8) 0; if n odd: Recall that a 2 .A; '/ is called positive if there exists an c 2 .A; '/ with a D c  c. Thus a is self-adjoint. Define the free multiplicative convolution of two compactly supported measures a ; b , of positive free elements a; b 2 .A; '/, say a  b , as follows by specifying its moments. Since in a C  -probability space A

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positive square roots a1=2 resp. b 1=2 of a resp. b as well as the positive element pa;b WD a1=2 ba1=2 are again in A, we may define a  b by: Z k x k da b .x/ WD '.pa;b /; k 2 N: (11.2.9) k k Since '.pa;b / D '.pb;a /; k 2 N, because ' is tracial, i.e. '.ba/ D '.ab/, we conclude that the free convolution  is commutative. By the same tracial property k and the relation of freeness, we show that '.pa;b / D '..ab/k / and this implies the associativity of . Moreover it follows from this representation that the multiplicative convolution measure a  b is uniquely determined by the distributions of a and b . In order to effectively compute both additive and multiplicative convolution of measures, one needs more properties of the lattice of partitions of 1; : : : ; n into blocks and the subset of non-crossing partitions together with the notion of multi-linear cumulant functionals. As above let Bj ; j D 1; : : : s denote the blocks of a partition p 2 NC.n/ of 1; : : : ; n. For p 2 NC.n/, the free mixed cumulants are multi-linear functionals p W An ! C defined in terms of a moment decomposition using the Möbius function .q; p/ of the lattice of non-crossing partitions NC.n/. We define the general mixed cumulant functionals p as follows: X p Œa1 ; : : : ; an  D 'q Œa1 ; : : : ; an  .p; q/; where (11.2.10) q2NC.n/;pq

0

'q Œa1 ; : : : ; an  WD ' @ Q

Y

k2B1

1

0

ak A


' @

Y

k2Bs

1

ak A ;

and the products k2Bj ak repeat the order of indices within the block Bj . Note that by Hall’s theorem, the coefficient .p; q/ can also be written as  .q; p/ using the relation of reversed refinement (see Remark 11.1.7). Then one shows, see [44, Prop. 11.4], that X '.a1


an / D p Œa1 ; : : : ; an : (11.2.11) p2NC.n/

In the special case p D 1n we write n instead of 1n . The following lemma is proved by induction on n. Lemma 11.2.4 ([44, Thm 11.20]). The elements a1 ; : : : ; an 2 A are free if and only if all mixed cumulants satisfy n Œaj1 ; : : : ; ajk  D 0; whenever aj1 ; : : : ajk , 1 6 jl 6 n; 1 6 k 6 n contains at least two different elements.

245

Non-crossing partitions

In contrast to (11.2.3), this characterisation of freeness holds even if the '.aj / are non-zero. For a partition p 2 NC.n/, recall that pc denotes its Kreweras complement in NC.n/. Then, one shows that for free elements a; b the following recursion involving the Kreweras complement holds: X n Œab; : : : ; ab D p Œa; : : : ; apc Œb; : : : ; b: (11.2.12) p2NC.n/

See [44, Rem. 14.5]. This entails that the cumulants of ab and thus by (11.2.11) the moments of ab are indeed determined by multi-linear functionals of a and b alone which again by virtue of (11.2.10) are determined by the moments of a together with the moments of b. The recursive Equation (11.2.12) and the Definition (11.2.10) of cumulants may be conveniently encoded as algebraic relations the following formal genP1 between n n erating series. For a 2 A let Ma .z/ D '.a /z denote moment gennD1 Pthe 1 n erating series and with n .a/ WD n Œa; : : : ; a let Ra .z/ WD nD1 n .a/z and 1 Ra .z/ WD z Ra .z/ denote cumulant generating series. In particular, for free selfadjoint a; b 2 A we get by binomial expansion of n .a C b/ and Lemma 11.2.4 that n .a C b/ D n .a/ C n .b/ and furthermore, as shown in [44, Lect. 12], Lemma 11.2.5. One has the following identities:

where

RaCb .z/ D Ra .z/ C Rb .z/;  1 C R .z/  a D z; Ra .zMa .z/ C z/ D Ma .z/; Ga z

(11.2.13) (11.2.14)

 1  1 X '.an / 1 Ga .z/ WD C 1 C M ; D a z nD1 z nC1 z z 1

can be identified with the Cauchy transform of the corresponding spectral measure a , that is Z da .t/ Ga .z/ D : t R z Hence the so-called R-transform R of a spectral measure a , introduced by Voiculescu in [50], is determined analytically by the inverse function of the Cauchy transform of a on the complex plane which is the starting point of the complex analytic theory of the asymptotic approximations of free additive convolution as developed in [22, 20, 21, 23]. Assuming that 1 D m1 ¤ 0, Ra .z/ WD Ra .z/ admits a formal inverse power series Ra. 1/ .z/. This may be defined via the inverse function of the Cauchy transform of a , which is well defined in a certain region in C. The so-called S -transform Sa .z/ WD

1 . R z a

1/

.z/ D

1Cz . Ma z

1/

.z/;

(11.2.15)

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B. Baumeister, K.-U. Bux, F. Götze, D. Kielak, H. Krause

of Voiculescu is a multiplicative homomorphism for free multiplicative convolution. That is, see [44, Lect. 18], one has the following result. Lemma 11.2.6. For two free self-adjoint positive elements a; b 2 A, one has Sab .z/ D Sa .z/Sb .z/

(11.2.16)

Since Sa is determined by the spectral measure of a, this means with Sa WD Sa for measures  D a ,  D b we have S .z/ D S .z/S .z/, which uniquely determines the multiplicative free convolution    in terms of the measures  and  on the positive reals via the characterising property of the S -transform. Let s be a semi-circular element as in (11.2.8). Then the moment generating 2 functions of s and s 2 are p given by Ms .z/ D f .z / and Ms 2 .z/ D f .z/ respectively, where f .z/ D .1 1 4z/=.2z/ 1. The corresponding distribution of s 2 is called Marchenko-Pastur or free Poisson law; it is given by the density p.x/ WD p 1 4=x 1 on the interval Œ0; 4. Via the inverse function f . 1/ .z/ D z.1 C z/ 2 2 of f we obtain in view of (11.2.15), Ss 2 .z/ D f .

1/

.z/

1 1Cz D z 1Cz

(11.2.17)

1 ; .1 C z/l

(11.2.18)

z and hence in view of (11.2.15) again Rs. 2 1/ .z/ D 1Cz or Rs 2 .z/ D 1 z z , whereas z from (11.2.14) we deduce with g.z/ WD z.1 C Ms .z// and g . 1/ .z/ D 1Cz 2 that Rs .z/ D g . z1/ .z/ 1 D z 2 . From here, we obtain for free variables t1 ; : : : ; tl with identical distribution given by s 2 , the so-called Marchenko–Pastur distribution, in view of (11.2.17)

St1 :::tl .z/ D St1 .z/l D

which determines the so-called free Bessel distributions, l with support in Œ0; Kl , Kl D .l C 1/lC1 = l l . Their moments are given by the so called Fuss–Catalan num1 bers, that is, if an element a 2 A has S -transform Sa .z/ D .1Cz/ l we have ! 1 lk C 1 k '.a / D DW Ck;l ; for all k > 1: (11.2.19) lk C 1 k The proof is based on combinatorial properties of non crossing partitions, see [5]. Proposition 11.2.7. For a sequence of N  N independent non-Hermitian random matrices, G1 ; : : : Gl , with independent Gaussian centered entries with variance 1=N , let W WD G1


Gl . Consider the normalised moments of W W  . As N ! 1 they converge as follows: Z Kl Z 1 x k dl D Ck;l (11.2.20) tr.W W  /k dP D lim N !1 N  0

Non-crossing partitions

247

This can be shown by induction, using tr.W W  /k D tr.G1 .G2


Gl Gl


G1 G1 /k

1

G2


Gl


Gl


G1 /; (11.2.21)

which by moving G1 to the right yields tr..G2


Gl Gl


G1 G1 /k

1

G2


Gl


Gl


G1 G1 /

D tr..G2


Gl Gl


G1 G1 /k /  k D tr .G2


Gl Gl


G2 /.G1 G1 / :

Since .G2


Gl Gl


G2 / and G1 G1 are asymptotically free (see Section 1.1 of this volume), we get by induction for the asymptotic distribution of l the recursion l D l 1  1 , where 1 can be identified with the limiting Marchenko–Pastur distribution of G1 G1 . For arbitrary N N independent Wigner matrices (which are Hermitian matrices with entries which are independent random variables unless restricted by symmetry) the relation (11.2.20) has been shown by combinatorial techniques after an appropriate regularisation in [2]. For more details on the asymptotic spectral distribution of products of so-called Girko–Ginibre matrices (having independent and identically distributed random entries) and their inverses using the free probability calculus, see [30]. Strictly speaking one needs to extend the non-commutative C  -probability spaces to spaces of unbounded operators to include distributions with non-compact support like those of Gaussian matrices see e.g. [22]. Remarkably, the same results hold for powers instead of products. Since G1l 1 .G1l 1 / and G1 G1 are also asymptotically free, a similar argument as above shows that the asymptotic distribution of .G1l /.G1l / is also given by l . Similarly as above, these results also extend to powers of non-Gaussian random matrices. The calculus of S -transforms may even be used to describe the asymptotic spectral measure of W W  when some of the factors in W D G1


Gl are inverted, after appropriate regularisation of the inverse matrices [30]. For instance, for W D G1 G2 1 , the limiting distribution of W W  is given by the square of a Cauchy distribution. Moreover, the calculus of R-transforms makes it possible, at least in principle, to deal with the case where W is a sum of independent products as above [40]. For instance, for W D G1 G2 1 C G3 G4 1 , the limiting distribution of W W  is also given by the square of a Cauchy distribution. This is related to the Cauchy distribution being “stable” under free additive convolution.

11.3 Braid groups Let D be the unit disk. The braid group Bn on n strands can be defined as the fundamental group of the configuration space Xn WD f f z1 ; : : : ; zn g  D j zi ¤ zj for i ¤ j g

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B. Baumeister, K.-U. Bux, F. Götze, D. Kielak, H. Krause

v3 v4

v2

v5

v1 v6

v8 v7

Figure 11.3.1. The path (braid) that is associated to the partition f f 1 g; f 2; 6; 7; 8 g; f 3; 5 g; f 4 g g.

of unordered n-point-subsets in D. One can visualise a path in Xn as a collection of n distinct points moving continuously in D subject only to the restriction that points are not allowed to collide. Since Xn is connected, the braid group (up to isomorphism) does not depend on the choice of a base point. We find it convenient to choose as the base point a set S D f v1 ; : : : ; vn g of n points on the boundary circle @ D numbered in counter-clockwise order. Then, we regard NC.n/ as the poset of non-crossing partitions of the set S , i.e., for any two distinct blocks of the partition, their convex hulls do not intersect. A noncrossing partition p 2 NC.n/ can be interpreted as a braid on n strands as follows: for each block B D f v˛1 ; : : : ; v˛k g, consider the counter-clockwise rotation of the block by one step: %B W v˛1 7! v˛2 7!


7! v˛k 7! v˛1 The product p WD

Y

%B

B W block of p

describes a loop in the configuration space Xn , which does not depend (up to homotopy relative to the basepoint) on the order of factors. We identify it with the corresponding element of the fundamental group Bn . Fact 11.3.1. The braid group Bn is generated by the braids i corresponding to the counter-clockwise rotations vi 7! vi C1 7! vi for i D 1; : : : ; n 1. In terms of these generators, the braid group Bn admits the following presentation:   i j D j i for ji j j > 2 Bn D 1 ; : : : ; n 1 i j i D j i j for ji j j D 1 There is an obvious homomorphism  W Bn ! Sn

Non-crossing partitions

v1

249

v5

v2

v3

v4

Figure 11.3.2. The generator 2 in the braid group on five strands. On the left, the “top view” representation is shown whereas and on the right we have the “front view” given by a strand diagram.

from the braid group on n strands to the symmetric group on n letters. A braid corresponds to a motion of the n points v1 ; : : : ; vn , and at the end of this motion, the dots may have changed positions. This way, each braid induces a permutation. Fact 11.3.2. The homomorphism  W Bn ! Sn is onto. On the level of presentations, it amounts to making the generators i involutions. Formally: the symmetric group has the presentation * + si sj D sj si for ji j j > 2 Sn D s1 ; : : : ; sn 1 si sj si D sj si sj for ji j j D 1 si D si 1 for all i and the homomorphism  is sending i to si . Strand diagrams are another frequently used visual representation of braids. Recall that a braid is given by a path in configuration space, i.e. the simultaneous motion of n points in the disk D. Parametrizing time by a real number in Œ0; 1, each of those moving points traces out a “strand” in D  Œ0; 1. The diagrams we have used so far can be regarded as a “top view” onto the cylinder D  Œ0; 1. A strand diagram is a view from the front. Here, it is useful to put the initial configuration U with the hemicircle fully visible from the front. Figure 11.3.2 shows the two representations of the generator 2 in B5 . Here, the generator i corresponds to a crossing of the i th and the .i C 1/th strands. The left strand runs over the right strand. We call such a crossing positive. The inverses of the generators correspond to negative crossings. 11.3.1 A classifying space for the braid group Tom Brady [15] has given a construction of a classifying space for braid groups that is strongly related to non-crossing partitions and has found some interesting applications. Recall that the Cayley graph CG† .G/ of a group G relative to a specified generating set † is the graph with vertex set G and edges connecting g to gx for any g 2 G and x 2 † n f 1 g. Note that the requirement x ¤ 1 rules out loops. Obviously, there

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is more structure here: the edge is oriented from g to gx and should be regarded as labeled by the generator x. Observation 11.3.3. Since † is a generating set for G, the Cayley graph CG† .G/ is connected: if we can write an element g as a word "

g D x1"1


xkk in the generators and their inverses, then 1

x1"1

x1"1 x2"2

x1"1 x2"2 x3"3






g

is an edge path connecting the identity element 1 to g. Note that the exponents of the generators tell us whether to traverse edges with or against their orientation. There are two generating sets for the braid group (and the symmetric group) of particular interest to us. First, we consider the digon generators ij corresponding to the counter-clockwise rotation vi 7! vj 7! vi . Let Bn be the Birman–Ko–Leemonoid [12, Section 2], i.e., the monoid generated by all the ij . We remark that Bn is strictly larger than the submonoid of positive braids (those that can be drawn using positive crossings only), which is the monoid generated by the i . We define a partial order on the braid group by: ˇ 6 ˇ0

W”

ˇ

1 0

ˇ 2 Bn

The image sij 2 Sn of ij in the symmetric group is a transposition. Consider the Cayley graph of the symmetric group Sn with respect to the generating set T  Sn of all transpositions. We define a partial order, called the absolute order, on Sn as follows: For permutations ; 2 Sn we declare  6T if there is a geodesic (i.e., shortest possible) path in the Cayley graph connecting the identity 1 to and passing through . Our largest generating set is: €n WD f p j p 2 NC.n/ g  Bn which is in 1-1 correspondence to the non-crossing partition lattice. Let sp denote the image of p in the symmetric group Sn . It turns out that the subset f sp j p 2 NC.n/ g  Sn is the order ideal of the n-cycle 1 7! 2 7!


7! n 7! 1 with respect to the partial order 6T just defined, that is the subset consists of all elements in Sn bounded above by the n-cycle. In fact, we have isomorphisms of various posets: Fact 11.3.4. [see [11, 15]] Let p; q 2 NC.n/. Then the following are equivalent: 1. In NC.n/, we have p  q. 2. In €n , the element p is a left-divisor of q , i.e., there exists r 2 NC.n/ such that q D p r

251

Non-crossing partitions

3. In €n , the element p is a right-divisor of q , i.e., there exists r 2 NC.n/ such that q D r p 4. In the braid group Bn , we have p 6 q . 5. In the symmetric group Sn , we have sp 6T sq . Thus, on €n the three partial orderings given by left-divisibility, right-divisibility, and the partial order 6 from Bn coincide. Moreover, we have isomorphisms NC.n/ Š f p j p 2 NC.n/ g Š f sp j p 2 NC.n/ g of posets. Example 11.3.5. Consider the non-crossing partititions p WD f f 1; 2; 8 g; f 3; 5 g; f 4 g; f 6 g; f 7 g g and q WD f f 1; 2; 6; 7; 8 g; f 3; 5 g; f 4 g g in NC.8/. Here, p  q holds and we expect p to be a left- and right-divisor of q within €8 . Figure 11.3.3 shows the corresponding factorisations. One can interpret the complementary divisors as the blockwise Kreweras complements. In particular, the Kreweras complement yields factorisations of the maximal element in €n .

D

ı

D

ı

Figure 11.3.3. Left and right divisibility in €8 .

252

B. Baumeister, K.-U. Bux, F. Götze, D. Kielak, H. Krause

The braid group Bn has a particularly nice presentation over the generating set €n : Fact 11.3.6 ([15, Thm. 4.8]). The valid equations 1 2 D 3 for 1 ; 2 ; 3 2 €n n f 1 g

(11.3.1)

are a defining set of triangular relations for the braid group Bn with respect to the generating set €n n f 1 g. Let €Q n be the Cayley graph of the braid group Bn with respect to the generating set €n n f 1 g. A clique in €Q n is a set of vertices that are pairwise connected via an edge. As a directed graph, €Q n does not have oriented cycles and each clique is totally ordered by the orientation of edges. Thus, a clique is of the form f ˇ; ˇp1 ; ˇp2 ; : : : ; ˇpk g where p1  p2 


 pk is an ascending chain in NC.n/, and ˇ 2 Bn is some element. We denote by YQn the simplicial complex of cliques (also known as the flag complex induced by the graph) in €Q n . In particular, €Q n is the 1-skeleton of YQn . Observation 11.3.7. All maximal chains in NC.n/ have length n. Hence, all maximal simplices in YQn have dimension n. The most important fact about YQn is its contractibilty. Theorem 11.3.8 ([15, Thm. 6.9 and Cor. 6.11]). The clique complex YQn is contractible, and the braid group Bn acts freely on it. Consequently, the orbit space Yn WD Bn n YQn is a classifying space for the braid group Bn . 11.3.2 Higher generation by subgroups For a subset I  f 1; : : : ; n g let BnI be the subgroup of Bn D 1 .Xn / given by those paths, where the points in f vi j i 2 I g fv g do not move at all. For k 2 f 1; : : : ; n g, we put Bnk WD Bn k , i.e., Bnk is the group th of braids where the k strand is rigid. It is, one might say, a group on n 1 strands and one rod. However, since vk is a point on the boundary @ D, braiding with the rod is impossible. Thus, Bnk really is just an isomorphic copy of Bn 1 inside of Bn . T Similarly, BnI D k2I Bnk is isomorphic to Bn #I . Let NCk .n/ be the lattice of those non-crossing partitions in NC.n/ where the sinT gleton f k g is a block. For a subset I  f 1; : : : ; n g, put NCI .n/ WD k2I NCk .n/. Then, €nI WD f p j p 2 NCI .n/ g is a generating set for BnI . Note that the inclusion BnI ,! Bn induces a bijection €n #I Š €nI . Recall that €n #I is a poset with respect to divisibility. A priory, there are two poset structures on

Non-crossing partitions

253

€nI : one from intrinsic divisibility with quotients again in €nI and one induced from the ambient poset €n , i.e., divisibility where quotients are allowed to be anywhere in €n . However, since €nI D €n \ BnI , the two poset structures coincide. Then, €n #I Š €nI is an isomorphism of posets. Moreover, the order preserving bijection f 1; : : : ; n #I g ! f 1; : : : ; n g n I induces an isomorphism NC.n #I / Š NCI .n/. This isomorphism is compatible with the poset isomorphism from Fact 11.3.4, and we have a commutative square of poset isomorphisms: €nI €n #I



NC.n

NCI .n/

#I /

The identity €nI D €n \ BnI has another consequence:

Observation 11.3.9. Let YQnI be the full subcomplex spanned by BnI as a set of vertices in YQn . Then, YQnI is isomorphic to YQn #I , whence it is contractible by Theorem 11.3.8. For any coset ˇ BnI , regarded as a set of vertices in YQn , the full subcomplex spanned by ˇ BnI is the translate ˇ YQnI and also contractible. Observation 11.3.10. Assume that two coset complexes ˇ YQnI and ˇ 0 YQnJ intersect, N Then ˇ YQ I D ˇN YQ I and ˇ 0 YQ J D ˇN YQ J . In this case, the intersection say in ˇ. n n n n ˇN YQnI \ ˇN YQnJ D ˇN YQnI [J is contractible. Let U WD .U˛ /˛2A be a family of sets. For a subset   A let \ U WD U˛ ˛2

denote the associated intersection. The simplicial complex N. U / WD f   A j ¿ ¤ U g of all index sets whose associated intersection is non-empty is called the nerve of the family U . If U is a family of subcomplexes in a CW complex, one has the following: Theorem 11.3.11 (Nerve Theorem, see [35, Cor. 4G.3]). Suppose U D .U˛ /˛2A is a covering of a simplicial complex X by a family of contractible subcomplexes. Suppose further that, for each  2 N. U /, the intersection U is contractible. Then, the nerve N. U / is homotopy equivalent to X .

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According to Observation 11.3.10, the Nerve Theorem applies in particular to the union: [ [ XQ n WD ˇ YQnk k ˇ 2B n

We deduce:

Proposition 11.3.12. The complex XQn is homotopy equivalent to the nerve N of the family f ˇ Bnk j ˇ 2 Bn ; 1 6 k 6 n g

of cosets.

This relates to higher generation by subgroups as defined by Abels and Holz. Definition 11.3.13 ([1, 2.1]). Let G be a group and let H be a family of subgroups. We say that H is m-generating for G if the coset nerve NG .H/ WD N.f gH j g 2 G; H 2 H g/ is .m

1/-connected.

From Proposition 11.3.12, we conclude immediately: Corollary 11.3.14. The family Bn WD f Bn1 ; : : : ; Bnn g is m-generating for the braid group Bn if and only if XQ n is .m 1/-connected. Recall that Bn acts freely on the simplicial complex YQn . The projection YQn ! Yn is a covering space map. In fact, YQn is the universal cover of Yn and the braid group Bn acts as the group of deck transformations. The subcomplex XQ n is Bn -invariant. Let Xn be its image in Yn . Proposition 11.3.15. The family Bn WD f Bn1 ; : : : ; Bnn g is m-generating for the braid group Bn if and only if the pair .Yn ; Xn / is m-connected. Proof. First, consider the long exact sequence of homotopy groups for the inclusion XQ n 6 YQn :


! 1 .XQ n / ! 1 .YQn / ! 1 .YQn ; XQ n / ! 0 .XQ n / ! 0 .YQn /

Since YQn is contractible, we obtain isomorphisms:

d C1 .YQn ; XQ n / Š d .XQ n /

On the other hand, YQn ! Yn is a covering space projection and therefore enjoys the homotopy lifting property. Moreover, XQ n is the full preimage of Xn . Therefore any map   Bd C1 ; Sd ;  ! .Yn ; Xn ; 1/

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255

lifts uniquely to a map

inducing a map




Bd C1 ; Sd ;  ! YQn ; XQ n ; 1

d C1 .Yn ; Xn / ! d C1 .YQn ; XQ n /

which is inverse to the map d C1 .YQn ; XQ n / ! d C1 .Yn ; Xn / coming from the covering space projection. Thus, we have isomorphisms d C1 .Yn ; Xn / Š d C1 .YQn ; XQ n / Š d .XQ n / 

and the claim follows from Corollary 11.3.14. We can detect 1-generating and 2-generating families by hand.

Remark 11.3.16. For n > 3, the family Bn is 1-generating for Bn , and for n > 4, it is 2-generating. S Proof. A family H is 1-generating for G if and only if H 2H H generates G. It is 2-generating for G if G is the product of the H 2 H amalgamated along their intersections [1, 2.4]. Note that the braid group Bn is generated by counter-clockwise rotations ˇij WD vi 7! vj 7! vi around digons. Thus, Bn WD f Bn1 ; : : : ; Bnn g generates as long as n > 3 since then each digon-generator is contained in some Bnk . Considering the digon-generators for Bn , defining relations are given by braid relations, visible in isomorphic copies of B3 inside Bn , and commutator relations, visible in isomorphic copies of B4 inside Bn . Hence all necessary defining relations are visible in the amalgamated product of the Bnk Š Bn 1 provided n > 5. For n D 4, the challenge is to derive the commutator relations: ˇ12 ˇ34 D ˇ34 ˇ12

and

ˇ23 ˇ41 D ˇ41 ˇ23

We do the first, the second is done analogously. Calculating with only three strands at a time, we find: ˇ12 ˇ34 ˇ24 D ˇ12 ˇ23 ˇ34 D ˇ23 ˇ13 ˇ34 D D ˇ23 ˇ34 ˇ14 D ˇ34 ˇ24 ˇ14 D ˇ34 ˇ12 ˇ24 The desired commutator relation follows.



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Remark 11.3.17. The little computation at the end of the preceeding proof shows that the commutator relations are redundant in the braid group presentation given in [15, Lem. 4.2]. Accordingly, they are also redundant in the analoguous presentation from [12, Prop. 2.1]. Theorem 11.3.18. For n > 4, the family Bn is m-generating for Bn if and only if the homology groups Hd .Yn ; Xn / are trivial for 1 6 d 6 m. Proof. As n > 4, the pair .Yn ; Xn / is 1-connected by Propositions 11.3.15 and 11.3.16. Thus, it follows from the relative Hurewicz theorem that m-connectivity of the pair is equivalent to m-acyclicity. By Proposition 11.3.15, this translates into higher generation of Bn by Bn .  As the pair .Yn ; Xn / consists of finite complexes that can be described explicitly, Theorem 11.3.18 implies that it is a finite problem to determine the higher connectivity properties of Bn relative to the family Bn . In particular, the question whether the bounds derived in Example 13.5.4 for higher generation in braid groups are sharp becomes amenable to empirical investigation. 11.3.3 Curvature in braid groups Definition 11.3.19. For an n  n symmetric matrix .mij / with entries in f2; 3; : : : g [ f1g we define the associated Artin group to be * + s1 ; : : : ; sn

si sj si


D sj si sj


„ ƒ‚ … „ ƒ‚ … mij factors

mij factors

Here, mij D 1 indicates that there is no defining relation for si and sj . We will refer to the relations appearing above as braid relations (even though some authors reserve this term for the relation with mij D 3). If one additionally forces the generators si into being involutions, one obtains the associated Coxeter group. A pair consisting of a Coxeter group together with the generating set fs1 ; : : : ; sn g is called a Coxeter system; its rank is defined to be the cardinality of the generating set. If the Coxeter group is spherical, the Coxeter system is said to be spherical as well. A Coxeter group is spherical if it is finite; an Artin group is spherical if the corresponding Coxeter group is spherical. Note that the braid group Bn is an Artin group and the symmetric group Sn is the associated Coxeter group. Here, mij D 3 for ji j j D 1 and mij D 2 otherwise. See Fact 11.3.1 Artin groups form a rich class of groups of importance in geometric group theory and beyond. From geometric group theory perspective they remain in focus largely due to the following conjecture.

Non-crossing partitions

257

Conjecture 11.3.20 (Charney). Every Artin group is CAT(0), i.e. it acts properly and cocompactly on a CAT(0) space. A CAT(0) space is a metric space with curvature bounded from above by 0; for details see the book by Bridson–Haefliger [19]. From the current perspective let us list some properties of CAT(0) groups: algorithmically, such groups have quadratic Dehn functions and hence soluble word problem; geometrically, all free-abelian subgroups thereof are undistorted; algebraically, the centralisers of infinite cyclic subgroups thereof split; topologically, the space witnessing CAT(0)-ness of a group G is a finite model for EG and thus, for example, allows to compute the K-theory of the reduced C  -algebra Cr .G/ provided the Baum–Connes conjecture is known for G. Conjecture 11.3.20 has been verified by Charney–Davis for right-angled Artin groups (RAAGs), that is for Artin groups with each mij equal to 2 or 1. Outside of this class, the conjecture is mostly open. In particular, it is open (in general) for the braid groups Bn . To prove that a group G is CAT(0), one has to first construct a space X on which G acts properly and cocompactly, and then prove that the space is indeed CAT(0). We shall use the space YQn from above, on which Bn acts freely and with compact quotient. What is missing, however, is a metric structure on YQn . Such a metric can be specified by realising the simplices in euclidean space, i.e., by endowing each simplex in YQn with the metric of a euclidean polytope. Instead of the standard one, we will follow Brady–McCammond [16]. Definition 11.3.21. Let e1 ; : : : ; em denote the standard basis of Rm . The m-orthoscheme is the convex hull of f 0; e1 ; e1 Ce2 ; : : : ; e1 Ce2 C


Cem g. The orthoscheme has the structure of an m-simplex and the vertices come with a grading: the vertex e1 C


C ek is declared to be of rank k. We now endow each maximal simplex in YQn with the orthoscheme metric. Let † D f ˇ; ˇ1 ; : : : ; ˇn g be a maximal simplex. Here, ˇ is a braid in Bn and 1 < 1 < 2 <


< n is a maximal chain in €n Š NC.n/, which has length n by Observation 11.3.7. We endow † with the metric of the standard n-orthoscheme by identifying ˇk with the vertex of rank k in the orthoscheme. It is easy to see that if two maximal simplices intersect, they induce identical metric on their common face. Thus we have turned YQn into a metric simplicial complex. Note that YQn is obtained by gluing copies of a single shape, the n-orthoscheme, and so YQn is a geodesic metric space by a result of Bridson (finitely many shapes of cells would suffice). Since the shape is euclidean, we may use Gromov’s link condition and deduce the following: Lemma 11.3.22. YQn is CAT(0) if and only if the link of each vertex in YQn is CAT(1).

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Here CAT(1) means that the curvature of the space is bounded above by that of the unit sphere; again, for details see [19]. The poset €n Š NC.n/ has a unique maximal element, which is the braid corresponding to the full counter-clockwise rotation: v1 7! v2 7!


7! vm 7! v1

The nth power n is central in the braid group Bn . In fact, it generates the infinite cyclic center of Bn . Brady–McCammond observed in [16] that this algebraic fact has a geometric counterpart: YQn splits as a cartesian product of the real line R and another metric space. The R-factor inside YQn points in the direction of the edges labelled by

. Because of this, instead of looking at the link of a vertex u in YQn , one can look at the link of a midpoint of the (long) edge .u; u /; every two such links are isometric (since Bn acts transitively on the vertices of YQn ), and so let L denote any such link. To compute the curvature of L, it is enough to study the subcomplex of YQn spanned by all simplices containing the edge .u; u /. Clearly, this is the subcomplex spanned by L and u with 1 6  6 , with simplices defined by the chain condition as before. Thus, such a link is isomorphic as a simplicial complex to the realisation of NC.n/; the subcomplex also comes with a metric, and it is clear that this coincides with the realisation of NC.n/ being endowed with its own orthoscheme metric defined as before by identifying each maximal simplex with the n-orthoscheme. We will refer to the realisation of NC.n/ with this metric simply as the orthoscheme complex of NC.n/. Note that if the orthoscheme complex of NC.n/ is CAT(0), then L, isometric to the link of the midpoint of the main diagonal, is CAT.1/, which implies that YQn , and so Bn , is CAT(0). In view of the above, Brady–McCammond formulate the following conjecture. Conjecture 11.3.23 ([16, Conj. 8.4]). For every n, the orthoscheme complex of NC.n/ is CAT(0), and so the braid group Bn is CAT(0). For n 6 4, the conjecture is easily seen to be true. If we know that the orthoscheme complexes of NC.m/ are CAT(0) for each m < n, then in fact the orthoscheme complex of NC.n/ is CAT(0) if and only if the link L is CAT(1). Thus, for n D 5, it is enough to study L, which is the realisation of the poset obtained from NC.n/ by removing the trivial and improper partitions, and endowing the realisation with the spherical orthoscheme metric. Knowing that the conjecture is true for all m < 5 tells us that L is locally CAT(1). Thus, using the work of Bowditch [14], it is enough to check whether any loop in L of length less than 2 can be shrunk, i.e., homotoped to the trivial loop without increasing its length in the process. Brady–McCammond use a computer to analyse all loops in L shorter than 2, and show that they are indeed shrinkable, thus establishing: Theorem 11.3.24 ([16, Thm. B]). For n 6 5, the braid group Bn is CAT(0).

259

Non-crossing partitions

Haettel, Kielak and Schwer go beyond that, proving Theorem 11.3.25 ([32, Cor. 4.18]). For n 6 6, the braid group Bn is CAT(0). Note that their proof is not computer assisted. The crucial improvement in the work of Haettel–Kielak–Schwer is to use the observation (present already in [16]), that the link L can be embedded into a spherical building, in the following way. First observe that the vertices of L are non-trivial proper partitions; let p be such a partition with blocks B1 ; : : : ; Bk . Let F be the field of two elements; we associate to p the subspace of Fn D hb1 ; : : : ; bn i which is the intersections of the kernels of the characters X bj D 0 j 2Bi

where 1 6 i 6 k, and bj is the j -th character in the basis dual to the bj . It is easy to see that this gives
sending each vertex of L to a proper nonPn a map  b trivial subspace of V WD ker j D1 j . But these subspaces are precisely the vertices of the spherical building of SLn 1 .F/, and it turns out that our bijection extends to a map sending each maximal simplex in L onto a chamber (i.e. maximal simplex) in the building in an isometric way. Thus we may view L as a subcomplex of the building. The spherical building is CAT(1), and this information gives the extra leverage used to prove Theorem 11.3.25.

11.4 Non-crossing partitions in Coxeter groups In this section, we introduce the general theory of non-crossing partitions and explain how non-crossing partitions appear in group theory. As already observed in the beginning of Section 11.3.3, the symmetric group Sn is a Coxeter group and .Sn ; Str / is a Coxeter system of rank n 1 where Str WD f.i; i C 1/ j 1 6 i 6 n

1g

is the set of neighbouring transpositions. Every Coxeter system .W; S / acts faithfully on a real vector space that is equipped with a symmetric bilinear form . ; / such that for every s 2 S there is a vector ˛s 2 V so that s acts as the reflection r˛s W v 7! v

2

.v; ˛s / ˛s .˛s ; ˛s /

on V . Thus every Coxeter group is a reflection group that is a group generated by a set of reflections on a vector space .V; . ; //. The vectors ˛s can be chosen so that the subset ˆ D fw.˛s / j s 2 S; w 2 W g of V is a so called root system. For a spherical Coxeter system a root system ˆ is characterised by the following three axioms

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(R1) ˆ generates V ; (R2) ˆ \ R˛ D f˙˛g for all ˛ 2 ˆ;

(R3) s˛ .ˇ/ is in ˆ for all ˛; ˇ 2 ˆ.

The spherical Coxeter groups W are precisely the finite real reflection groups. Coxeter classified the finite root systems which then also gives a classification of the spherical Coxeter systems: there are the infinite families of type An ; Bn ; Cn and Dn and some exceptional groups. For instance .Sn ; Str / is of type An 1 . Note that the groups of type Bn and Cn are isomorphic; and also that the root systems of type An ; Bn ; Cn and Dn are all crystallographic that is .˛; ˇ/ 2 Z for all ˛; ˇ 2 ˆ: .˛; ˛/ We call T WD [w2W w 1 S w the set of reflections of the Coxeter system .W; S /. If the system is spherical, then T is indeed the set of all reflections. For instance in the symmetric group Sn the set T is the conjugacy class of transpositions, see also Section 11.3. There the so called absolute order 6T on Sn has been introduced. Let Œid; .1; 2; : : : ; n/6T be the closed intervall in Sn with respect to 6T . In Fact 11.3.4 it has been stated that .NC.n/; / and .Œid; .1; 2; : : : ; n/6T ; 6T / are posets that are isomorphic. Therefore NC.n/ can be thought of being of type An 1 . Out of combinatorial interest, Reiner generalised the concept of non-crossing partitions to the infinite series of type Bn and Dn geometrically [45]. Independently of his work and of each other Brady and Watt [17] as well as Bessis [10] generalised the concept of non-crossing partitions to all the finite Coxeter systems. Their approach agrees with Reiner’s in type Bn [4]. Brady and Watt as well as Bessis started independently the study of the dual Coxeter system .W; T / instead of .W; S /. A dual Coxeter system .W; T / of finite rank n has the property that there is a subset S of T such that .W; S / is a Coxeter system [10]. It then follows that T is the set of reflections in .W; S /. This concept is called by Bessis dual approach to Coxeter and Artin groups. A (parabolic) standard Coxeter element in .W; S / is the product of all the elements in (a subset of) S in some order and a (parabolic) Coxeter element in .W; T / is a (parabolic) standard Coxeter element in .W; S / for some simple system S in T for W . For instance in type An 1 , so in the symmetric group Sn , the standard Coxeter elements with respect to S D Str are precisely those n-cycles in Sn that can be written as a first increasing and then decreasing cycle. All the n-cycles in Sn are the Coxeter elements in the dual system .Sn ; T / where T is the set of reflections, that is the conjugacy class of transpositions. The partial order 6T on the symmetric group Sn presented in Section 11.3 can be generalised to all the dual Coxeter systems .W; T /. We consider the Cayley graph CGT .W / of the group W with respect to the generating set T . For u; v 2 W we declare u 6T v if there is a geodesic path in the Cayley graph connecting the identity to v and passing through u. This partial order is also called the absolute order on W .

261

Non-crossing partitions

We also introduce a length function lT on W : for u 2 W we define lT .u/ D k if there is a geodesic path from the identity to u of length k in the Cayley graph. Notice, if lT .u/ D m then u is the product of m reflections, that is u D t1


tm with ti 2 T , and there is no shorter factorisation of u in a product of reflections. In this case we say that u D t1


tm is a T -reduced factorisation of u. In particular, if u 6T v, then there are k; m 2 N with k 6 m and reflections t1 ; : : : ; tm in T such that u D t1


tk and v D t1


tm . Thus u 6T v if and only if lT .u/ C lT .u

1

v/ D lT .v/:

Definition 11.4.1. For a dual Coxeter system .W; T / and a Coxeter element c in W the set of non-crossing partitions is NC.W; c/ D fu 2 W j u 6T cg: This definition is conform with the definition in type An , see Fact 11.3.4. The length function lT yields a grading on NC.W; c/ and the map d W NC.W; c/ ! NC.W; c/; x 7! x

1

c

a duality on NC.W; c/ that inverses the order relation. This implies the following. Fact 11.4.2. NC.W; c/ is a poset that is  graded

 selfdual

 [18, 10] a lattice if W is spherical. The number of elements in NC.W; c/ in a finite dual Coxeter system of type X is the generalised Catalan number of type X . In types Bn and Dn there are also nice geometric models for the posets of non-crossing partitions. Note that in a spherical Coxeter system always T  NC.W; c/. There is also a presentation of W with generating set T [10]. The relations are the so called dual braid relations with respect to a Coxeter element c 2 W : for every s; t; t 0 2 T set st D t 0 s whenever the relation st D t 0 s holds in W and st 6T c:

The Matsumoto property means if we have for some w 2 W two shortest factorisations as products of elements of S , or equivalently two geodesic paths from id to w in the Cayley graph CGS .W /, then we can transform one factorisation or path into the other one just by applying braid relations; that is W has a group presentation as given in Definition 11.3.19.

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The dual Matsumoto property for a Coxeter element c 2 W is the statement that if we have two shortest factorisations c D t1


tm D u1


um with ti ; ui 2 T as products of elements of T , that is two T -reduced factorisations of c in W , then one factorisation can be transformed into the other one just by applying dual braid relations. It follows that the dual Matsumoto property holds for c, since hT j dual braid relationsi is a presentation of W . We obtain the dual Matsumoto property for an arbitrary element w 2 W by replacing c by w in the definition of the dual braid relations and of the dual Matsumoto property above. For an element w 2 W , let RedT .w/ D f.t1 ; : : : ; tm / j ti 2 T and w D t1


tm is T -reducedg: The dual Matsumoto property for w 2 W is equivalent to the transitive Hurwitz action of the braid group BlT .w/ on the set of T -reduced factorisations RedT .w/ of w. For the braid i 2 BlT .w/ , see Fact 11.3.1, the action is given by i .t1 ; : : : ; tn / D .t1 ; : : : ; ti

1 ; ti

1

ti C1 ti ; ti ; ti C2 ; : : : ; tn /:

We will discuss this action in more detail in the next section. The dual approach can also be applied to Artin groups; given a Coxeter system .W; S /, we will denote the corresponding Artin group by A.W; S /. If in the following the Coxeter system .W; S / is of type X , then we abbreviate A.W; S / either by A.W / or by AX . Further we take a copy Sa of S in A.W; S / and write

A.W; S / WD hSa j .s1 /a .s2 /a .s1 /a


D .s2 /a .s1 /a .s2 /a


for s1 ; s2 2 S i in order to distinguish between W and A.W /. We call an Artin group A.W / spherical if the Coxeter group is spherical. And in the rest of this section, we always consider spherical Artin groups. Notice that the Matsumoto property implies that one can lift every w 2 W to an element in A.W / just by mapping w to .s1 /a


.sk /a 2 AW whenever w D s1


sk is a reduced factorisation of w into elements of S . We denote this section of W in A.W / by W . The non-crossing partitions are a good tool for the better understanding of the spherical Artin groups; for instance they can be used to construct a finite simplicial classifying space for the spherical Artin groups (see Section 11.3.1), or to solve the word or the conjugacy problem in them, see [17, 10]. The basic idea of this solution of the word and the conjugacy problem in the spherical Artin group A.W / is to give a new presentation of A.W / as follows. Let

Non-crossing partitions

263

NC.W; c/a be a copy of the set of non-crossing partitions NC.W; c/ with respect to a standard Coxeter element c, that is there is a bijection a W NC.W; c/ ! NC.W; c/a : Then the new generating set is NC.W; c/a ; and the new relations are the expressions .w1 /a


.wr /a whenever w1 ; w2 ; : : : ; wr are the vertices of a circuit in Œid; c6T  CGNC.W;c/ .W /: Then this presentation can be used to obtain a new normal form for the elements in A.W / [10]. Notice that this presentation generalises the presentation of the braid group given by Birman, Ko and Lee [12] to all the spherical Artin groups, see also Fact 11.3.6 in Section 11.3.1. Next, we explain this new presentation. Denote the group given by the presentation above by A.W; c/. The strategy to prove that A.W; c/ and A.W / are isomorphic is to use Garside theory. As a first step the presentation above can be transformed into a presentation with set of generators a copy Ta D fta j t 2 T g of T and set of relations the dual braid relations with respect to c. The next step is to consider the monoid A.W; c/ generated by Ta and the dual braid relations, and to show that this is a Garside monoid. Then using Garside theory one shows that the group of fractions Frac.A.W; c// of A.W; c/ equals A.W; c/. The last step is to prove that the group of fractions Frac.A.W; c// and the Artin group A.W / are isomorphic. Theorem 11.4.3 ([10]). Let AW be a spherical Artin group. Then,

AW Š hTa j ta ta0 D .t t 0 t/a ta if t; t 0 2 T and t t 0 6T ci: Note also that a basic ingredient in the proof of Theorem 11.4.3 is the dual Matsumoto property for c, that is the transitivity of the Hurwitz action of the braid group BlT .c/ on RedT .c/. The isomorphism between A.W; c/ and AW given by Bessis is difficult to understand explicitly. So an immediate question is what the elements of NC.W; c/a are expressed in the generating set Sa ? The rational permutation braids, that is, the elements xy 1 where x; y 2 W , are also called Mikado braids as they satisfy in type An 1 a topological condition and are therefore easy to recognise. This condition on an element in the Artin group A.W / of type An 1 , that is on a braid in the braid group Bn , is that we can lift and remove continuously one strand after the next of the braid without disturbing the remaining strands until we reach an empty braid [26]. Theorem 11.4.4. If AW is spherical Artin group and c 2 W a standard Coxeter element, then the dual generators of A.W; c/, that is the elements of NC.W; c/a , are Mikado braids in AW . Proof. This is [26] for those groups of type different from Dn and [8] for those of type Dn . 

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Figure 11.4.1. A Mikado braid in AB8 whose image in AB8 is a Mikado braid in AD8 .

Notice that Licata and Queffelec [43] have a proof of Theorem 11.4.4 in types A,D,E with a different approach using categorification. In order to be able to find a topological property that characterises the Mikado braids as in type An 1 topological models for the series of spherical Artin groups AW are needed. There is an embedding of Artin groups of type Bn into those of type A2n 1 . The situation in type Dn is as follows [8]: The root system of type Dn embeds into the root system of type Bn , which implies that the Coxeter system of type Dn is a subsystem of that one of type Bn . But there is not an embedding of the Artin group of type Dn into that one of type Bn that satisfies a certain natural condition. Let .W; S / be a Coxeter system of type Bn . Then there is precisely one element s 2 S that is a reflection corresponding to a short root. Let

ABn WD ABn =hhs 2 ii; where hhs 2 ii is the normal closure of s 2 in ABn . Then the following holds. Proposition 11.4.5 ([8, Lem. 2.5 and Prop. 2.7]). There is a natural embedding of ADn onto an index-2 subgroup of ABn . More precisely, there is the following commutative diagram

A Bn



A Bn



! A Bn ? ? y B ! WBn

B

ht1 ; : : : ; tn i ? ? D y

Š

ADn

WDn

The embedding of ADn into ABn makes it possible to associate braid pictures to the ADn -elements and to characterise Mikado braids in type Dn geometrically. A reader familiar with Hecke algebras will find it interesting that the Mikado braids satisfy a positivity property involving the canonical Kazhdan-Lusztig basis C WD fCw j w 2 W g of the Iwahori–Hecke algebra H.W / related to the Coxeter

Non-crossing partitions

265

system .W; S /, see [39, 26]. There is a natural group homomorphism a W AW ! H.W / from AW into the multiplicative group H.W / of H.W /. The image of a Mikado braid, that is of a rational permutation braid, in H.W / has as coefficients Laurent polynomials with non-negative coefficients when expressed in the canonical basis C by a result by Dyer and Lehrer (see [28, 26]).

11.5 The Hurwitz action Hurwitz action in Coxeter systems. Deligne showed the dual Matsumoto property in spherical Coxeter systems, that is he showed that the Hurwitz action of the braid group BlT .c/ on RedT .c/ is transitive for every Coxeter element c in .W; S / [25]; and Igusa and Schiffler proved it for arbitrary Coxeter systems [37]. In [6] a new, more general and first of all constructive proof of this property is given: Theorem 11.5.1 ([6, Thm. 1.3]). Let .W; T / be a (finite or infinite) dual Coxeter system of finite rank n and let c D s1


sm be a parabolic Coxeter element in W. The Hurwitz action on RedT .c/ is transitive. Theorem 11.5.1 is also more general than Theorem 1.4 in [37], as in [6] dual Coxeter systems are considered while in [37] Coxeter systems, and in general the set of Coxeter elements is in a dual system larger than that one in a Coxeter system. The proof of Thereom 11.5.1 is based on a study of the Cayley graphs CGS .W / and CGT .W /. Using the same methods one can also show that every reflection occurring in a reduced T -factorisation of an element of a parabolic subgroup P of W is already contained in that parabolic subgroup. Theorem 11.5.2 ([6, Thm. 1.4]). Let .W; S / be a (finite or infinite) Coxeter system, P a parabolic subgroup and w 2 P . Then RedT .w/ D RedT \P .w/. This basic fact was not known before and can be seen as a founding stone towards a general theory for ‘dual’ Coxeter systems. Hurwitz action in the spherical Coxeter systems and quasi-Coxeter elements. In the rest of the section, .W; T / is a finite dual Coxeter system. In order to understand the dual Coxeter systems .W; T / one also needs to know for which elements in W the Hurwitz action is transitive. The answer to that question is as follows [7]. A parabolic quasi-Coxeter element is an element w 2 W that has a reduced factorisation into reflections such that these reflections generate a parabolic subgroup of W . Note if one reduced T -factorisation of w 2 W generates a parabolic subgroup P then every reduced T -factorisation of w is in P by Theorem 11.5.2. It also follows that every such factorisation generates P [7, Thm. 1.2].

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If a factorisation of w generates the whole group W , it is a quasi-Coxeter element. Clearly every Coxeter element is a quasi-Coxeter element. In type An and Bn every quasi-Coxeter element is already a Coxeter element. The smallest Coxeter system containing a proper quasi-Coxeter element is of type D4 . Now we can answer the question above. Theorem 11.5.3 ([7, Thm. 1.1]). Let .W; S / be a spherical Coxeter system and let w 2 W . The Hurwitz action is transitive on RedT .w/ if and only if w is a parabolic quasi-Coxeter element. Recently, Wegener showed that the dual Matsumoto property holds for quasiCoxeter elements in affine Coxeter systems as well [52]. These two results have the following consequence. Corollary 11.5.4. Let .W; T / be a dual Coxeter system, w 2 W and w D t1


tm a reduced T -factorisation, then the Hurwitz action is transitive on RedT .w/ in the Coxeter group W 0 WD ht1 ; : : : ; tm i whenever W 0 is a spherical or an affine Coxeter group. Proof. According to Theorem 3.3 of [27], W 0 WD ht1 ; : : : ; tm i is a Coxeter group. Theorem 11.5.3 and the main result in [52] then yield the statement.  The (parabolic) quasi-Coxeter elements are interesting for more reasons; for instance also for the following. Let ˆ be the root system related to .W; S / and let L.ˆ/ WD Zˆ and L.ˆ_ / WD Zˆ_ where ˛ _ WD 2˛=.˛; ˛/ be the root and the coroot lattices, respectively. Quasi-Coxeter elements are also intrinsic in the dual Coxeter systems as they generate the root as well as the coroot lattice: Let w D t1


tn be a reduced T -factorisation of w 2 W and let ˛i 2 ˆ be the root related to the reflection ti for 1 6 i 6 n. Theorem 11.5.5 ([9, Thm. 1.1]). Let ˆ be a finite crystallographic root system of rank n. Then w is a quasi-Coxeter element if and only if 1. f˛i j 1 6 i 6 ng is a Z-basis of the root lattice L.ˆ/, and

2. f˛i_ j 1 6 i 6 ng is a Z-basis of the coroot lattice L.ˆ_ /. Thus if all the roots in ˆ are of the same length, then L.ˆ/ D L.ˆ_ / and the quasi-Coxeter elements correspond precisely to the basis of the root lattice. Quasi-Coxeter elements and Coxeter elements share further important properties beyond Hurwitz transitivity. Theorem 11.5.6 ([7, Cor. 6.11]). An element x 2 W is a parabolic quasi-Coxeter element if and only if x 6T w for a quasi-Coxeter element w.

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Finally, Gobet observed that, in a spherical Coxeter system, every parabolic quasiCoxeter element can be uniquely written as a product of commuting parabolic quasiCoxeter elements [31]. This factorisation of a quasi-Coxeter element can be thought of as a generalisation of the unique disjoint cycle decomposition of a permutation.

11.6 Non-crossing partitions arising in representation theory In this section, we explain how non-crossing partitions arise naturally in representation theory. For any finite dimensional algebra A over a field k we consider the category mod A of finite dimensional (right) A-modules and denote by K0 .A/ its Grothendieck group. This group is free abelian of finite rank, and a representative set of simple A-modules S1 ; : : : ; Sn provides a basis e1 ; : : : ; en if one sets ei D ŒSi  for all i . As usual, we denote for any A-module X by ŒX  the corresponding class in K0 .A/. The Grothendieck group comes equipped with the Euler form K0 .A/  K0 .A/ ! Z given by X hŒX ; ŒY i D . 1/n dimk ExtnA .X; Y / n>0

which is bilinear and non-degenerate (assuming that A is of finite global dimension). The corresponding symmetrised form is given by .x; y/ D hx; yi C hy; xi. For a class x D ŒX  given by a module X , one defines the reflection sx W K0 .A/ ! K0 .A/;

a 7! a

2

.a; x/ x; .x; x/

(11.6.1)

assuming that .x; x/ ¤ 0 divides .ei ; x/ for all i . Let us denote by W .A/ the group of automorphisms of K0 .A/ that is generated by the set of simple reflections S.A/ D fse1 ; : : : ; sen g; it is called the Weyl group of A. From now on, assume that A is hereditary, that is, of global dimension at most one. Then, one can show that the Weyl group W .A/ is actually a Coxeter group. For example, the path algebra kQ of any quiver Q is hereditary and in that case kQ-modules identify with k-linear representations of Q. Proposition 11.6.1 ([36, Thm. B.2]). A Coxeter system .W; S / is of the form .W .A/; S.A// for some finite dimensional hereditary algebra A if and only if it is crystallographic in the following sense: 1. mst 2 f2; 3; 4; 6; 1g for all s ¤ t in S , and

2. in each circuit of the Coxeter graph not containing the edge label 1, the number of edges labelled 4 (resp. 6) is even.

We may assume that the simple A-modules are numbered in such a way that hei ; ej i D 0 for i > j , and we set c D se1


sen . Note that c D c.A/ is a Coxeter

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element which is determined by the formula hx; yi D hy; c.x/i

for

x; y 2 K0 .A/:

We are now in a position to formulate a theorem which provides an explicit bijection between certain subcategories of mod A and the non-crossing partitions in NC.W .A/; c/. Call a full subcategory C  mod A thick if it is closed under direct summands and satisfies the following two-out-of-three property: any exact sequence 0 ! X ! Y ! Z ! 0 of A-modules lies in C if two of fX; Y; Zg are in C . A subcategory is coreflective if the inclusion functor admits a right adjoint. Theorem 11.6.2. Let A be a hereditary finite dimensional algebra. Then, there is an order preserving bijection between the set of thick and coreflective subcategories of mod A (ordered by inclusion) and the partially ordered set of non-crossing partitions NC.W .A/; c/. The map sends a subcategory which is generated by an exceptional sequence E D .E1 ; : : : ; Er / to the product of reflections sE D sE1


sEr . The rest of this article is devoted to explaining this result. In particular, the crucial notion of an exceptional sequence will be discussed. This result goes back to beautiful work of Ingalls and Thomas [38]. It was then established for arbitary path algebras by Igusa, Schiffler, and Thomas [37], and we refer to [36] for the general case. Observe that path algebras of quivers cover only the Coxeter groups of simply laced type (via the correspondence A 7! W .A/); so there are further hereditary algebras. We may think of Theorem 11.6.2 as a categorification of the poset of non-crossing partitions. There is an immediate (and easy) consequence which is not obvious at all from the original definition of non-crossing partitions; the first (combinatorial) proof required a case by case analysis. Corollary 11.6.3. For a finite crystallographic Coxeter group, the corresponding poset of non-crossing partitions is a lattice. Proof. Any finite Coxeter group can be realised as the the Weyl group W .A/ of a hereditary algebra of finite representation type. In that case any thick subcategory is coreflective. On the other hand, it is clear from the definition that the intersection of any collection of thick subcategories is again thick. This yields the join, but also the meet operation; so the poset of thick and coreflective subcategories is actually a  lattice; see Remark 11.1.1 This categorification provides some further insight into the collection of all posets of non-crossing partitions. This is based on the simple observation that any thick and coreflective subcategory C  mod A (given by an exceptional sequence E D .E1 ; : : : ; Er /) is again the module category of a finite dimensional hereditary algebra, say C D mod B. Then the inclusion mod B ! mod A induces not only an inclusion K0 .B/ ! K0 .A/, but also an inclusion W .B/ ! W .A/ for the corresponding Weyl groups, which identifies W .B/ with the subgroup of W .A/ generated by sE1 ; : : : ; sEr ,

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and identifies the Coxeter element c.B/ with the non-crossing partition sE in W .A/. Moreover, the inclusion W .B/ ! W .A/ induces an isomorphism 

NC.W .B/; c.B// ! fx 2 NC.W .A/; c.A// j x 6 sE g: The following result summarises this discussion; it reflects the fact that there is a category of non-crossing partitions. This means that we consider a poset of noncrossing partitions not as a single object but look instead at the relation with other posets of non-crossing partitions. Corollary 11.6.4 ([36, Cor. 5.8]). Let NC.W; c/ be the poset of non-crossing partitions given by a crystallographic Coxeter group W . Then, any element x 2 NC.W; c/ is the Coxeter element of a subgroup W 0 6 W that is again a crystallographic Coxeter group. Moreover, NC.W 0 ; x/ D fy 2 NC.W; c/ j y 6 xg:

11.7 Generalised Cartan lattices Coxeter groups and non-crossing partitions are closely related to root systems. The approach via representation theory provides a natural setting, because the Grothendieck group equipped with the Euler form determines a root system; we call this a generalised Cartan lattice and refer to [36] for a detailed study. The following definition formalises the properties of the Grothendieck group K0 .A/. A generalised Cartan lattice is a free abelian group € Š Z n with an ordered standard basis e1 ; : : : ; en and a bilinear form h ; iW €  € ! Z satisfying the following conditions. 1. hei ; ei i > 0 and hei ; ei i divides hei ; ej i for all i; j . 2. hei ; ej i D 0 for all i > j . 3. hei ; ej i 6 0 for all i < j . The corresponding symmetrised form is .x; y/ D hx; yi C hy; xi

for x; y 2 €:

The ordering of the basis yields the Coxeter element cox.€/ WD se1


sen : We can define reflections sx as in (11.6.1) and denote by W D W .€/ the corresponding Weyl group, which is the subgroup of Aut.€/ generated by the simple reflections se1 ; : : : ; sen . We write NC.€/ D NC.W; c/ with c D cox.€/ for the poset of noncrossing partitions, and the set of real roots is ˆ.€/ WD fw.ei / j w 2 W .€/; 1 6 i 6 ng  €:

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A real exceptional sequence of € is a sequence .x1 ; : : : ; xr / of elements that can be extended to a basis x1 ; : : : ; xn of € consisting of real roots and satisfying hxi ; xj i D 0 for all i > j . A morphisms € 0 ! € of generalised Cartan lattices is given by an isometry (morphism of abelian groups preserving the bilinear form h ; i) that maps the standard basis of € 0 to a real exceptional sequence of €. This yields a category of generalised Cartan lattices. What is this category good for? One of the basic principles of category theory is Yoneda’s lemma which tells us that we understand an object € by looking at the representable functor Hom. ; €/ which records all morphisms that are received by €. In our category all morphisms are monomorphisms, so Hom. ; €/ amounts to the poset of subobjects (equivalence classes of monomorphisms € 0 ! €). Theorem 11.7.1 ([36, Thm 5.6]). The poset of subobjects of a generalised Cartan lattice € is isomorphic to the poset of non-crossing partitions NC.€/. The isomorphism sends a monomorphism W € 0 ! € to s.e1 /


s.er / where cox.€ 0 / D se1


ser . Moreover, the assignment w 7! wj€ 0 induces an isomorphism 

W .€/  hs.e1 / ; : : : ; s.er / i ! W .€ 0 /:

11.8 Braid group actions on exceptional sequences The link between representation theory and non-crossing partitions is based on the notion of an exceptional sequence and the action of the braid group on the collection of complete exceptional sequences. This will be explained in the following section. There are two sorts of abelian categories that we need to consider. This follows from a theorem of Happel [33, 34] which we now explain. Fix a field k and consider a connected hereditary abelian category A that is k-linear with finite dimensional Hom and Ext spaces. Suppose in addition that A admits a tilting object. This is by definition an object T in A with Ext1A .T; T / D 0 such that HomA .T; A/ D 0 and Ext1A .T; A/ D 0 imply A D 0. Thus the functor HomA .T; /W A ! mod ƒ into the category of modules over the endomorphism algebra ƒ D EndA .T / induces an equivalence  Db .A/ ! Db .mod ƒ/ of derived categories [3]. There are two important classes of such hereditary abelian categories admitting a tilting object: module categories over hereditary algebras, and categories of coherent sheaves on weighted projective lines in the sense of Geigle and Lenzing [29]. Happel’s theorem then states that there are no further classes. Theorem 11.8.1 (Happel). A hereditary abelian category with a tilting object is, up to a derived equivalence, either of the form mod A for some finite dimensional hereditary algebra A or of the form coh X for some weighted projective line X.

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It is interesting to observe that these abelian categories form a category: Any thick and coreflective subcategory is again an abelian category of that type; so the morphisms are given by such inclusion functors. Now, fix an abelian category A which is either of the form A D mod A or A D coh X, as above. Note that in both cases the Grothendieck group K0 .A/ is free of finite rank and equipped with an Euler form, as explained before. An object X in A is called exceptional if it is indecomposable and Ext1A .X; X / D 0. A sequence .X1 ; : : : ; Xr / of objects is called exceptional if each Xi is exceptional and HomA .Xi ; Xj / D 0 D Ext1A .Xi ; Xj / for all i > j . Such a sequence is complete if r equals the rank of the Grothendieck group K0 .A/. Let n denote rank of K0 .A/. Then, the braid group Bn on n strands is acting on the collection of isomorphism classes of complete exceptional sequences in A via mutations, and it is an important theorem that this action is transitive (due to Crawley-Boevey [24] and Ringel [46] for module categories, and Kussin–Meltzer [42] for coherent L sheaves). Any tilting object T admits a decomposition T D niD1 Ti such that .T1 ; : : : ; Tn / is a complete exceptional sequence. We denote by W .A/ the group of automorphisms of K0 .A/ that is generated by the corresponding reflections sT1 ; : : : ; sTn ; it is the Weyl group with Coxeter element c D sT1


sTn and does not depend on the choice of T . Thus we can consider the poset of non-crossing partitions and we have the Hurwitz action on factorisations of the Coxeter element as product of reflections. But it is important to note that W .A/ is not always a Coxeter group when A D coh X, and it is an open question whether the Hurwitz action is transitive. The key observation is now the following. Proposition 11.8.2. The map .E1 ; : : : ; Er / 7 ! sE1


sEr which assigns to an exceptional sequence in A the product of reflections in W .A/ is equivariant for the action of the braid group Br . The proof is straightforward. But a priori it is not clear that the product sE1


sEr is a non-crossing partition. In fact, the proof of Theorem 11.6.2 hinges on the transitivity of the Hurwitz action on factorisations of the Coxeter element. So the analogue of Theorem 11.6.2 for categories of type A D coh X remains open. A proof would provide an interesting extension of the theory of crystallograpic Coxeter groups and non-crossing partitions, which seems very natural in view of Happel’s theorem since the Grothendieck group K0 .A/ is a derived invariant. Partial results were obtained recently by Wegener in his thesis [51]. In fact, when a weighted projective line X is of tubular type (that is, the weight sequence is up to permutation of the form .2; 2; 2; 2/; .3; 3; 3/; .2; 4; 4/ or .2; 3; 6/), then the Grothendieck group gives rise to a tubular elliptic root system [47, 48]. Wegener showed the transitivity of the Hurwitz action in this case. Thus, one has in particular the analogue of Theorem 11.6.2 for coh X in the tubular case.

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References [1] H. Abels and S. Holz, Higher generation by subgroups, J. Algebra 160 (1993), 310–341. [2] N. Alexeev, F. Götze and A. Tikhomirov, Asymptotic distribution of singular values of powers of random matrices, Lith. Math. J. 50 (2010), 121–132. [3] L. Angeleri Hügel, D. Happel and H. Krause (eds.), Handbook of Tilting Theory, Cambridge University Press, Cambridge, 2007. [4] D. Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc. 202 (2009), No. 949. [5] T. Banica, S. Belinschi, M. Capitaine and B. Collins, Free Bessel laws, Canad. J. Math. 63 (2011), 3–37. [6] B. Baumeister, M. Dyer, C. Stump and P. Wegener, A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements, Proc. Amer. Math. Soc. Ser. B 1 (2014), 149–154. [7] B. Baumeister, T. Gobet, K. Roberts and P. Wegener, On the Hurwitz action in finite Coxeter groups, J. Group Th. 20 (2017), 103–131. [8] B. Baumeister and T. Gobet, Simple dual braids, noncrossing partitions and Mikado braids of type Dn , Bull. London Math. Soc., 49 (2017), 1048–1065. [9] B. Baumeister and P. Wegener, A note on Weyl groups and root lattices, Archiv der Mathematik 111 (2018), 469–477. [10] D. Bessis, The dual braid monoid, Ann. Sci. École Normale Sup. 36 (2003), 647–683. [11] P. Biane, Some properties of crossings and partitions, Discr. Math. 175 (1997), 41–53. [12] J. Birman, K.H. Ko and S.J. Lee, A new approach to the word and conjugacy problems in the braid groups, Adv. Math. 139 (1998), 322–353. [13] A. Blass and B.E. Sagan, Möbius functions of lattices, Adv. Math. 127 (1997), 94–123. [14] B.H. Bowditch, Notes on locally CAT(1) spaces. In Geometric Group Theory (R. v. Charney, M. Davis, and M. Shapiro, eds.), de Gruyter, Columbus, OH (1995), 3, 1–48. [15] T. Brady, A partial order on the symmetric group and new K.1/’s for the braid groups, Adv. Math. 161 (2001), 20–40. [16] T. Brady and J. McCammond, Braids, posets and orthoschemes, Algebr. Geom. Topol., 10 (2010), 2277–2314. [17] T. Brady and C. Watt, K.; 1/’s for Artin groups of finite type, Geom. Dedicata 94 (2002), 225–250. [18] T. Brady and C. Watt, Noncrossing partitions lattices in finite real reflection groups, Trans. Amer. Math. Soc. 360 (2008), 1983—2005. [19] M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999. [20] G.P. Chistyakov and F. Götze, Limit theorems in free probability theory. I, Ann. Probab. 36 (2008), 54–90. [21] G.P. Chistyakov and F. Götze, Limit theorems in free probability theory. II, Cent. Eur. J. Math. 6 (2008), 87–117.

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[43] A. Licata and H. Queffelec, Braid groups of type ADE, Garside monoids, and the categorified root lattice, preprint, arXiv:1703.06011. [44] A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, Cambridge University Press, Cambridge, 2006. [45] V. Reiner, Non-crossing partitions for classical reflection groups, Discr. Math 177 (1997), 195–222. [46] C.M. Ringel, The braid group action on the set of exceptional sequences of a hereditary Artin algebra. In Abelian group theory and related topics (Oberwolfach, 1993) (R. Göbel, P. Hill, and W. Liebert, eds.), Contemp. Math. 171, Amer. Math. Soc., Providence, RI (1994), pp. 339–352. [47] K. Saito, Extended affine root systems. I. Coxeter transformations, Publ. Res. Inst. Math. Sci. 21(1) (1985), 75–179. [48] Y. Shiraishi, A. Takahashi and K. Wada, On Weyl groups and Artin groups associated to orbifold projective lines, J. Algebra 453 (2016), 249–290. [49] R.P. Stanley, Enumerative Combinatorics, Volume 1, Cambridge University Press, Cambridge, 1997. [50] D. Voiculescu, Lectures on free probability theory. In Lectures on Probability Theory and Statistics (P. Bernard, ed.), LNM 1738, Springer, Berlin, 2000, 279–349. [51] P. Wegener, Hurwitz Action in Coxeter Groups and Elliptic Weyl Groups, PhD thesis, Univ. Bielefeld, 2017. [52] P. Wegener, On the Hurwtiz action in affine groups, preprint, arXiv:1710.06694.

Chapter 12

The derived category of the projective line H. Krause and G. Stevenson In this chapter, we discuss the localising subcategories of the derived category of quasi-coherent sheaves on the projective line over a field. We provide a complete classification of all such subcategories which arise as the kernel of a cohomological functor to a Grothendieck category.1

12.1 Introduction Ostensibly, this chapter is about the projective line over a field, but secretly it is an invitation to a discussion of some open questions in the study of derived categories. More specifically, we are thinking of localising subcategories and to what extent one can hope for a complete classification. The localising subcategories are of interest for various reasons. For instance, one can localise a derived category in the sense of Verdier [26] by annihilating the objects of a localising subcategory; it is a formal analogue of localisation in ring theory. The case of noetherian affine schemes is by now quite well understood, having been settled by Neeman in his celebrated chromatic tower paper [19]. However, surprisingly little is known in the simplest non-affine case, namely the projective line over a field. We seek to begin to rectify this state of affairs and to advertise this and similar problems. Let us start by recalling what is known. We write QCoh P 1k for the category of quasi-coherent sheaves on the projective line P 1k over a field k, and Coh P 1k denotes the full subcategory of coherent sheaves. There is a complete description of the objects of Coh P 1k , due to Grothendieck [9]; see [5, §2] for a detailed discussion. The colimit closed Serre subcategories of QCoh P 1k are known by work of Gabriel [8], and are parametrised by specialisation closed collections of points of P 1k . When one passes to the derived category D.QCoh P 1k /, the situation becomes considerably more complicated. Several new phenomena appear as a result of the fact that one can no longer non-trivially talk about subobjects and it remains a challenge to provide a complete classification of localising subcategories, which are the derived analogue of colimit closed Serre subcategories. An enticing aspect of this problem is that it not only represents the first stumbling block for those coming from algebraic geometry, but also for the representation

1 Project

C10

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theorists. There is an equivalence of triangulated categories 

D.QCoh P 1k / ! D.Mod A/ where Mod A denotes the module category of the finite dimensional k-algebra i h 2 A D k0 kk

given by 2  2 matrices with the obvious addition and multiplication [6]. The algebra A is isomorphic to the path algebra of the Kronecker quiver
! !
and is known to be of tame representation type. The ring A is one of the simplest algebras that is not of finite representation type, and so understanding its derived category is also a key question from the point of view of representation theory. In particular, it is known by work of Ringel [21, 22] that Mod A, the category of all representations, is wild and so it is very natural to ask if, as in the case of commutative noetherian rings, localisations can nonetheless be classified. In this chapter we make a contribution toward this challenge in two different ways. First of all, one of the main points of this work is to highlight this problem, provide some appropriate background, and set out what is known. To this end the first part of the chapter discusses the various types of localisation one might consider in a compactly generated triangulated category and sketches the localisations of D.QCoh P 1k / which are known. Our second contribution is to provide a new perspective and new tools. The main new result is that the subcategories we understand admit a natural intrinsic characterisation: it is shown in Theorem 12.4.12 that they are precisely the cohomological ones. In the final section, we provide a discussion of the various restrictions that would have to be met by a non-cohomological localising subcategory. Here, our main results are that such subcategories come in Z-families and consist of objects with full support on P 1k .

12.2 Preliminaries This section contains some background on localisations, localising subcategories, purity, and the projective line. It also serves to fix notation and recall general terminology, and thus may be safely skipped, especially by experts, and referred back to as needed. We assume familiarity with the notion of a triangulated category and will not give the basic definitions. But we seek to provide a brief glossary of some of the specialised terminology, belonging to the world of triangulated categories with small coproducts, that will be used throughout. For more fundamental definitions and facts we refer the reader to the book of Neeman [20].

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12.2.1 Localising subcategories and localisations Let T be a triangulated category with all small coproducts and products. The case we have in mind is that T is either well-generated or compactly generated. Definition 12.2.1. A full subcategory L of T is localising if it is closed under suspensions, cones, and coproducts. This is equivalent to saying that L is a coproduct closed triangulated subcategory of T. Remark 12.2.2. It is a consequence of closure under (countable) coproducts that L is closed under direct summands and hence thick (which means closed under finite sums, summands, suspensions, and cones). Given a collection of objects X of T we denote by Loc.X / the localising subcategory generated by X , i.e. the smallest localising subcategory of T containing X . The collection of localising subcategories is partially ordered by inclusion, and forms a lattice (with the caveat it might not be a set) with meet given by intersection. We next present the most basic reasonableness condition a localising subcategory can satisfy. Definition 12.2.3. A localising subcategory L of T is said to be strictly localising if the inclusion i W L ! T admits a right adjoint i Š , i.e. if L is coreflective. Some remarks on this are in order. First of all, it follows that i Š is a Verdier quotient because it has a fully faithful left adjoint, and that there is a localisation sequence

L o

i iŠ

/ T o

j

/ T=L

j

inducing a canonical equivalence 

L? WD fX 2 T j Hom.L; X / D 0g ! T=L: Next we note that in nature localising subcategories tend to be strictly localising. This is, almost uniformly, a consequence of Brown representability; if T is well-generated and L has a generating set of objects then L is strictly localising. Now let us return to the localisation sequence above. From it we obtain two endofunctors of T, namely i i Š and j j  ; which we refer to as the associated acyclisation and localisation respectively. They come together with a counit and a unit which endow them with the structure of an idempotent comonoid and monoid respectively. The localisation (or acyclisation) is equivalent information to L. One can give an abstract definition of a localisation functor on T (or in fact any category) and then work backward from such a functor

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to a strictly localising subcategory. Further details can be found in [16]. We will use the language of (strictly) localising subcategories and localisations interchangeably. 12.2.2 Purity Let T be a compactly generated triangulated category and let Tc denote the thick subcategory of compact objects. We denote by Mod Tc the Grothendieck category of modules over Tc , i.e. the category of contravariant additive functors Tc ! Ab. There is a restricted Yoneda functor H W T ! Mod Tc

defined by HX D Hom. ; X /jTc ;

(12.2.1)

which is cohomological, conservative, and preserves both products and coproducts. Definition 12.2.4. A morphism f W X ! Y in T is a pure-monomorphism (resp. pure-epimorphism) if Hf is a monomorphism (resp. epimorphism). An object I 2 T is pure-injective if every pure-monomorphism I ! X is split, i.e. it is injective with respect to pure-monomorphisms. It is clear from the definition that if I 2 T with HI injective then I is pureinjective. It turns out that the converse is true and so I is pure-injective if and only if HI is injective. Moreover, Brown representability allows one to lift any injective object of Mod Tc uniquely to T and thus one obtains an equivalence of categories 

fpure-injectives in Tg ! finjectives in Mod Tc g: Further details on purity, together with proofs and references for the above facts, can be found, for instance, in [15]. 12.2.3 The projective line Throughout, we will work over a fixed base field k which will be supressed from the notation. For instance, P 1 denotes the projective line P 1k over k. We will denote by  the generic point of P 1 . The points of P 1 that are different from  are closed. A subset V  P 1 is specialisation closed if it is the union of the closures of its points. In our situation, this just says that V is specialisation closed if  2 V implies V D P 1 . As usual, QCoh P 1 is the Grothendieck category of quasi-coherent sheaves on P 1 and Coh P 1 is the full abelian subcategory of coherent sheaves. We use standard notation for the usual ‘distinguished’ objects of QCoh P 1 . The i th twisting sheaf is denoted O .i / and for a point x 2 P 1 we let k.x/ denote the residue field at x. In particular, k./ is the sheaf of rational functions on P 1 . For an object X 2 D.QCoh P 1 / or a localising subcategory L we will often write X.i / and L.i / for X ˝ O.i / and L ˝ O.i / respectively. All functors, unless explicitly mentioned otherwise, are derived. In particular, ˝ denotes the left derived tensor product of quasi-coherent sheaves and Hom the right derived functor of the internal hom in QCoh P 1 . For an object X 2 D.QCoh P 1 / we set supp X D fx 2 P 1 j k.x/ ˝ X ¤ 0g:

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This agrees with the notion of support one gets as in [4] by allowing D.QCoh P 1 / to act on itself; the localising subcategories generated by k.x/ and €x O coincide, where €x denotes the local cohomology functor with respect to x. Remark 12.2.5. Let A be a hereditary abelian category, for example QCoh P 1 . Then Extn .X; Y / vanishes for all n > 1 and therefore every object of the derived category D.A/ decomposes into complexes that are concentrated in a single degree. It follows that the functor H 0 W D.A/ ! A induces a bijection between the localising subcategories of D.A/ and the full subcategories of A that are closed under kernels, cokernels, extensions, and coproducts.

12.3 Types of localisation In this section we give a further review of the notions of localisation, or equivalently localising subcategory, that naturally arise and that we treat in this chapter. These come in various strengths and what is known in general varies accordingly. We take advantage of this review to give a whirlwind tour of certain aspects of the subject and to expose some technical results that are absent from the literature. Unless otherwise specified we will denote by T a compactly generated triangulated category. One also can, and should, consider the well-generated case which arises naturally even when one starts with a compactly generated category. However, our focus will, eventually, be on those categories controlled by pure-injectives which more or less binds us to the compactly generated case. 12.3.1 Smashing localisations In this section we make some brief recollections on the most well understood class of localising subcategories. Definition 12.3.1. A localising subcategory L of T is smashing if it is strictly localising and satisfies one, and hence all, of the following equivalent conditions [16, §5.5]:  the subcategory L? is localising;

 the corresponding localisation functor preserves coproducts, i.e. the right adjoint to T ! T=L preserves coproducts;

 the quotient functor T ! T=L preserves compactness;

 the corresponding acyclisation functor preserves coproducts, i.e. the right adjoint to L ! T preserves coproducts.

The smashing subcategories always form a set. Amongst the smashing subcategories, there is a potentially smaller distinguished set of localising subcategories. Unfortunately, there is no standard way to refer to such categories; the snappy nomenclature only exists for the corresponding localisations.

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Definition 12.3.2. A localisation is finite if its kernel is generated by objects of Tc , i.e. the corresponding localising subcategory is generated by objects which are compact in T. If L is the kernel of a finite localisation then it is smashing. It is also compactly generated, although there are in general many localising subcategories of T which are, as abstract triangulated categories, compactly generated but are not generated by objects compact in T. The smashing conjecture for T asserts that every smashing localisation is a finite localisation. This is true in many situations, for instance it holds for the derived category D.Mod A/ of a ring A when it is commutative noetherian [19] or hereditary [17]. On the other hand, it is known to fail for certain rings (see for instance [14]) and is open in many cases of interest, for example the stable homotopy category. 12.3.2 Cohomological localisations We now come to the next types of localising subcategories in our hierarchy, which are defined by certain orthogonality conditions. This gives a significantly weaker hierarchy of notions than being smashing. First a couple of reminders. An abelian category A is said to be (AB5) if it is cocomplete and if filtered colimits are exact. If in addition A has a generator then it is a Grothendieck category. An additive functor H W T ! A is cohomological if it sends triangles to long exact sequences i.e. given a triangle X

f

g

/ Y

/ Z

h

/ †X

the sequence




/ H.†

1

Z/

†

1h

/ H.X /

H.f /

/ H.Y /

H.g/

/ H.Z/

H.h/

/ H.†X /

/




is exact in A. Definition 12.3.3. A localising subcategory L  T is cohomological if there exists a cohomological functor H W T ! A into an (AB5) abelian category such that H preserves all coproducts and

L D fX 2 T j H.†n X / D 0 for all n 2 Zg; that is L is the kernel of H  . We can extend this definition to an analogue for arbitrary regular cardinals, with Definition 12.3.3 being the @0 or ‘base’ case. The idea is to relax the exactness condition on the target abelian category. This requires a little terminological preparation. Let J be a small category and ˛ a regular cardinal. We say that J is ˛-filtered if for every category I with jI j < ˛, i.e. I has fewer than ˛ arrows, every functor F W I ! J has a cocone. For instance, this implies that any collection of fewer than ˛ objects of

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J has an upper bound and any collection of fewer than ˛ parallel arrows has a weak coequaliser. If ˛ D @0 we just get the usual notion of a filtered category. Let A be an abelian category. We say it satisfies (AB5˛ ) if it is cocomplete and has exact ˛-filtered colimits. Definition 12.3.4. A localising subcategory L  T is ˛-cohomological if there exists an (AB5˛ ) abelian category A and a coproduct preserving cohomological functor H W T ! A such that

L D fX 2 T j H.†n X / D 0 for all n 2 Zg;

that is L is the kernel of H  .

If L is ˛-cohomological then it is clearly ˇ-cohomological for all ˇ > ˛. Remark 12.3.5. An @0 -cohomological localising subcategory is just a cohomological localising subcategory. We will usually stick to the shorter terminology for the sake of brevity and to avoid a proliferation of @’s. We now make a few observations on ˛-cohomological localising subcategories and then make some further remarks on the case ˛ D @0 . Lemma 12.3.6. Smashing subcategories are cohomological. Proof. Suppose L is smashing. Then T=L is compactly generated and for H we can take the composite T ! T=L ! Mod.T=L/c where the latter functor is the restricted Yoneda functor (12.2.1).



Theorem 12.3.7. Let L be an ˛-cohomological localising subcategory. Then L is generated by a set of objects and so it is, in particular, strictly localising. Proof. This follows by applying [16, Thm. 7.1.1] and then [16, Thm. 7.4.1].



Corollary 12.3.8. A localising subcategory L is generated by a set of objects of T if and only if there exists an ˛ such that L is ˛-cohomological. Proof. We have just seen that an ˛-cohomological localising subcategory has a generating set. On the other hand if L is generated by a set of objects then L is wellgenerated, and so is strictly localising, and the quotient T=L is also well-generated (see [16, Thm. 7.2.1]). One can then compose the quotient T ! T=L with the universal functor from T=L to an (AB5˛ ) abelian category, for a sufficiently large ˛, to get the required cohomological functor.  Let us now restrict to cohomological localisations and make the connection to purity in triangulated categories.

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Proposition 12.3.9. A localising subcategory L  T is cohomological if and only if there is a suspension stable collection of pure-injective objects .Yi /i 2I in T such that L D fX 2 T j Hom.X; Yi / D 0 for all i 2 I g. Proof. Recall from (12.2.1) the restricted Yoneda functor which we denote by HT , for clarity, for the duration of the proof. This functor identifies the full subcategory of pure-injective objects in T with the full subcategory of injective objects in Mod Tc as noted earlier (see [15, Cor. 1.9] for details). A cohomological functor H W T ! A that preserves coproducts admits a factorisation H D HN ı HT such that HN W Mod Tc ! A is exact and preserves coproducts; see [15, Proposition 2.3]. The full subcategory Ker HN D fM 2 Mod Tc j HN .M / D 0g is a localising subcategory, so of the form fM 2 Mod Tc j Hom.M; Ni / D 0 for all i 2 I g for a collection of injective objects .Ni /i 2I in Mod Tc . Now choose pure-injective objects .Yi /i 2I in T such that HT .Yi / Š Ni for all i 2 I .  12.3.3 When things are sets As has been alluded to in the previous sections, it is a significant subtlety that one does not usually know the class of all localising subcategories forms a set. In fact there is no example where one knows that there are a set of localising subcategories by ‘abstract means’; all of the examples come from classification results. If one does know there is a set of localising subcategories then life is much easier. The purpose of this section is to give some indication of this, and record some other simple observations. Everything here should be known to experts, but these observations have not yet found a home in the literature. As previously let T be a compactly generated triangulated category. Lemma 12.3.10. If the localising subcategories of T form a set then every localising subcategory is generated by a set of objects (and hence by a single object). Proof. Suppose, for a contradiction, that L is a localising subcategory of T which is not generated by a set of objects. We define a proper chain of proper localising subcategories L0 ¨ L1 ¨


¨ L˛ ¨ L˛C1 ¨


¨ L;

each of which is generated by a set of objects, by transfinite induction. For the base case pick any object X0 of L and set L0 D Loc.X0 /. This is evidently generated by a set of objects, namely fX0 g. By assumption L is not generated by a set of objects so L0 ¨ L. Suppose we have defined a proper localising subcategory L˛ of L which is generated by a set of objects. Since L˛ is proper we may pick an object X˛C1 in L but not in L˛ and set L˛C1 D Loc.L˛ ; X˛C1 / © L˛ :

This is clearly still generated by a set of objects and hence is still a proper subcategory of L. For a limit ordinal  we set

L D Loc.L j  < /: Again this is generated by a set of objects (and so is still not all of L).

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This gives an ordinal indexed chain of distinct localising subcategories of T. However, this is absurd since the collection of ordinals is not a set and so cannot be embedded into the set of all localising subcategories of T. Hence L must have a generating set (i.e. the above construction must terminate).  Remark 12.3.11. The above argument does not use that T is compactly generated. It is valid for any triangulated category with infinite coproducts. One then deduces that all localisations are cohomological for an appropriate cardinal. Lemma 12.3.12. If the localising subcategories of T form a set then every localising subcategory of T is ˛-cohomological for some regular cardinal ˛. Proof. By the previous lemma, the hypothesis imply that every localising subcategory of T is generated by a set of objects. It then follows from Corollary 12.3.8 that they are all cohomological.  One can, to some extent, also work in the other direction. Then the following result is of interest. Proposition 12.3.13. For every regular cardinal ˛ the collection of localising subcategories of T that are ˛-cohomological forms a set. 

Proof. See Theorem 2.3 in [12]. Corollary 12.3.14. The following conditions are equivalent for T: 1. The collection of all localising subcategories of T forms a set. 2. The collection

[

fL j L is ˛-cohomologicalg

˛2Card

forms a set.

3. There exists a regular cardinal  such that every localising subcategory of T is -cohomological. Proof. (1) ) (2): Use Lemma 12.3.12. (2) ) (3): We have seen in Corollary 12.3.8 that being ˛-cohomological for some ˛ is the same as being generated by a set of objects. Thus the hypothesis asserts that there are a set of localising subcategories which have generating sets. From this perspective it is clear we can pick a regular cardinal  such that every localising subcategory of T which is generated by a set is generated by -compact objects. Moreover, since the union in the statement of the lemma is both a set and indexed by a class, we conclude that the chain stabilises and so, taking  larger if necessary, we may also assume every ˛-cohomological localising subcategory of T

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is -cohomological. Thus it is enough to show that, under the hypothesis of (2), every localising subcategory is ˛-cohomological for some ˛, i.e. is generated by a set of objects. Suppose then there is a localising subcategory L which is not generated by a set of objects. In particular L0 D Loc.L \ T / ¨ L:

But this is nonsense. Since L0 is a proper localising subcategory of L we can find some object X in L but not in L0 and consider L00 D Loc.L0 ; X /. Clearly L00 is still contained in L, it properly contains L0 , it is generated by a set and hence -cohomological, and it contains the -compact objects of L. These are not compatible statements: we have assumed  large enough so that L00 must be generated by the -compact objects it contains but this contradicts L0 ¨ L00 .  (3) ) (1): Use Proposition 12.3.13. Remark 12.3.15. Corollary 12.3.14 and the preceding lemmas apply, verbatim, to a well-generated triangulated category. In fact the proofs go through, essentially without modification. The results we use from [16] apply to well-generated categories. 12.3.4 State of the art Let us conclude this section with a brief description of what is known concerning the various conditions we have discussed. It is perhaps more honest to say that we pose a number of questions concerning these definitions. Question 12.3.16. Is there a compactly generated triangulated category admitting a proper class of localising subcategories? No such example is known. However, our collection of examples is very limited in the sense that we only know there are a set of localising subcategories in the instances where we can explicitly parameterise them. There are no abstract techniques to show that a compactly generated triangulated category has a set of localising subcategories. As noted in Lemma 12.3.10 an affirmative answer to this question would imply that every localising subcategory is generated by a set of objects. In particular, every localising subcategory would be strictly localising and, in fact, ˛-comological for some ˛ by Lemma 12.3.12. This leads to a pair of natural questions. Question 12.3.17. Is every localising subcategory of a compactly generated triangulated category T strictly localising? Question 12.3.18. Let T be a compactly generated triangulated category. Is there a regular cardinal ˛ such that every localising subcategory of T is ˛-cohomological? Again the answers are not known. The former question is known to have an affirmative answer (in even greater generality than we have asked it), as proved in [7], provided Vopˇenka’s principle is assumed to hold. For the uninitiated, Vopˇenka’s principle is a very strong large cardinal axiom. This is a remarkable achievement

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and little is known in the absence of strong set-theoretic assumptions outside of cases where we have a classification. As far as we are aware there is no example of a localising subcategory which is not cohomological with respect to some cardinal and nothing is known concerning our final question outside of cases where there is a classification of localising subcategories.

12.4 Cohomological localisations for the projective line We now turn to the example we have in mind, namely D.QCoh P 1 / the unbounded derived category of quasi-coherent sheaves on P 1 . We first describe the thick subcategories of Db .Coh P 1 /, the bounded derived category of coherent sheaves on P 1 . We then recall the classifications of smashing subcategories and of tensor ideals in D.QCoh P 1 /. Finally, we classify the (@0 -)cohomological localising subcategories—there are no surprises and they are exactly the ones which have been understood for some time. It is of course possible that there are ˛-cohomological localising subcategories for ˛ > @0 which we are not aware of. It is in some sense tempting to guess that this is not the case, i.e. that our list is already a complete list of localising subcategories, but there is no real evidence for this. We close by making some remarks on the hurdles that such an ‘exotic’ localisation would have to clear. Before getting on with this let us recapitulate the connection with representation theory. By a result of Beilinson [6], the object T D O ˚ O .1/ is a tilting object in Coh P 1 which induces an exact equivalence 

RHom.T; /W D.QCoh P 1 / ! D.Mod A/ where Mod A denotes the module category of A D End.T / Š

h

k k2 0 k

i

:

We refer to [2] for details about tilting. Note that A is isomorphic to the path algebra of the Kronecker quiver
! !
and this algebra is known to be of tame representation type. In fact, the representation theory of this algebra amounts to the classification of pairs of k-linear maps, up to simultaneous conjugation. The finite-dimensional representations were already known to Kronecker [18]. 12.4.1 Thick subcategories of the bounded derived category The structure of the lattice of thick subcategories of Db .Coh P 1 /, which we recall in this section, has been known for some time; it can be computed by hand using the fact that Coh P 1 is tame and hereditary. The structure of the coherent sheaves on P 1 is well known: there is a Z-indexed family of indecomposable vector bundles and a 1-parameter family of torsion sheaves for each closed point on P 1 .

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For each i 2 Z one has a thick subcategory

Thick.O .i // D add.†j O .i / j j 2 Z/ Š Db .k/

where the identifications follow from the computation of the cohomology of P 1 . These are the only proper non-trivial thick subcategories which are generated by vector bundles and are also the only thick subcategories which are not tensor ideals. Thus we have a lattice isomorphism 

fthick subcategories of Db .Coh P 1 / generated by vector bundlesg ! Z where Z denotes the lattice given by the following Hasse diagram: ❅PP ♥♥  P ♥♥♥⑦⑦⑦⑦ ❅❅❅PPPPP ♥ ♥ ♥





PPP  ❅   PPP❅❅❅ ⑦⑦⑦♥♥♥♥♥ PP ⑦♥♥♥



This is a special case of a general result because the indecomposable vector bundles are precisely the exceptional objects of Db .Coh P 1 /. For any hereditary artin algebra A the thick subcategories of Db .mod A/ that are generated by exceptional objects form a poset which is isomorphic to the poset of non-crossing partitions given by the Weyl group W .A/; see [10, 11] and Theorem 11.6.2. Note h thati W .A/ is an 2 Q affine Coxeter group of type A1 for the Kronecker algebra A D k0 kk , keeping in mind the derived equivalence 

Db .Coh P 1 / ! Db .mod A/:

The thick tensor ideals are classified by Spc Db .Coh P 1 / Š P 1 , where the space Spc Db .Coh P 1 / is meant in the sense of Balmer [3], and its computation is a special case of a general result of Thomason [25]. What all this boils down to is that for any set of closed points V of P 1 there is a thick tensor ideal

DbV .Coh P 1 / WD fE j supp E  V g D Thick.k.x/ j x 2 V / consisting of complexes of torsion sheaves supported on V . Moreover, together with 0 and Db .Coh P 1 /, this is a complete list of thick tensor ideals. One can make this uniform by considering subsets of P 1 which are specialisation closed. In this language, by extending the above notation to allow Db¿ .Coh P 1 / D 0 and DbP 1 .Coh P 1 / D Db .Coh P 1 /, we have a lattice isomorphism 

fthick tensor ideals of Db .Coh P 1 /g ! fspc subsets of P 1 g; where ‘spc’ is an abbreviation for ‘specialisation closed’, which is given by [ I 7! supp I D supp E and V 7! DbV .Coh P 1 / E 2I

for I a thick tensor ideal and V a specialisation closed subset.

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We know every object of Db .Coh P 1 / is a direct sum of shifts of line bundles and torsion sheaves and so one can readily combine these classifications to obtain a lattice isomorphism2 

fthick subcategories of Db .Coh P 1 /g ! fspc subsets of P 1 g q Z: The verification that the evident bijection is indeed a lattice map as claimed is elementary: the twisting sheaves are supported everywhere so are not contained in any proper ideal, and any twisting sheaf and a torsion sheaf, or any pair of distinct twisting sheaves, generate the category. Thus for i ¤ j and V proper non-empty and specialisation closed in P 1 we have

DbV .Coh P 1 / _ Thick.O.i // D Db .Coh P 1 / D Thick.O.i // _ Thick.O.j // and

DbV .Coh P 1 / ^ Thick.O.i // D 0 D Thick.O.i // ^ Thick.O.j //: 12.4.2 Ideals and smashing subcategories We now describe the localising subcategories that one easily constructs from our understanding of the compact objects Db .Coh P 1 / in D.QCoh P 1 /. By [17], the smashing conjecture holds for D.QCoh P 1 / (our computations will also essentially give a direct proof of this fact). Thus the finite localisations one obtains by inflating the thick subcategories listed above exhaust the smashing localisations i.e. 

fthick subcategories of Db .Coh P 1 /g !

fsmashing subcategories of D.QCoh P 1 /g

as lattices. The localising ideals are also understood. Again this is known more generally (there is such a classification for any locally noetherian scheme, see [1]) but can easily be computed by hand for P 1 . The precise statement is that there is a lattice isomorphism 

flocalising tensor ideals of D.QCoh P 1 /g ! 2P

1

1

where 2P denotes the powerset of P 1 with the obvious lattice structure. The bijection is given by the assignments

L 7! fx 2 P 1 j k.x/ ˝ L ¤ 0g 2 Let L0 ; L00 be a pair of lattices with smallest elements 00 ; 000 and greatest elements 10 ; 100 . Then L0 qL00 denotes the new lattice which is obtained from the disjoint union L0 [ L00 (viewed as sum of posets) by identifying 00 D 000 and 10 D 100 .

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for a localising ideal L and

V 7! €V D.QCoh P 1 / WD fX 2 D.QCoh P 1 / j X ˝ k.y/ Š 0 for y … V g for a subset V of points on P 1 . This agrees with the classification of smashing subcategories in the sense that the smashing ideals are precisely those inflated from the compacts, i.e. those corresponding to specialisation closed subsets of points. Since P 1 is 1-dimensional the only new localising ideals that occur are obtained by throwing the residue field of the generic point, k./, into a finite localisation. Thus we have identified a sublattice consisting of a copy of Z and the powerset of P 1 inside the lattice of localising subcategories of D.QCoh P 1 /. The lattice structure extends that of the lattice of thick subcategories of Db .Coh P 1 / in the expected way. The naive guess is that this is, in fact, the whole lattice. While we do not know if this is the case, we can give an intrinsic definition of the localisations we have stumbled into so far. This description is the goal of the next two subsections. 12.4.3 An aside on continuous pure-injectives In order to describe the localisations we have listed so far a word on continuous pure-injectives is required. Definition 12.4.1. A pure-injective object I is continuous (or superdecomposable) if it has no indecomposable direct summands. We say that T has no continuous pure-injective objects if every non-zero pureinjective object has an indecomposable direct summand or, in other words, if there are no continous pure-injectives. An equivalent condition is that every pure-injective object is the pure-injective envelope of a coproduct of indecomposable pure-injective objects. Proposition 12.4.2. The category D.QCoh P 1 / has no continuous pure-injective objects. i h 2 Proof. Let A D k0 kk denote the Kronecker algebra. We use the derived equiva

lence D.QCoh P 1 / ! D.Mod A/. Let X be a pure-injective object in D.Mod A/. ` Observe that X decomposes into a coproduct X D n2Z Xn of complexes with cohomology concentrated in a single degree, since A is a hereditary algebra. Thus we may assume that X is concentrated in degree zero and identifies with a pure-injective A-module. Now the assertion follows from the description of the pure-injective Amodules in [13, Thm. 8.58]. 

Corollary 12.4.3. A localising subcategory L  D.QCoh P 1 / is cohomological if and only if there is a collection of indecomposable pure-injective objects .Yi /i 2I in QCoh P 1 such that L D fX 2 D.QCoh P 1 / j Hom.X; †j Yi / D 0 for all i 2 I; j 2 Zg.

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Proof. By Proposition 12.3.9 being cohomological is equivalent to being the left perpendicular of a collection of pure-injective objects. By the last proposition D.QCoh P 1 / has no continuous pure-injectives and so we may replace such a collection of pure-injectives with the collection of its indecomposable summands without changing the left perpendicular. These are all honest sheaves since QCoh P 1 is hereditary.  12.4.4 Classifying cohomological localisations In this section we give a classification of the cohomological localising subcategories of D.QCoh P 1 /. As we will show in Theorem 12.4.12 they are precisely the subcategories described in Section 12.4.2. Our strategy is to use Corollary 12.4.3 and the classification of indecomposable pureinjectives for D.QCoh P 1 / to compute everything explicitly; we can compute the set of indecomposable pure-injectives associated to each of the localising subcategories described in Section 12.4.2 and show any suspension stable set of pure-injectives has the same left perpendicular as one of these. To this end we first recall the description of the indecomposable pure-injective objects of QCoh P 1 . Let us set up a little notation: given a closed point x 2 P 1 we can consider the corresponding map of schemes ix W Spec OP 1 ;x ! P 1 : We denote the maximal ideal of OP 1 ;x by mx and the residue field OP 1 ;x =mx by k.x/. Let E.x/ be the injective envelope of the residue field k.x/, and A.x/ the mx adic completion of OP 1 ;x , which is the Matlis dual of E.x/. Pushing these forward along ix gives objects in QCoh P 1 which we denote by

E .x/ D ix  E.x/ and A.x/ D ix  A.x/: Proposition 12.4.4. The indecomposable pure-injective quasi-coherent sheaves are given by the following disjoint classes of sheaves:  the indecomposable coherent sheaves;

 the Prüfer sheaves E .x/ for x 2 P 1 closed;

 the adic sheaves A.x/ for x 2 P 1 closed;

 the sheaf of rational functions k./.

Proof. The indecomposable pure-injective quasi-coherent sheaves correspond to the indecomposable pure-injective modules over the Kronecker algebra via the derived  equivalence D.QCoh P 1 / ! D.Mod A/. The latter have beeen classified in [13, Thm. 8.58].  Remark 12.4.5. The Prüfer sheaves and k./ are precisely the indecomposable injective quasi-coherent sheaves.

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Having recalled the indecomposable pure-injective sheaves we now determine how they interact with the localisations described in Section 12.4.2. Let us begin by recording their supports. Lemma 12.4.6. We have supp E .x/ D fxg; supp k./ D fg; and supp A.x/ D fx; g: Proof. All of these sheaves are pushforwards along the inclusions of the spectra of local rings at points, and so their supports are contained in the relevant subset Spec OP 1 ;x . Having reduced to computing the support over OP 1 ;x this is then a standard computation.  As one would expect the localisations Loc.O .i // are particularly simple. Lemma 12.4.7. The only indecomposable pure-injective quasi-coherent sheaf in Loc.O .i //? is O .i 1/. Proof. There is a localisation sequence for the compacts Thick.O .i // o o

/ Db .Coh P 1 / o

/ Thick.O .i

1//

o

 

Db .k/

Db .k/

where the adjoints exist since O .i / is exceptional and the computation of the right orthogonal follows from the computation of the cohomology of P 1 . Applying Thomason’s localisation theorem shows that Loc.O .i //? D Loc.O .i

1// D Add.†j O .i

1/ j j 2 Z/

and the claim is then immediate.



We next compute the pure-injectives lying in the right perpendicular of the localising ideals. Lemma 12.4.8. Let V be a set of closed points with complement U . Then, the indecomposable pure-injective sheaves in €V D.QCoh P 1 /? are:  the indecomposable coherent sheaves supported at closed points in U ;  the Prüfer sheaves E .x/ for x 2 U ;

 the adic sheaves A.x/ for x 2 U ;

 the sheaf of rational functions k./.

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Proof. By the classification of localising ideals of D.QCoh P 1 /, we know that the category €V D.QCoh P 1 /? consists of precisely those objects supported on U . Since V consists of closed points we know U contains the generic point . The list is then an immediate consequence of Lemma 12.4.6.  Lemma 12.4.9. Let V be a subset of P 1 containing the generic point and denote its complement by U . Then the indecomposable pure-injective sheaves in €V D.QCoh P 1 /? are:  the indecomposable coherent sheaves supported at closed points in U ;  the adic sheaves A.x/ for x 2 U .

Proof. The sheaf of rational functions k./ has a map to every indecomposable injective sheaf and so no E .x/ is contained in the right perpendicular category (and clearly k./ is not). The category €V D.QCoh P 1 / contains the torsion and adic sheaves for points in V so the only indecomposable pure-injective sheaves which could lie in the right perpendicular are those indicated; it remains to check they really don’t receive maps from objects of €V D.QCoh P 1 /. This is clear for the residue fields k.x/ for x 2 U , as they cannot receive a map from any of the residue fields generating €V D.QCoh P 1 /. Since the right perpendicular is thick it thus contains all the indecomposable coherent sheaves supported on U . Moreover, the right perpendicular is closed under homotopy limits and so contains the corresponding adic sheaves A.x/.  We now know which subsets of indecomposable pure-injectives occur in the right perpendiculars of the localising subcategories we understand. It’s natural to ask for the minimal set giving rise to one of these categories, as in Corollary 12.4.3. Let us make the convention that for an object E 2 D.QCoh P 1 / ?

E D fF 2 D.QCoh P 1 / j Hom.F; †j E/ D 0 8j 2 Zg:

We can, without too much difficulty, compute all of the left perpendiculars of the indecomposable pure-injectives. Lemma 12.4.10. The left perpendicular categories to the suspension closures of the indecomposable pure-injectives are as follows: 1.

?

2.

?

3.

?

4.

?

5.

?

F D €P 1 nfxg D.QCoh P 1 / for any F 2 Coh P 1 supported at x 2 P 1 ;

O.i / D Loc.O.i C 1//; E .x/ D €P 1 nfx;g D.QCoh P 1 /; A.x/ D €P 1 nfxg D.QCoh P 1 /; k./ D €P 1 nfg D.QCoh P 1 /.

Proof. These are all (more or less) straightforward computations.



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Knowing this it is not hard to write down minimal sets of pure-injectives determining the ideals. Corollary 12.4.11. Let V be a subset of P 1 . Then, we have €V D.QCoh P 1 / D

?

fk.x/ j x … V g:

We also now have enough information to confirm that we have a complete list of the cohomological localising subcategories. Theorem 12.4.12. There is a lattice isomorphism 

1

fcohomological localising subcategories of D.QCoh P 1 /g ! 2P q Z; 1

where 2P is the powerset of P 1 , with inverse defined by

V 7! €V D.QCoh P 1 / and i 7! Loc.O.i //: That is, the cohomological localising subcategories are precisely the localising ideals and the Loc.O .i // for i 2 Z. Proof. By Corollary 12.4.3, the cohomological localising subcategories are precisely the localising subcategories which are left perpendicular to a set of indecomposable pure-injectives. Taking the left perpendicular of a set of pure-injectives corresponds to intersecting the corresponding left perpendiculars. By Lemma 12.4.10 we thus see that any such localising subcategory is of the form claimed.  Remark 12.4.13. Denote by Ind P 1 the set of isomorphism classes of indecomposable pure-injective sheaves. The subsets of the form L? \ Ind P 1 for some cohomological localising subcategory L are listed in Lemmas 12.4.7, 12.4.8, and 12.4.9.

12.5 Exotic localisations As noted in Section 12.4.2, we have a classification both of ideals and of smashing localisations for D.QCoh P 1 /. Moreover, we have just shown in Theorem 12.4.12 that together these are precisely the cohomological localisations. It is obvious to ask if there are non-cohomological localisations; we do not know the answer to this question and don’t hazard a guess. In this section, we at least provide some foundation for future work in this direction by presenting some criteria to guarantee a localising subcategory is an ideal. This is relevant as any non-cohomological localisation could not be an ideal—we proved that all ideals are cohomological. As we shall see, this dramatically restricts the possible form of a potential ‘exotic’ localising subcategory.

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12.5.1 A restriction on supports We begin by analysing support theoretic conditions that ensure a localising subcategory is an ideal. Since P 1 is 1-dimensional the consequences we obtain are quite strong. However, the ideas present in the arguments should be of more general interest. The first observation is that if the support of an object does not contain some closed point then that object generates an ideal. Lemma 12.5.1. Let y be a closed point of P 1 and let X 2 D.QCoh P 1 / be such that y … supp X . Then, L D Loc.X / is an ideal. Proof. By definition we have k.y/ ˝ X Š 0. Since y is a closed point the torsion sheaf k.y/ is compact, and hence rigid, so we deduce that

Hom.k.y/; X / Š 0: In particular, X 2 Loc.k.y//? Š D.QCoh A1 /, where we have made an identification of P 1 n fyg with the affine line. Since k.y/ is compact the subcategory Loc.k.y//? is localising and so

L  Loc.k.y//? Š D.QCoh A1 /: It just remains to note that every localising subcategory of D.QCoh A1 / is an ideal and that Loc.k.y//? is itself an ideal, from which it is immediate that L is an ideal in D.QCoh P 1 /.  Let V D P 1 n fg denote the set of closed points of P 1 . Corresponding to this Thomason subset there is a smashing subcategory €V D.QCoh P 1 / which comes with a natural action of D.QCoh P 1 /, in the sense of [23], via the corresponding acyclisation functor. Moreover, €V D.QCoh P 1 / is a tensor triangulated category in its own right, with tensor unit €V O (which is, however, not compact). Lemma 12.5.2. The category €V D.QCoh P 1 / is generated by its tensor unit and hence every localising subcategory contained in it is an ideal in it, and thus a submodule for the D.QCoh P 1 / action. In particular, every localising subcategory of D.QCoh P 1 / contained in €V D.QCoh P 1 / is an ideal of D.QCoh P 1 /. Proof. The subset V is discrete, in the sense that there are no specialisation relations between any distinct pair of points in it. It follows from [24] that €V D.QCoh P 1 / decomposes as Y €x D.QCoh P 1 /: €V D.QCoh P 1 / Š x2V

L With respect to this decomposition, the monoidal unit €V O is just x2V €x O , which clearly generates. It then follows that every localising subcategory of €V D.QCoh P 1 / is an ideal (see for instance [23, Lemma 3.13]) and from this the remaining statements are clear. 

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As a particular consequence, we get the following statement, which is more in the spirit of Lemma 12.5.1. Lemma 12.5.3. Let X 2 D.QCoh P 1 / be an object such that  … supp X . Then, Loc.X / is an ideal. Proof. Since  … supp X we have X 2 €V D.QCoh P 1 /. Thus Loc.X / is contained in €V D.QCoh P 1 / and therefore an ideal by the previous lemma.  We have shown that for any object X with proper support the category Loc.X / is an ideal. Next we will show that any localising subcategory containing such an object is automatically an ideal. This requires the following technical lemma. Lemma 12.5.4. If L is a non-zero localising ideal of D.QCoh P 1 / then the quotient T D D.QCoh P 1 /=L is generated by the tensor unit. Proof. Since the property of being generated by the tensor unit is preserved under taking quotients it is enough to verify the statement when L has support a single point. If supp L is a closed point, we can identify T with the derived category of the open complement, which is isomorphic to A1 . Having made this observation the conclusion follows immediately. It remains to verify the lemma in the case that supp L D fg. In this situation there is a recollement / / 1 €V D.QCoh P 1 / o / D.QCoh P / o / L where, as above, V denotes the set of closed points of P 1 . The bottom four arrows identify €V D.QCoh P 1 / with the quotient T and the desired conclusion is given by Lemma 12.5.2.  Combining all of this we arrive at the following proposition. Proposition 12.5.5. If L is a localising subcategory of D.QCoh P 1 / such that there is a non-zero X 2 L with supp X ¨ P 1 , then L is an ideal. Proof. Let X 2 L as in the statement of the proposition. The object X generates a non-zero localising subcategory Loc.X /  L. Since the support of X is proper and non-empty it fails to contain some point of P 1 and so, by one of Lemma 12.5.1  or 12.5.3, it is an ideal. We thus have a monoidal quotient functor D.QCoh P 1 / ! D.QCoh P 1 /= Loc.X / and an induced localising subcategory L= Loc.X / in the quotient. By Lemma 12.5.4 the quotient D.QCoh P 1 /= Loc.X / is generated by the tensor unit and so L= Loc.X / is a tensor ideal in it. But then L D  1 .L= Loc.X // is also an ideal, which completes the proof. 

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Example 12.5.6. The non-ideals we know, namely the Loc.O .i //, are of course compatible with the proposition: every object of Loc.O .i // is just a sum of suspensions of copies of O .i /, and these are all supported everywhere. The following interpretation is the most striking in our context. Corollary 12.5.7. If L is a localising subcategory which is not cohomological then every non-zero object of L is supported everywhere. 12.5.2 Twisting sheaves and avoiding compacts We next make a few comments concerning the interactions between localising subcategories and the twisting sheaves. Lemma 12.5.8. If L is a localising subcategory which is not an ideal then

L \ L.i / D 0 for all i 2 Z n f0g. Proof. Without loss of generality we may assume i > 0. Suppose, for a contradiction, that X 2 L \ L.i / is non-zero. Pick a closed point y and consider a triangle

O. i / ! O ! Z.y/ ! †O. i / where Z.y/ is the cyclic torsion sheaf of length i supported at y. We can tensor with X to get a new triangle X. i / ! X

! X ˝ Z.y/ ! †X. i /;

where both X and X. i / lie in L by hypothesis. Thus, since L is localising, we see that X ˝ Z.y/ lies in L. By Proposition 12.5.5 we know that X is supported everywhere and so X ˝Z.y/ ¤ 0. But on the other hand, X ˝Z.y/ is supported only at y which, by the same Proposition, implies that L is an ideal yielding a contradiction.  Remark 12.5.9. The lemma implies that non-cohomological localising subcategories would have to come in Z-indexed families. Lemma 12.5.10. If L is a localising subcategory such that Loc.O .i // ¨ L for some i 2 Z, then L D D.QCoh P 1 /. Proof. Localising subcategories containing Loc.O .i // are in bijection with localising subcategories of D.QCoh P 1 /= Loc.O .i //. This quotient is just D.k/ and so, since we have asked for a proper containment, the result follows. 

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We can now conclude that any non-cohomological localising subcategory must intersect the compact objects trivially. Proposition 12.5.11. If L is a localising subcategory which is not cohomological, then L contains no non-zero compact object. Proof. The indecomposable compact objects are just the indecomposable torsion sheaves at each point and the twisting sheaves. By Lemma 12.5.1, we know L cannot contain a torsion sheaf and, by the last lemma, it cannot contain a twisting sheaf. 

References [1] L. Alonso Tarrío, A. Jeremías López and M.J. Souto Salorio, Bousfield localization on formal schemes, J. Algebra 278 (2004), 585–610. [2] L. Angeleri Hügel, D. Happel and H. Krause (eds.) Handbook of tilting theory, London Mathematical Society Lecture Note Series 332, Cambridge University Press, Cambridge, 2007. [3] P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. (Crelle) 588 (2005), 149–168. [4] P. Balmer and G. Favi, Generalized tensor idempotents and the telescope conjecture, Proc. Lond. Math. Soc. 102 (2011), 1161–1185. [5] P. Baumann and C. Kassel, The Hall algebra of the category of coherent sheaves on the projective line, J. Reine Angew. Math. (Crelle) 533 (2001), 207–233. [6] A.A. Be˘ılinson, Coherent sheaves on Pn and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), 68–69. [7] C. Casacuberta, J. Gutiérrez and J. Rosický, Are all localizing subcategories of stable homotopy categories coreflective?, Adv. Math. 252 (2014), 158–184. [8] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448. [9] A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphere de Riemann, Amer. J. Math. 79 (1957), 121–138. [10] A. Hubery and H. Krause, A categorification of non-crossing partitions, J. Eur. Math. Soc. (JEMS) 18 (2016), 2273–2313. [11] C. Ingalls and H. Thomas, Noncrossing partitions and representations of quivers, Compos. Math. 145 (2009), 1533–1562. [12] S.B. Iyengar and H. Krause, The Bousfield lattice of a triangulated category and stratification, Math. Z. 273 (2013), 1215–1241. [13] C.U. Jensen and H. Lenzing, Model-Theoretic Algebra with Particular Emphasis on Fields, Rings, Modules, Gordon and Breach, New York, 1989. [14] B. Keller, A remark on the generalized smashing conjecture, Manuscripta Math. 84 (1994), 193–198. [15] H. Krause, Smashing subcategories and the telescope conjecture—an algebraic approach, Invent. Math. 139 (2000), 99–133.

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[16] H. Krause, Localization theory for triangulated categories. In Triangulated Categories (T. Holm, P. Jørgensen and R. Rouquier, eds.), 161–235, London Mathematical Society Lecture Note Series 375, Cambridge University Press, Cambridge, 2010, 161–235. [17] H. Krause and J. Št’ovíˇcek, The telescope conjecture for hereditary rings via Ext-orthogonal pairs, Adv. Math. 225 (2010), 2341–2364. [18] L. Kronecker, Algebraische Reduktion der Scharen bilinearer Formen, Sitzungsber. Akad. Berlin (1890), 1225–1237. [19] A. Neeman, The chromatic tower for D.R/, Topology 31 (1992), 519–532. [20] A. Neeman, Triangulated categories, Annals of Mathematics Studies 148, Princeton University Press, Princeton, 2001. [21] C.M. Ringel, Infinite-dimensional representations of finite-dimensional hereditary algebras. In Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977), Academic Press, London, 1979, 321–412. [22] C.M. Ringel, Tame algebras are WILD, Algebra Colloq. 6 (1999), 473–480. [23] G. Stevenson, Support theory via actions of tensor triangulated categories, J. Reine Angew. Math. (Crelle) 681 (2013), 219–254. [24] G. Stevenson, Filtrations via tensor actions, Int. Math. Res. Not. (2017), in press. [25] R.W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1997), 1–27. [26] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996).

Chapter 13

Higher finiteness properties of braided groups K.-U. Bux The properties of a group to be finitely generated or finitely presentable are the first two instances in a sequence of so called higher finiteness properties defined in terms of skeletons of classifying spaces. The study of higher finiteness properties is a prime example of how one can use a nice action of a group on a topological space to better understand the group. We shall illustrate this method in detail using the braided Thompson group V br as an example.1

13.1 Introduction: From group theory to topology The main connection between group theory and topology is via the fundamental group. To each topological space X with a chosen base point x, the fundamental group 1 .X; x/ is the set of loops in X based at x considered up to homotopy. Clearly, loops based at x cannot leave the path component of x. Hence, the fundamental group 1 .X; x/ only detects phenomena within the connected component of x. Multiplication in the fundamental group is given by concatenation of loops. This defines a functor 1 W Top ! Gr from the category of pointed spaces to the category of groups. Moving the base point x within its connected component also yields an isomorphic fundamental group. Therefore, we restrict ourselves to path connected spaces. A homotopy equivalence of spaces induces an isomorphism of their fundamental groups. Thus, the fundamental group of a connected space X depends only on the homotopy type of X and we get a functor 1 W HTop ! Gr on the category of homotopy types of pointed connected spaces. It turns out that this functor is surjective, i.e., every group arises as the fundamental group of a topological space. Even more is true. Fact 13.1.1 ([20, Prop. 7.1.5 and Cor. 7.1.7]). Let G be a group. There exists a connected CW-complex Y with contractible universal cover such that G D 1 .Y /. Such a cell complex Y is called a classifying space for the group G. A classifying space for G is unique up to homotopy equivalence. As we can obtain G from any classifying space of G by taking the fundamental group, the classifying space is a complete topological invariant of the group. Coarser 1 Projects

B1, C13

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invariants can therefore be obtained from it. In particular, the group (co)homology of G can be regarded as the (co)homology of its classifying space. See [10, Prop. II.4.1] or [20, Prop. 8.1.4] for homology and [10, Sec. III.1, page 59] or [20, Prop. 13.1.1] for cohomology. Numerical measures of complexity for groups can be obtained by leveraging the fact that the classifying space is only unique up to homotopy equivalence. Definition 13.1.2. A group G is said to have geometric dimension d if it admits a classifying space of dimension d but none of smaller dimension. It is said to be of type Fm if it admits a classifying space with finite m-skeleton. We say that G is of type F1 if it has a classifying space that has finite skeleta in all dimensions. The finiteness length .G/ of G is the largest m 2 N [ f1g for which G is of type Fm . So, if G has geometric dimension d , then its (co)homology coincides with the (co)homology of a CW-complex of dimension d . In particular, the (co)homology of G is trivial in dimensions above d . Similarly, if G is of type Fm , its (co)homology can be computed from a CW-complex with finite m-skeleton. In particular, the (co)homology is finitely generated up to dimension d . Often, but not always, the converses hold. So, one a heuristic level, one might think of the dimension of a group G as marking the dimension where the homology collapses for good, whereas .G/ C 1 marks the first dimension where the homology explodes. Note that such an explosion may very well be followed by an ultimate collaps. So, finiteness length and dimension may both be finite. Since a CW-complex with finite 5-skeleton automatically has a finite 4-skeleton, we observe that a group of type F5 automatically has type F4 . Hence, the types form a sequence of finiteness properties of groups of increasing strength. The first few of these finiteness properties are familiar. Fact 13.1.3. Every group has type F0 . A group is of type F1 if and only if it is finitely generated; and is it of type F2 if and only if it is finitely presented. For a finitely presented group G, it is possible to characterise whether it is of type F3 by generators, relations, and so called identities among relations. This explicit description enables one sometimes to handcraft the 3-skeleton of a classifying space for G and deduce that G has type F3 . For finiteness properties above F4 , however, no such arguments are feasible. Finiteness length and dimension are among the weakest pieces of (co)homological information one might have about a group. Regrettably, there are quite a few groups of interest, where even these small amounts of information are hard to obtain and nothing else in known about the homology. In this paper, we focus on finiteness properties and describe the current state of the art method of establishing the finiteness length. We shall use the braided Thompson group as a nontrivial example.

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13.2 Brown’s criterion There are only very few established methods to deduce higher finiteness properties. The most important and most widely applicable was developed by K. Brown in a seminal paper [11]. To illustrate the criterion, let us consider a G of type Fm . Let Y be a classifying CW-complex for G with finite m-skeleton. Its universal cover YQ is a contractible CW-complex on which G acts freely. The action on the m-skeleton YQ .m/ is also free, and the orbits space is compact. Moreover, the homotopy groups d .YQ .m/ / are trivial for d < m. Conversely, if we are given a free and cocompact action on a CW-complex whose homotopy groups vanish in dimensions below m, we can attach cells in higher dimensions (equivariantly so as to extend the free G-action) and obtain the universal cover of classifying space that witnesses G to be of type Fm . Thus, we obtain a first criterion. Observation 13.2.1. The group G is of type Fm if and only it acts freely and cocompactly on a CW-complex X whose homotopy groups d .X / vanish in dimensions d < m. A space whose first homotopy groups vanish is called highly connected. More precisely, X is called l-connected if d .X / is trivial for d 6 l. We adopt the convention that every non-empty space is considered . 1/-connected. Note the difference between the strict inequality in the criterion (13.2.1) and the non-strict inequality in the definition of connectivity. This unfortunate difference causes a shift by one in many theorems. Brown’s criterion improves upon Observation 13.2.1 in two ways. First, he removes the restriction to free actions. Instead, cell stabilisers are just required to have suitable finiteness properties in their own right. In addition, the action is not required to have a compact quotient. This condition is replaced by considering a filtration by invariant and cocompact subcomplexes. So let G act on a CW-complex X by cell-permuting homeomorphisms. We call X m-good if the following conditions are satisfied: 1. The space X is .m 1/-connected, i.e., the homotopy groups d .X / are trivial for d < m. 2. The stabiliser G of any d -cell of dimension d 6 m is of type Fm

d.

Note that X is not assumed to have a compact quotient mod G. Instead, we consider a filtration X0 6 X1 6 X2 6


S of X by G-invariant subcomplexes X˛ 6 X with X D ˛ X˛ where each X˛ has a compact quotient mod G.

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Of course, some conditions on homotopy groups will be relevant. We call the filtration .X˛ /alpha essentially homotopically trivial in dimension d if for every filtration index ˛ 2 N there is a filtration index ˇ > ˛ so that the induced homomorphism d .X˛ /

! d .Xˇ /

of homotopy groups in dimension d is trivial, i.e., if each d -sphere in X˛ bounds a ball in Xˇ . Note that this is stronger than saying that the limit of the homotopy groups d .X˛ / is trivial, which would just mean that any d -sphere in X˛ bounds a ball in some Xˇ , which by compactness of spheres is just equivalent to the sphere bounding a ball in X . We call the filtration .X˛ /˛ essentially l-connected if it is essentially homotopically trivial in dimensions d for all d 6 l. Brown proves the following: Theorem 13.2.2 ([11, Thms 2.2 and 3.2]). Assume that X is an m-good G-complex with a filtration by G-invariant cocompact subcomplexes X˛ as above. The group G is of type Fm if and only if the filtration .X˛ /˛ is essentially .m 1/-connected. As a corollary, Brown obtains a criterion for groups of type F1 in more familiar terms of relative connectivity. Corollary 13.2.3 ([11, Cor. 3.3]). Assume that X is a contractible G-complex with cells stabilisers of type F1 and a filtration by G-invariant cocompact subcomplexes X˛ . Suppose the connectivity of the pair of spaces .X˛C1 ; X˛ / tends to infinity as ˛ tends to 1. Then, G is of type F1 . The criterion clearly suggests a strategy for establishing higher finiteness properties for a group G. An established pattern runs as follows: 1. Find a highly connected, usually a contractible, space X upon which G acts with cell stabilisers that are of type F1 . Not only finiteness properties can be deduced this way. Often, more detailed (co)homological information is hidden in this action. Important examples include: symmetric spaces and affine buildings for arithmetic groups [6, 7, 13], Teichmüller space and Harer’s arc complex for mapping class groups [24, 22, 23], outer space for outer automorphism groups of free groups [14]. As these examples show, often a good deal of work is spent on the construction of a good space for a given group to act on and establishing its topological structure. 2. Find a filtration by G-invariant subspaces X˛ that have compact quotient mod G. This often takes the form of finding a G-invariant height function h W X ! R, mimicking those on locally-symmetric spaces defined by (14.3.3). The filtration is then given by sublevel complexes, i.e., full subcomplexes spanned by the set of vertices u 2 X with height h.u/ 6 s.

3. Show that the filtration .X˛ /˛ is essentially .m 1/-connected. If the filtration is given by a height function h W X ! R this part of the argument usually

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employs combinatorial Morse theory developed by M. Bestvina and N. Brady in [3]. It reduces the problem to local considerations about so called descending links, which are certain subcomplexes of links in X . For the analysis of descending links, various tools from topology are available. We shall give an account of combinatorial Morse theory in the following section. In Section 13.6, we describe the braided Thompson group V br , and in the Sections 13.7 through 13.10, we illustrate the above proof strategy by outlining its implementation for V br . A major point is the analysis of descending links which we can reduce to arc matching complexes, which we introduce in Section 13.4. As an aside, Section 13.5 draws a connection of arc matching complexes to higher generation of braid groups by subgroups.

13.3 Combinatorial Morse theory A piecewise Euclidean complex is a CW-complex where all closed cells carry the structure of a Euclidean polyhedron, all attaching maps are injective and restrict on faces to isometric identifications with cells of smaller dimension. In particular, different cells intersect in faces, on which they induce the same metric. Injectivity of attaching maps prevents identification of faces of the same cell. The notion of a piecewise spherical complex is a CW-complex is analogously defined, except that closed cells are spherical polyhedra. See [3, Sec. 2] for a slightly more general point of view, where piecewise affine complexes are discussed. The link of a point x in Euclidean space can be viewed as the sphere of directions issuing from x. If x is a corner of a polyhedron, the subset of directions pointing into the polyhedron is a spherical polyhedron. If x is a vertex of a piecewise Euclidean complex X , the link lkX .x/ is a piecewise spherical complex: each Euclidean cell containing x contributes a spherical polyhedron in lkX .x/. Links in piecewise spherical complexes are also piecewise spherical for the same reasons. Example 13.3.1. A piecewise Euclidean complex where all cells are unit cubes is called a cube complex. The link of any vertex in a cube complex is a piecewise spherical complex where all cells are right angled spherical simplices. Definition 13.3.2. Let X be a piecewise Euclidean complex. A function h W X ! R is called a Morse function if the following hold: 1. The restriction of h to each closed cell is an affine map. 2. The map h is not constant on edges. 3. The image h.X .0/ / of the vertex set is a discrete subset of R. For a level s 2 R, we define the level set f h D s g as the preimage f h D s g WD h 1 .s/. The corresponding sublevel set is f h 6 s g WD h 1 .. 1; s/. The maximal subcomplex Xs fully contained in the sublevel set f h 6 s g is a sublevel complex.

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Fact 13.3.3. The sublevel set f h 6 s g deformation retracts onto the sublevel complex Xs . Hence, both are homotopy equivalent. Let X be a piecewise Euclidean complex with a Morse function h W X ! R. Consider a closed cell  in X . Since h is affine on  , the max set of h on  is a face of  . Since h is not constant on edges, this top face consists of a single vertex, the [combinatorial Morse theory]top vertex of  . Definition 13.3.4. For a Morse function h W X ! R and a vertex u 2 X , we define the descending star of u to be the collection of those cells  for which u is the top vertex (this set is partially ordered by inclusion). The descending link lk#X .u/ of u is the union of all the spherical polyhedra lk .u/ where  ranges over the cells in the descending star of u. The analogy to Morse theory on smooth manifolds is as follows. The condition that h is not constant on edges says that the vertices are the critical points. Then the image of the critical points is the discrete set of critical values. The descending link replaces the descending sphere at a critical point. However, the main thrust is the same: the connectivity of descending links control the change in homotopy type of sublevel sets as the level increases. Lemma 13.3.5 (Morse Lemma [3, Lemma 2.5]). Assume that there is no critical value in the interval .s; t. Then the inclusion of sublevel sets f h 6 s g  f h 6 t g is a homotopy equivalence. Therefore, the same holds for the inclusion Xs  Xt of sublevel complexes. Assume that there is precisely one critical value t in the interval .s; t, then the sublevel set f h 6 t g is homotopy equivalent to the sublevel set f h 6 s g with the descending links of vertices of height t coned off. As the Morse Lemma 13.3.5 allows one to control how the homotopy type of sublevel complexes changes, one can use it to deduce connectivity properties of a filtration by sublevel complexes. Corollary 13.3.6 ([3, Cor. 2.6]). Let h W X ! R be a Morse function and sup# pose that the descending link lkX .u/ is l-connected for each vertex u 2 X with s < h.u/ 6 t. Then, the inclusion of sublevel complexes Xs 6 Xt induces isomorphisms in d for d 6 l and an epimorphism in lC1 . Example 13.3.7. Let be a tripod. We define the Morse function on sending the central vertex to 0 one terminal vertex to 1 and the other two to 1. The product X WD  of two tripods is a square complex shown in Figure 13.3.1. On X , we consider the Morse function obtained by adding the heights of the two coordinates. Some preimages of non-critical values are shown as well as the link and descending link of the central vertex.

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h

!

R Figure 13.3.1. The product of two tripods is a square complex. Here it is is drawn according to a height function. Point preimages are horizontal sections, shown on the left for some non-critical values. The sublevel sets start at the bottom a union of four contractible components. Between the lowest two critical values, sublevel sets are homotopy equivalent to a circle. This circle is coned off as the descending link of the middle vertex (shown as solid arcs; the dotted arcs show the non-descending parts of the link). Above this critical value, sublevel sets are contractible.

13.4 Matching complexes for graphs and surfaces Let € be a graph. We shall speak of nodes and lines of € and save the terminology of vertices and edges for CW-complexes. So let N be the set of nodes of € and let L be its set of lines. We allow loops and multiple edges. The matching complex M.€/ of € is the simplicial complex that has L as its vertex set and wherein a subset   L is a simplex if its elements are pairwise disjoint lines in €. The name derives from the notion of a matching, which is a collection of pairwise disjoint lines in a graph. So the simplices of the matching complex M.€/ are the matchings in €. We shall discuss connectivity of matching complexes for the complete and complete bipartite graphs. The results are well-known, see e.g., [2] or [5]. Example 13.4.1 (e.g. [5, Thm. 4.1]). For the complete graph € D Kn on n ˘nodes, the matching complex Mn WD M.€/ is .n/-connected where .n/ D nC1 2. 3

Example 13.4.2 (e.g. [5, Thm. 1.1]). For the complete bipartite graph € D Km;n on m white and n black nodes, the matching complex Mm;n WD M.€/, called the chess board complex, is .m; n/-connected where    mCnC1 .m; n/ D min m; n; 2: 3

Remark 13.4.3. Matching complexes are just one example of complexes based upon graphs. A construction similar in spirit is to consider sub-forests of €, i.e., vertices are non-loop lines and a collection of lines forms a simplex if their union does not

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contain a cycle. The connectivity of the forest complex F .€/ is V C where V is the number of vertices of € and C is the number of its components. Various proofs of this fact are known, see, e.g., [25], [4, Thm. 7.3.3] or [28, Prop. 2.2]. Now, we turn to arc matching complexes, which encode the possible ways of drawing a matching onto a given surface. Let † be a connected surface possibly with punctures and boundary components. Let N be a set of marked points, possibly on the boundary, and let € be a graph with N as its set of nodes. An arc is a curve connecting to marked points in N . A drawing of € onto † is continuous injective map 'W € !†

such that ' restricts to the identity on N and '.€nN / does not contain boundary points nor marked points. Each line of € is mapped to an arc on †. The injectivity condition says that arcs do not self-intersect and do not meet each other apart from their endpoints. We consider arcs and drawings up to isotopy relative to marked points, punctures and boundary. As soon as its Euler characteristic is negative, the surface † can be endowed with a hyperbolic metric with geodesic boundary components. Since we consider isotopy relative to marked points, we may also think of them as punctures. Then, a hyperbolic metric exists in any non-boring situation, e.g., as soon as we have three marked points (now thought of as punctures metrically turned into cusps). In the examples below, three marked points are necessary to have more than one arc. In the presence of a hyperbolic metric, we can represent each arc by the unique geodesic in its homotopy class (relative endpoints). This uniqueness implies that two different geodesic arcs between the same points cannot bound a disk. Also geodesic arcs coincide or intersect transversally as geodesic germs extend uniquely in both directions. By the bigon criterion [17, 1.2.4], geodesic representatives of arcs are in minimal position, i.e., the geometric intersection number of their homotopy classes coincides with the number of intersection points of the geodesic representatives. Thus: Observation 13.4.4. A drawing is uniquely determined by (the homotopy classes of) its arcs. The arc matching complex A.€; †/ is the simplicial complex whose simplices are drawings (up to relative homotopy) of matchings in € onto †. In this definition, Observation 13.4.4 is implicitly used: in a simplicial complex, a simplex is determined by its vertices. If drawings were not uniquely determined by their arcs, we could only define a CW-complex whose cells have the geometric shape of simplices. We remark that A. Hatcher and K. Vogtmann consider very similar complexes in [21]. They do not require the arcs to be fully disjoint but allow that arcs meet at their end points. For the arc matching complex on a complete graph, one can deduce the same connectivity as for its matching complex.

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Figure 13.4.1. An arc matching in A† .5/ on the disk with its corresponding matching in M5 .

Example 13.4.5 (see [12, Prop. 3.6]). Let € WD Kn be the complete graph on n nodes, all of which shall be interior points of the connected surface †. Then,  the ˘ arc matching complex A† .n/ WD A.€; †/ is .n/-connected, where .n/ D nC1 2 3 as in Example 13.4.1. Remark 13.4.6. One might consider other arc matching complexes also based upon the complete graph Kn , the difference being the positioning of the nodes in the surface. We place all marked points in the interior of the surface. One can also put some (or even all) marked points on the boundary, and connectivity of the resulting complex will in general be different (and usually worse). We now turn to arc matching complexes based on complete bipartite graphs. Again, one can place the marked points within the interior or the boundary of the surface; and connectivity of the resulting complexes may depend on how marked points are placed. We shall be concerned with a placement that admits a particularly nice action of the braid group on the resulting arc matching complex, namely placing the white nodes on the boundary and the black nodes in the interior. Example 13.4.7. Let † be a connected surface with one boundary component. Let € D Km;n be the complete bipartite graph on m white and n black nodes. Let A† .m; n/ be the arc matching complex in † based on € where all white nodes are placed on the boundary of † and all black nodes are placed in  the ˘interior of †. Then / 2. A† .m; n/ is .m; n/-connected, where .m; n/ WD min.m; nC1 2

Remark 13.4.8. Compared to Example 13.4.2, the arc matching complex A† .m; n/ has lower connectivity. This phenomenon is due to placing the white nodes on the boundary of †. If we put all nodes into the interior of †, the corresponding

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Figure 13.4.2. An arc matching in A† .4; 6/ on the disk with its corresponding matching in M4;6 .

arc matching complex has the same connectivity as its non-surface counterpart from Example 13.4.2. Remark 13.4.9. The arc matching complex A.€; †/ projects simplicially to the matching complex M.€/ simply by ignoring the way the matching is drawn. It is tempting to make use of this projection in establishing connectivity of arc matching complexes. Unfortunately, the fibers of this projection are badly behaved even in those cases where connectivity of M.€/ and A.€; †/ agree. So far, it appears that this approach is not feasible.

13.5 Higher generation in symmetric groups and braid groups In [1], H. Abels and S. Holz introduced the concept of higher generation by subgroups. Let G be a group and let H be a family of subgroups H 6 G. The associated coset complex is the simplicial complex CC.H/ whose vertex set f gH j g 2 G; H 2 H g is the set of left cosets of subgroups in H and where a collection of such cosets forms a simplex if their intersection is non-empty. We say that H is m-generating for G if CC.H/ is .m 1/-connected. Note that any non-empty S family of subgroups is 0generating. The family H is 1-generating if the union H 2H H generates G. Abels and Holz show that H is 2-generating if and only if G is the free product of the H 2 H amalgamated along their intersections. It turns out that matching complexes and arc matching complexes arise as coset complexes in symmetric groups and in braid groups. To see this, we recall a simple criterion that allows one to detect coset complexes.

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Proposition 13.5.1 (see [31, A.5]). Let G act on a simplicial complex X with a maximal simplex C as a strict fundamental domain. Then, X is isomorphic (as a simplicial complex) to the coset complex CC.F / where

F WD f StabG .u/ j u is a vertex of C g Example 13.5.2. The symmetric Sn group on n letters acts on the vertex set of Kn and therefore on the associated matching complex Mn and on its barycentric subdiV n . A simplex of the barycentric subdivision M V n is a flag of simplices in vision M Mn . Note that up to an action of Sn , a matching is uniquely characterised by the numV n has a maximal simplex as a strict ber of lines it uses. Hence the action on M fundamental domain, e.g., we can choose the "obvious" flag: f 1; 2 g f 1; 2 g f 3; 4 g f 1; 2 g f 3; 4 g f 5; 6 g :: : V n is the stabiliser of a simplex in Mn , i.e., the stabiliser The stabiliser of a vertex in M of a matching. By Proposition 13.5.1 and Example 13.4.1, the stabilisers of the above matchings form a family of subgroups in Sn that is ..n/ 1/-generating. Example 13.5.3. The symmetric group Sn on n letters acts on the bipartite complete graph Km;n on m white nodes w1 ; : : : ; wm and n black nodes b1 ; : : : ; bn by permuting the black nodes. We assume m 6 n. In this case, the matching w1

b1

w2

b2






wm

bm

is a fundamental domain for the action on the chess board complex Mm;n . The stabiliser of the vertex wi bi is the subgroup StabSn .i / isomorphic to Sn Thus, we conclude from Proposition 13.5.1 and Example 13.4.2:

1.

The family f StabSn .i / j 1 6 i 6 m g is ..m; n/ C 1/-generating for Sn . Example 13.5.4. Now consider the complete bipartite graph Km;n on m white and n black nodes. Let † be a disk. Consider the arc matching complex A† .m; n/ from Example 13.4.7. Assume m 6 n. Then, the braid group Bn acts on A† .m; n/ with a maximal simplex (a complete matching) as a strict fundamental domain. This action of the braid group Bn arises as follows. A braid records a motion of the n punctures within the disk. We think of the disk being made out of elastic material so that the moving punctures drag ambient points

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along. We keep points on the boundary fixed. This way, a braid defines a mapping class of the n-punctured disk relative to the boundary. This point pushing map induces an isomorphism of the braid group Bn and the mapping class group of an n-punctured disk, where the boundary is fixed pointwise and the punctures are fixed as a set. The mapping class group clearly acts on the arc matching complex A.m; n/. Stabilisers of its vertices (matchings between a single white node and a single black node) are isomorphic copies of braid groups Bn 1 inside Bn . A mapping class that stabilises an arc can be regarded as a mapping class on the surface obtained by cutting along the arc. This cut yields another dist, but we are missing a puncture. Let Bm;n be the family of these vertex stabilisers. Now, Proposition 13.5.1 applies, and we have: The family Bm;n is .l C 1/-generating for Bn if and only if A† .m; n/ is l-connected. As we know a lower bound on the connectivity of A† .m; n/ from Example 13.4.7, we conclude that Bm;n is ..m; n/ C 1/-generating for the braid group Bn . For a different perspective on higher generation of the family Bn WD Bn;n see Theorem 11.3.18 in this volume and the preceding discussion.

13.6 The braided Thompson group V br Elements of the braid group Bn can be represented by braid diagrams on n strands. Two such diagrams represent the same element if one can be transformed into the other by an ambient isotopy. Alternatively, one may restrict attention to diagrams in normal form: only one pair of adjacent strands is allowed to cross at any given time. Then, finitely many rules of equivalence for diagrams suffice: one requires the braid relation for adjacent crossings; and for crossings with disjoint support, one requires that order does not matter: D

D

Of course, these rules are taken from the well known presentation of Bn . Now, we consider diagrams where strands are also allowed to split and merge. Since we do not want strands to twist (invisibly in the picture) we require that splits and merges only take place in the xz-plane. Also, a strand can only split in two; and one of the resulting strands is the left branch whereas the other is the right branch. The same rule applies to merges. Equivalence is defined by ambient isotopy with two additional rules. A split followed immediately by a merge of its resulting branches is as good as not splitting at all. That is, one may remove an eye. An eye is a disk lying entirely in the xz-plane bounded by two branches following a split and merging afterwards that is not intersected by any other strand of the diagram. Similarly, a

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Figure 13.6.1. These diagrams do not have eyes. In the left picture, the disk bounded by the branches twists; in the right example, another strand passes through the disk.

merge can be undone by a split that follows. The two allowed moves are called eye removal and X removal: D eye removal

D X removal

With splits and merges interfering, the number of strands in a diagram is not constant. An .m; n/-diagram has m strands at the top (heads) and n strands at the bottom (feet). Clearly, .n; n/-diagrams can be stacked. They form a group, whose isomorphism type does not depend on the number of heads and feet. More generally, we can consider the category B whose objects are counting numbers 1; 2; 3; : : : and where the .m; n/-diagrams are the morphisms from m to n. The braided Thompson group V br is defined as the automorphism group of the object 1 in this category. Its elements are thus represented by .1; 1/-diagrams. An example is shown in Figure 13.6.2. This braided version of Thompson’s group V has been introduced independently by M. Brin [9] and P. Dehornoy [16]. Any .1; 1/-diagram can be put into standard form. By applying eye and X removals, one can achieve that the diagram consists of a sequence of splits (forming a binary tree in the xz-plane) followed by a braid and concluded by a sequence of merges (again forming a binary tree in the xz-plane). As in the case of braid diagrams, one can restrict the form of allowed diagrams and replace the equivalence of ambient isotopy by purely combinatorial requirements. In addition to the braid moves and removal of eye and X sub-diagrams, we then need moves that allow us to move splits and merges past one another and past crossings. Here is a pictorial representation of some new moves one needs to consider:

D

D

Whereas in the case of braid groups the moves correspond to the relations of a finite presentation, it is not obvious that one can extract a finite presentation for V br from

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Figure 13.6.2. A .1; 1/-diagram representing an element of V br .

these rules. Not even a finite generating set suggests itself: diagrams can grow to arbitrarily many strands; and splits, merges and crossings far to the right need to be expressed in terms of moves closer to the left. Nonetheless, M. Brin [8, Thm. 2] gave a finite presentation for V br . We can strengthen his result. Theorem 13.6.1 (see [12, Main Thm.]). The braided Thompson group V br is of type F1 , i.e., the group V br admits a classifying space with finite skeleta in each dimension. We also remark that V br has torsion and therefore cannot admit a finite classifying space. In fact, it does not have a torsion free subgroup of finite index. Thus, a classifying space that is finite in each dimension is the best one can hope for. In the remainder of this paper, we shall outline how Brown’s criterion can be used to prove Theorem 13.6.1. For various Thompson groups, this method has been carried out successfully. The complexes we shall introduce below are inspired by the complexes used by M. Stein [27] for Thompson groups and D. Farley [18] for the larger class of diagram groups.

13.7 A cube complex for V br Elements of V br are represented by .1; 1/-diagrams and multiplication corresponds to stacking. Similarly, a .1; 1/-diagram can be put on top of a .1; n/-diagram resulting in a .1; n/-diagram. This defines an action of V br on the the set of .1; n/-diagrams, an action most naturally considered not as a left or right action but as an action from above. In the language of the category B , this is just the action of V br D AutB .1/ on the set MorB .1; n/ of morphisms. Similarly, there is an action from below of the group AutB .n/ on MorB .1; n/. Note that the braid group Bn consists of .n; n/-diagrams. Thus, Bn acts on MorB .1; n/ as a subgroup of AutB .n/. Since the two actions commute, V br acts on the set Dn of

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Bn -orbits of MorB .1; n/. Informally, modding out the action of Bn on MorB .1; n/ amounts to consider .1; n/-diagrams up to dangling feet. Observation 13.7.1. The group V br acts simply transitively on MorB .1; n/. The induced action on Dn is transitive and its point stabilisers are isomorphic to Bn . We shall now construct a cube complex X with the set of all diagrams with dangling feet D WD D1 t D2 t D2 t


as its set of vertices. We glue in edges corresponding to the splitting of feet. We call the splitting of a single foot in a diagram from Dn an elementary expansion. As the number of feet increases by one, an elementary expansion yields a diagram in DnC1 . We fill in all the edges from a diagram to its elementary expansions. A diagram with n feet allows for exactly n different elementary expansions, thus n edges issue from it. Of course, it might also have incoming edges as it can be obtained as an elementary expansion from other diagrams. Higher dimensional cubes arise from commuting elementary expansions. Consider a diagram ‚ with n feet. For any selection of them, the diagram obtained from ‚ by splitting all the chosen feet does not depend on the order in which the strands are split. This means that the total of ! ! ! ! n n n n 2n D C C


C C 0 1 n 1 n diagrams that can be obtained from ‚ by commuting elementary expansions are connected by edges that form the 1-skeleton of a cube. We call ‚ the bottom corner of the cube. Of course, this terminology foreshadows a Morse function where diagrams with more feet will be higher than those with fewer feet. We obtain X by filling in all those cubes. Figure 13.7.1 shows how a square in X arises. Since V br acts on D from above whereas elementary expansions take place at the bottom, the action of V br on D induces an action on X . Cell stabilisers of this action are braid groups. More precisely, we make the following: Observation 13.7.2. The stabiliser of a cube coincides with the stabiliser of its bottom corner. Thus, cell stabilisers of the V br -action on X are isomorphic to braid groups on finitely many strands and therefore of type F1 . In order to derive finiteness properties from the action on X , we also need that X has vanishing homotopy groups. Proposition 13.7.3. The complex X is contractible. Sketch of proof. Consider the trivial .1; 1/-diagram j and let Yn be the subcomplex spanned by all diagrams with at most n feet that can be obtained from j by (iterated)

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Figure 13.7.1. A square in X.

expansions. Note that YnC1 deformation retracts onto Yn . It follows that the union S Y WD n Yn is a contractible subcomplex of X . The vertices in Y can be represented by diagrams that are just binary planar trees, i.e., free of braiding and merges. Splitting all feet of a diagram simultaneously takes the bottom corner of each cube to its top corner. By interpolation, we can extend this to a continuous map. This map, however, does not preserve the cell structure of X : for instance, it takes edges to square diagonals. In fact, it is even possible to extend it to a continuous flow within X . Note that iterated splitting of all feet will ultimately undo any merge that happened somewhere up in the diagram; and as diagrams are considered up to dangling feet, everything untangles at the bottom when all merges are gone. Thus, we end in the subcomplex Y . By compactness, a sphere in X involves only finitely many diagrams. Thus, flowing a sphere up the complex, it will eventually be moved completely into the subcomplex Y that consists of merge-free diagrams. Once inside the contractible subspace Y , the sphere can be crushed. 

13.8 The Morse function and its descending links To implement the remaining parts of the proof, we need a Morse function on X . On the vertices, we just take the number of feet. This extends to an affine map on each cube. Thus, we obtain the height function h W X ! R. Let Xn be the sublevel complex spanned by all diagrams with at most n feet. Note that the critical values of the Morse function h are integers.

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Observation 13.8.1. The action of V br on D preserves the number of feet. Hence, the induced action on X leaves the Morse function h W X ! R invariant. The sublevel complexes Xn have compact quotient modulo the action of V br since the action on Dn is transitive for any number n. Since edges correspond to elementary expansions and elementary expansions always change the number of feet, there are no horizontal edges. Therefore, vertices are the only critical points of this combinatorial Morse function and it suffices to consider descending vertex links. As X is a cube complex, the link lk.‚/ of a vertex ‚ is a piecewise spherical complex whose cells are right-angled spherical simplices, i.e., each cube that has ‚ as a corner contributes a (right-angled spherical) simplex to the link. In fact, not only do the cells have the shape of simplices but the link is indeed a simplicial complex. According to Definition 13.3.4, the descending link lk# .‚/ of a vertex ‚ 2 X (i.e., a diagram with dangling feet) is the subcomplex of the link spanned by those cells that have ‚ as their top corner. The pertinent question is how to describe the descending link of a diagram ‚n with n feet. It suffices to consider only one diagram. Observation 13.8.2. The group V br acts transitively on the set Dn of vertices at height n. Thus all descending links of such vertices are isomorphic as simplicial complexes. #

Therefore, we shall define a simplicial complex Ln that is isomorphic to the descending link lk# .‚n / of any diagram at height n. A braid-merge diagram is a diagram without splits. A braid-merge diagram is thin if any strand can merge at most once. Let Cn be the set of thin braid-merge diagrams with n heads up to the equivalence relation of dangling feet. We can organise the thin braid-merge diagrams from Cn into a cube complex Zn using the same rule as for X : there is an edge from one diagram to another if the latter can be obtained from the former by an elementary expansion. Since the resulting diagram is required to be split-free, only those expansions are allowed that undo merges already present in the diagram. Thus, the trivial diagram |||n with n heads and feet is the unique top vertex that has only incoming edges. An example of a cube below |||5 is shown in Figure 13.8.1. Define L#n to be the link of |||n in Zn . The simplices in L#n are in one-to-one correspondence to the cubes containing |||n which in turn can be uniquely identified by their bottom corner. The cell poset of L#n is therefore given by the set Cn n f |||n g of non-trivial thin braid-merge diagrams and faces of a cell are obtained by undoing merges. #

Observation 13.8.3. The cell complex Ln is isomorphic to the descending link lk# .‚n / of any vertex ‚n 2 X with n D h.‚n / feet.

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Figure 13.8.1. A square in Z5 below |||5 .

Proof. The vertex ‚n is a .1; n/-diagram. Sticking this diagram on top of each diagram in Zn realises the subcomplex of X consisting of all cubes with ‚n as their top corner. 

13.9 Connectivity of descending links Let C .n; k/ be the arc matching complex of the complete graph Kn on n marked points in a k-punctured disk. Let En;k be the plane with n marked points and k punctures, and let C .n; k/ be the arc matching complex of the complete graph Kn on the n marked points in En;k . We suppress counting the punctures in the notation when there are none, i.e., we put C .n/ WD C .n; 0/. A simplex in C .n; k/ consists of a collection of pairwise disjoint arcs connecting marked points where such collections are considered up to isotopy. We have seen in Example 13.4.5 that C .n; k/ is .n/-connected. # We shall derive connectivity properties of Ln by projecting it to the arc matching # complex C .n/. A simplex of dimension k in Ln is given by a thin braid-merge diagram with k C 1 elementary contractions, i.e., a diagram without splits and exactly k C 1 merges where each strand merges at most once. We put the diagram into standard formn by moving merges to the bottom and arranging that merging strands are adjacent. At the picture level, we map it to a k-simplex in C .n/ in two steps: first, we draw the merges as horizontal arcs connecting neighboring strands; in a second step, we

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Figure 13.9.1. Unwinding sequence.

straighten all the strands and record what this does to the horizontal arcs. Formally, the first step is just looking at a particularly obvious k-simplex in C .n/ whose arcs keep track which pairs of feet merge. The second step uses the interpretation of the braid group Bn as the mapping class group of the n-punctured plane. So undoing the braid can be realised as a mapping class and we consider how this mapping class acts on the simplex obtained in the first step. We shall refer to this construction as unwinding. The process is best understood visually: 7! This mapping is witnessed by the ‘unwinding sequence’ shown in Figure 13.9.1. Observation 13.9.1. The unwinding construction induces a well-defined simplicial # mapˆ W Ln ! C .n/. Intuition instead of proof. We have to address two issues. The first is the problem of ˆ being well-defined. The unwinding construction takes place at the level of dia# grams, but cells in Ln are given by diagrams up to dangling of feet. As indicated in Figure 13.9.2, any additional dangling (drawn with thick strokes) is undone automatically during the unwinding procedure. The remaining question is compatibility with the simplicial structure, i.e., if  is a simplicial cell in L#n and  6  is a face, then unwinding  yields a face of the arc matching obtained by unwinding  . This, however, is obvious: undoing a merge in  amounts to deleting the corresponding arc in ˆ. /.  In the same way that dangling is undone during unwinding, twisting strands before merging is undone during unwinding, as shown in Figure 13.9.3. Consequently, the

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Figure 13.9.2. Automatic undangling during unwinding. The top row shows in thick strokes additional (compared to the bottom row) dangling and how it is undone at the beginning of the the unwinding procedure.

Figure 13.9.3. Two diagrams unwinding to the same arc.

unwinding map is not an isomorphism. However, this phenomenon is the only source of non-injectivity. Fact 13.9.2. The fiber ˆ 1 .˛/ of a single arc is an infinite discrete set (a Z worth of diagrams). The preimage ˆ 1 . / of a k-simplex is the full join of its vertex fibers: ˆ In particular, the fiber ˆ

1

1

. / D

. / is .k

ˆ

1

.˛/

˛2

1/-connected.

Example 13.9.3. We consider the edge  D #

in the arc matching complex C .5/. The edges in L5 unwinding to  are given by diagrams of the form shown in Figure 13.9.4. The fiber over the closed simplex  is therefore the complete bipartite graph over two countable infinite sets. Hence, it is connected.

Higher finiteness properties of braided groups

k

319

l

Figure 13.9.4. The numbers k; l 2 Z count how many times the pair of strands is twisting. Each of # these edges in L5 unwinds to the edge .

In [26], D. Quillen proved several very useful results that relate connectivity properties of the domain of a projection to connectivity of the base, provided one has good control over the fibers of the map. We shall make use of the following criterion, which Quillen phrases in terms of maps between posets. We specialise it for simplicial complexes (representing their poset of cells). Recall that a complex X is d -spherical if it has dimension d and is .d 1/-connected. Theorem 13.9.4 ([26, Thm. 9.1]). Let ' W X ! Y be a simplicial map. Assume that Y is d -spherical, and for each simplex   Y that the link lkY . / is .d dim. / 1/-spherical. About the fibers assume that the preimage ' 1 . / of a closed simplex is dim. /-spherical. Then, X is d -spherical. Our complexes are not spherical: they are highly connected but have cells above the allowed dimension. Removing the cells above the critical dimension yields the following: Corollary 13.9.5. Let ' W X ! Y be a simplicial map. Assume that Y is lconnected, and for each simplex   Y that the link lkY . / is .l dim. / 1/connected. About the fibers, assume that the preimage ' 1 . / of a closed simplex is .dim. / 1/-connected. Then X is l-connected.  ˘ Recall that we defined .n/ WD nC1 2. 3

Proposition 13.9.6. The cell complex L#n is .n/-connected. Consequently, the descending link lk# .‚/ of any n-feet vertex is .n/-connected.

Proof. We consider the projection ˆ W L#n ! C .n/. The base space C .n/ is .n/connected. Moreover, the link of a k-simplex  in C .n/ is isomorphic to the arc matching complex C .n 2k 2; k C 1/, which is .n 2k 2/-connected. Since .n

2k

2/ > .n/

k

1;

the claim follows from Corollary 13.9.5 and Fact 13.9.2.



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13.10 Finiteness properties of V br : Proof of Theorem 13.6.1 We want to apply Brown’s criterion in the form of Corollary 13.2.3. We consider the action of V br on the cube complex X . The complex is contractible by Proposition 13.7.3 and the stabilisers of cubes are all of type F1 by Observation 13.7.2. The subcomplexes Xn are V br -invariant and have compact quotient mod V br by Observation 13.8.1. In order to apply Corollary 13.2.3, we need to see that the connectivity of the pair .Xn ; Xn 1 / tends to infinity as the filtration index n tends to infinity. It is here, where combinatorial Morse theory is used. By Proposition 13.9.6, descending links of vertices at height n are .n/-connected. By Corollary 13.3.6, it follows that the inclusion Xn 1  Xn induces isomorphisms of homotopy groups d in dimensions d 6 .n/. From the long exact sequence of homotopy groups of a pair we deduce that the pair .Xn ; Xn 1 / is .n/-connected. Thus, connectivity tends to infinity as n tends to infinity. This concludes the proof of Theorem 13.6.1. Remark 13.10.1. We did not use the fact that the inclusion Xn 1  Xn induces an epimorphism in d for d D .n/ C 1. From this, we obtain one additional degree of connectivity for the pair .Xn ; Xn 1 /. Remark 13.10.2. The arc matching complexes for complete bipartite graphs studied in Example 13.4.7 can be used to prove a conjecture of F. Degenhardt that the braided Houghton groups, introduced in [15], have the same finiteness properties as their unbraided counterparts, which were already treated by Brown in [11, Sec. 5]. These groups have been given an alternative description as mapping class groups of certain infinite surfaces in [19]. Details on their finiteness properties as well as a complete treatment of the related arc matching complexes will be given elsewhere.

References [1] H. Abels and S. Holz, Higher generation by subgroups, J. Algebra 160 (1993), 310–341. [2] C.A. Anastasiadis, Decompositions and connectivity of matching and chessboard complexes, Discrete Comput. Geom. 31 (2004), 395–403. [3] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445–470. [4] A. Björner, The homology and shellability of matroids and geometric lattices. In [30], 226– 283. [5] A. Björner, L. Lovász, S.T. Vre´cica and R.T. Živaljevi´c, Chessboard complexes and matching complexes, J. London Math. Soc. 49 (1994), 25–39. [6] A. Borel and J-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. [7] A. Borel and J.-P. Serre, Cohomologie d’immeubles et de groupes S-arithmétiques, Topology 15 (1976), 211–232.

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[8] M. Brin, The algebra of strand splitting II: A presentation for the braid group on one strand, Int. J. Algebra Comput. 16 (2006), 203–219. [9] M. Brin, The algebra of strand splitting I: A braided version of Thompson’s group V , J. Group Th. 10 (2007), 757–788. [10] K.S. Brown, Cohomology of Groups, GTM 87, Springer, New York, 1982. [11] K.S. Brown, Finiteness properties of groups, J. Pure Appl. Alg. 44 (1987), 45–75. [12] K.-U. Bux, M. Fluch, M. Schwandt, S. Witzel and M. Zaremsky, The braided Thompson’s groups are of type F1 , J. reine angew. Math. (Crelle) 718 (2016), 59–101.

[13] K.-U. Bux, R. Köhl and S. Witzel, Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem, Ann. of Math. 177 (2013), 311–366.

[14] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91–119. [15] F. Degenhardt, Endlichkeitseigenschaften gewisser Gruppen von Zöpfen unendlicher Ordnung, PhD thesis, Univ. Frankfurt, 2000. [16] P. Dehornoy, The group of parenthesized braids, Adv. Math. 2005 (2006), 354–409. [17] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton University Press, Princeton, 2012. [18] D. Farley, Finiteness and CAT.0/ properties of diagram groups, Topology 42 (2003), 1065– 1082. [19] L. Funar, Braided Houghton groups as mapping class groups, An. S¸ tiin¸t. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 53 (2007), 229–240. [20] R. Geoghegan, Topological Methods in Group Theory, GTM 243, Springer, New York, 2008. [21] A. Hatcher and K. Vogtmann, Tethers and homology stability for surfaces, Algebr. Geom. Topol. 17 (2017), 1871–1916 [22] J.L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985), 215–249. [23] J.L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), 157–176. [24] W.J. Harvey, Geometric structure of surface mapping class groups. In [29], 255–269. [25] J.S. Povran, Decompositions, shellings and diameters of simplicial complexes and convex polytopes, Thesis, Cornell University, 1977. [26] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math. 28 (1978), 101–128. [27] M. Stein, Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc. 332 (1992), 477–514. [28] K. Vogtmann, Local structure of some OUT(Fn )-complexes, Proc. Edinburgh Math. Soc. 33 (1990), 367–379. [29] C.T.C. Wall (ed.), Homological Group Theory, LMS Lecture Notes 35, Cambridge University Press, Cambridge, 1979. [30] N. White, Matroid Applications, Encyclopedia Math. Appl., volume 40, Cambridge University Press, Cambridge, 1990. [31] M. Zaremsky, Higher generation for pure braid groups, Appendix to arXiv version of [12], arXiv:1210.2931.

Chapter 14

Zeta functions and the trace formula W. Hoffmann Prehomogeneous vector spaces provide a framework for the method of analytic continuation of zeta integrals due to Hecke and Tate. We will describe instances where convergence can only be achieved by truncation. Special values of such zeta integrals appear in the Arthur-Selberg trace formula, and their study is relevant in connection with recent ideas in the Langlands program.1

14.1 Introduction The initial motivation for introducing zeta functions was the study of the asymptotic distribution of a sequence of real numbers, e.g., the sequence of primes, of closed geodesics on a Riemann surface, or of the eigenvalues of a self-adjoined operator. The larger the half-plane to which a zeta function can be meromorphically continued, the smaller the error term in that asymptotics. The best one can expect in this respect is meromorphic continuation to the entire complex plane. Zeta functions with this property often satisfy a functional equation. The classical prototype is the Riemann zeta function or, more generally, the Dedekind zeta function of a number field. In this case, the functional equation is a consequence of the Poisson summation formula, as was shown by Hecke and Tate. The most general zeta functions which owe their functional equation to the Poisson summation formula are those associated with prehomogeneous vector spaces. They will be the subject of the first section. A seemingly unrelated topic is the Arthur–Selberg trace formula, which plays an important role in the theory of automorphic forms. Roughly speaking, that formula computes the trace of a certain intgral operator by integrating its kernel function over the diagonal. In order to ensure convergence, that kernel function has to be truncated in most cases. The two topics are related because truncation is also necessary in some cases to ensure the convergence of zeta functions associated with prehomogeneous vector spaces. In fact, we expect that the so-called unipotent contribution to the trace formula can be expressed in terms of such zeta functions. So far, this has been proved for groups of rank up to two.

1 Project

C7

324

W. Hoffmann

14.2 Zeta integrals and zeta functions 14.2.1 Defintions and basic properties We recall that an affine algebraic variety X over a field F can be given by a family of polynomial equations in several variables with coefficients in F . Not only for F itself, but for every commutative F -algebra E, one may consider the set X.E/ of solutions with coordinates in E. More generally, if A is a commutative unital ring and X is a scheme over A, one defines X.B/ for every extension ring B as the set of morphisms from the one-point scheme with coefficients in B to the scheme X (more precisely, to the result of changing the base ring from A to B). If G is an algebraic group acting algebraically on an algebraic variety X , everything defined over a field F , then a geometric G-orbit over F is a G-invariant subvariety Y of X defined over F with Y .F / ¤ ¿ and which, as such, is minimal under inclusion. If F is algebraically closed, then Y .F / is in fact a G.F /-orbit in the ordinary sense. Now we can introduce our basic object (cf. [18]). Definition 14.2.1. A prehomogeneous vector space over a field F is a finite dimensional vector space V endowed with a linear action of an algebraic group G, all defined over F , such that V contains a dense Zariski-open geometric G-orbit Vreg defined over F . A relative invariant is a nonzero rational function p on V with the property that, for every extension field E and every g 2 G.E/, there exists  2 E  D GL1 .E/ such that p.gx/ D p.x/:

Here V is considered as an affine variety, so that V .E/ D V ˝F E. Clearly,  is a function of g and defines a morphism G ! GL1 of algebraic groups over F . The group of relative invariants over F is generated by the nonzero constants and the defining functions of the F -irreducible hypersurfaces in the singular set V n Vreg (cf. [18, §2.2]). For simplicity, we assume that there is only one such hypersurfce with defining function p. Let us give some examples. 1. V is a simple associative algebra over F with reduced norm p and G.F / is the group of its invertible elements, so that  D pjG . We recall that, by Wedderburn’s theorem, V is isomorphic to a matrix algebra over a division algebra over F . 2. V is arbitrary, endowed with a non-degenerate quadratic form p, and G.F / D fg 2 GL.V / j 9  2 F  8 x 2 V W p.gx/ D p.x/g; the conformal group of p. 3. V D Sym2 W (the symmetric square) and G D GL.W /. One may identify V with the space of quadratic forms on the dual space W  , take for p.x/ the discriminant of x, and .g/ D det g 2 . 4. V D Sym3 W and G D GL.W /. This space is prehomogeneous only for dim W D 2, known as the space of binary cubic forms. Again, p.x/ is the discriminant of x and .g/ D det g 6 .

Zeta functions and the trace formula

325

5. V D VC ˚ V with a perfect pairing p between the subspaces and G.F / D f.gC ; g / 2 GL.VC /  GL.V / j 9 W p.gC xC ; g x / D p.xC ; x /g: Now the next basic object enters the stage (cf. [18, §5.3]). Definition 14.2.2. The zeta integral associated with a prehomogeneous vector space over the field Q is Z X !.g/ .g/ dg; Z.; !; ƒ/ D G.R/=€

2ƒ\Vreg .Q/

where ! W G.R/ ! C is a continuous homomorphism,  is a Schwartz function on V .R/, ƒ is a lattice in V .Q/ and € is the stabiliser of ƒ in G.R/. We may assume that G acts (almost) faithfully, so that € is discrete. Then the G.R/-invariant measure dg on the homogeneous space G.R/= € is determined by a fixed Haar measure on G.R/ and the counting measure on €. Often one chooses !.g/ D j.g/js , where s 2 C and the character  W G ! GL1 corresponds to a regular relatively invariant function p on V . Then the integral-sum is absolutely convergent for s in some right half-plane, which may be empty. Zeta-functions arise as follows. We write the contribution from a €-orbit Œ€ D f  j 2 €g in ƒ as a sum over 2 €= € , where € denotes the stabiliser of  in €. Assuming absolute convergence, we interchange the sum over the set of orbits with the integral and combine the latter with the inner sum to get Z X !.g/.g/dg: Z.; !; ƒ/ D Œ€ ƒ\Vreg .Q/

G.R/=€

Now we split the resulting integral into an integral over G.R/=G .R/ and one over G .R/= € . The integrand does not depend on the variable in G .R/ and can be taken out of the inner integral. On each G.R/-orbit O in Vreg .R/, we fix a G.R/-invariant measure dG x and define a nonvanishing function q up to a nonzero factor by the condition q.gx/ D !.g/q.x/:

In fact, if q0 corresponds in this sense to the character !0 .g/ D jdetV gj, we have dG x D q0 .x/dx for a Lebesgue measure dx on V .R/. The chosen measure on O fixes those on both subfactors of G.R/ for each , and the remaining integral can be reduced by a substitution to the weighted orbital integral Z .x/q.x/dG x: JO .; q/ D O

326

W. Hoffmann

As a consequence, the zeta integral can be written as a finite sum X Z.; !; ƒ/ D .q; ƒ \ O/JO .; q/

(14.2.1)

OVreg .R/

where the zeta function for the orbit O is defined by .q; ƒ \ O/ D

X

Œ€ ƒ\O

vol.G .R/= € / : q./

If !.g/ D j.g/js for the character  corresponding to a relative invariant p, we can take q.x/ D jp.x/js and write the zeta function as .s; ƒ \ O/. This is a certain Dirichlet series. In the special case of example (1) with V D Q, ƒ D Z, p.x/ D x and O D RC , we obtain the Riemann zeta function. The whole construction may also be applied to a prehomogeneous vector space defined over any number field F rather than just Q and to an O -lattice ƒ in V , where O is the ring of integers in F . Then one has to replace R by the F -algebra F1 , the product of all completions of F with respect to archimedean valuations, into which O embeds diagonally as a discrete subgroup. The supply of characters ! increases considerably if we consider Q all O -lattices ƒ in V at once. For this purpose, one has to use the ring A D 0v Fv of adeles of F . It consists of the elements of the product of the completions of F over all valuations v whose components for almost all non-archimedean v lie in the maximal compact subring Ov of Fv . Thus A is locally compact, and the diagonal embedding of F turns it into an F -algebra. It has a splitting A D F1 Afin , where the second factor corresponds to the non-archimedean valuations. For every variety X over F , the set X.A/ of adelic points is endowed with a locally compact topology such that An gets the product topology and every morphism between F -varieties extends to a continuous map between the sets of their adelic points. In particular, G.A/ can be endowed with a Haar measure. Now we present the adelic version of Definition 14.2.2 (cf. [18, §6.8]). Here, the discrete subgroups V D V .F / of V .A/ and G.F / of G.A/ play the roles of ƒ and €. Definition 14.2.3. The adelic zeta integral associated with a prehomogeneous vector space over a number field F is Z X !.g/ .g/ dg; Z.; !/ D G.A/=G.F /

2Vreg .F /

where ! W G.A/ ! C is a continuous homomorphism and  is a Schwartz–Bruhat function on V .A/. Unramified characters of G.A/ are those of the form g 7! j.g/js for a rational character , where we now use the idele norm defined by vol.aB/ D jaj vol.B/ for

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327

every a 2 A and every Borel set B  A. The zeta integral of the product of such a character, for  as above, and a general character ! is denoted by Z.; s; !/. In order to obtain zeta functions, we have to choose a finite set S of valuations of F containing all archimedean ones. There is a splitting of A into F -algebras Q Q FS D v2S Fv and AS D 0v…S Fv and correspondingly X.A/ D X.FS /  X.AS / Q for every variety X over F . If U is a lattice in V .AS / over the ring v…S Ov , the set ƒ D f 2 V .F / j  S 2 U g is a lattice over the ring OS of S -integers in F and, conversely, U is the closure of ƒ embedded diagonally into V .AS /. For a SchwartzBruhat function of the form .x/ D S .xS /1U .x S /; where 1U is the characteristic function of U , we get a decomposition X Z.; !/ D  S .q S ; O \ ƒ/JO .S ; qS /

(14.2.2)

OVreg .FS /

over the G.FS /-orbits O in Vreg .FS / generalising (14.2.1). A necessary condition for the convergence of the zeta integral is the finiteness of the volumes occurring as coefficients of the zeta functions. This is equivalent to the condition that for some (hence for any)  2 Vreg .F /, the stabiliser G has no nontrivial characters defined over F . A sufficient conditionQis given in [22]. For every variety X over F , there is a splitting X.A/ D 0v X.Fv / into a restricted product over all valuations of F . In particular, a character ! of G.A/ is determined by its restrictions !v to the factors G.Fv /. However, the above zeta functions do not admit Euler products in general. In example (1), they do. Here, we have G D Vreg , whence one can combine the integral in Definition 14.2.3 with the complete sum to obtain an integral over G.A/. In the one-dimensional case V D F , the corresponding zeta function is the Hecke L-function of the character !. For a general simple algebra V , one may replace ! by an automorphic representation  of G.A/ (see below) and get an operator-valued zeta integral. One may present  as the restricted tensor product of representations v of the factors G.Fv /, and if the Schwartz-Bruhat function  is the product of functions v on G.Fv /, one has a splitting O Z.; s; / D Zv .v ; s; v / v

into local zeta integrals. The automorphic L-function L.s; / is then defined as an Euler product, see [11]. 14.2.2 Functional equations A functional equation for a zeta function associated with a prehomogeneous vector space V over a number field F is usually a consequence of the Poisson summation formula. The adelic version of that formula reads X X O ./ ./ D 2V

2V 

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W. Hoffmann

and is valid for any Schwartz–Bruhat function  on V .A/, whose Fourier transform is the Schwartz–Bruhat function on V  .A/ given by Z O .x/ .hx; yi/ dx: .y/ D V .A/

Here W A ! C is a fixed nontrivial continuous character of the additive group, trivial on F , and the Haar measure dx is normalised by vol.V .A/=V .F // D 1. Definition 14.2.4. The prehomogeneous vector space V is called regular over F if there exists a relative invariant p defined over F such that the map x 7! p.x/ 1 dp.x/ 2 V  is dominant. In this case, the dual space V  is also prehomogeneous for the contragredient action characterised by hgx; gyi D hx; yi, and the map in the definition sets up an equivariant isomorphism between the regular sets. The basic relative invariant p of V  pulls back to a constant times 1=p, hence its corresponding character is  D 1=. All the examples above are regular prehomogeneous vector spaces, while a vector space V with the tautological action of GL.V / is not regular for dim V > 1, having no relative invariants at all. We will now recall how the functional equation emerges, whose proof also provides the meromorphic continuation. If we decompose Z.; !/ D ZC .; !/ C Z .; !/; where Z˙ .; !/ D

Z

!.g/

j.g/j> 1. For every g 2 G.A/, the Schwartz–Bruhat function g .x/ D .gx/ 1O  1 c has the Fourier transform  g D !0 .g/ g , where !0 .g/ D jdetV gj D !0 .g/ . Hence the Poisson summation formula implies O ! !0 / C I.; !/; Z .; !/ D ZC .;

provided the integral Z I.; !/ D

j.g/j  , which provides the meromorphic continuation. The contributions from the zero orbits S0 D f0g in V and SO0 D f0g in V  are given by s0 D sO0 D 0 and O D vol.G.A/1 =G.F //: †0 ./ D †0 ./ ˇ If  denotes the characteristic function of the set fg 2 G.A/ ˇ j.g/j < 1g, then Z.; s/

Z O †0 ./ j.g/js D s  G.A/=G.F /

X

.g/

2Vreg .F /

 .g/

Z

V .A/

!

.gx/ dx dg:

(14.2.5) In many cases (e.g., if G is anisotropic modulo center, S D f0g and SO D f0g), it is known that the contribution from SO0 is the principal part in the Laurent expansion of Z.; s/ at the abscissa of convergence s D  and that the integral on the righthand side is convergent for s in a half-plane strictly larger than the half-plane of convergence of Z.; s/. Its value at s D  gives a formula for the constant term in the Laurent expansion. Using (14.2.2), that constant term can also be expressed in terms of the principal parts of the zeta functions and the Taylor coefficients of the weighted orbital integrals. In preparation for the next example, we recall the notion of a height function. For a free module V over a commutative ring A, we denote by VP the scheme-theoretic complement of the origin. Then VP .A/ consists of all x 2 V for which there exists an A-linear form l on V such that l.x/ is invertible. If V is a vector space over the number field F , a height function on VP .A/ is defined as Y kxk D kxv kv ; v

where each k : kv is a norm on V .Fv / homogeneous with respect to the valuation j : jv , coming from a quadratic resp. hermitian form for archimedean v, and such that fx 2 VP .Afin / j kxk 6 1g is open in V .Afin /. Then kaxk D jajkxk for every a 2 A , so that the restriction to VP .F / induces a height function on the projective space VP .F /=F  . For the group G.F / D GL.V /, the stabiliser K  G.A/ of a height function is a maximal compact subgroup for which Kfin is open in G.Afin /, and every such subgroup arises in this way. In case V .F / D F n , one may take ( .jx1 jev C


C jxn jev /1=e if v is archimedean and e D ŒC W Fv , kxkv D max.jx1 jv ; : : : ; jxn jv / if v is non-archimedean, whose stabiliser Kv is O.n/, U.n/ or GL.n; Ov /, resp. The space V D Sym2 W of quadratic forms in Example 3 was studied by Siegel and by Shintani [25]. We have .g/ D .det g/2 , and there is a natural pairing between

331

Zeta functions and the trace formula

the algebras Sym W and Sym W  obtained by applying the elements of the latter, interpreted as differential operators on W  , to the elements of the former and evaluating at zero. In particular, we may identify V  with Sym2 W  . For each r 6 n D dim W , there is a geometric orbit Sr  V consisting of the elements of rank r. For n ¤ 2, the zeta integral Z.; s/ is convergent for Re s >  D nC1 and has a meromorphic 2 continuation to the complex plane satisfying the functional equation (14.2.4). For each 0 6 r < n, the orbit Sr (resp. SOr ) causes a simple pole at 2r (resp.  2r ), and formula (14.2.5) remains valid. However, the individual contributions of the orbits to I.; !/ do not converge. Shintani circumvented this problem by choosing a function  that vanishes on S.A/ and for which O vanishes on SO .A/, while the orbital integrals in (14.2.1) do not. The residues were fully determined only later in [16] using different methods. In the case n D 2, the zeta integral Z.; s/ is divergent due to the presence of split elements  D 1
2 2 Vreg .F / with 1 ¤ 2 2 W . Indeed, the i are eigenvectors of each g 2 G , and the ratio of the eigenvalues is a rational character of that group. In order to salvage convergence, one fixes a height function on W .A/ and chooses a family t of truncation parameters tU > 0 for all one-dimensional subspaces U  W in such a way that the characteristic functions U of the sets ˇ fg 2 G.A/ ˇ kgk > tU kk for  2 U g satisfy U .g / D  U .g/ for all 2 G.F /. Then the truncated zeta integral  Z X  X j.g/js 1 Z t .; s/ D U .g/ .g/dg; (14.2.6) G.A/=G.F /

2Vreg .F /

U j

where U runs through the two or none lines whose elements divide the given , converges for Re s >  D 32 . Since G.F / acts transitively on the set of split elements, we can choose one such element 0 with corresponding subspaces U1 , U2 and write Z t .; s/ D

Z

G.A/=G.F /

j.g/js

C

Z

X

.g/dg

2Vreg .F / irred.

G.A/=G0 .F /

j.g/js 1

U .g/ 1


U .g/ .g0 /dg: 2

For tU1 D tU2 D 0, Yukie [27] called this the adjusted zeta integral. It extends meromorphically to the complex plane and satisfies the functional equation (14.2.4). It has poles at 0 and 32 coming from zero orbits S0 , SO0 and poles at 12 and 1 from the orbits S1 , SO1 of nonzero squares, see [27].

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W. Hoffmann

Theorem 14.2.5. The truncated zeta integral for the space of binary quadratic forms satisfies Z t .; s/ D

Z

O †0 ./ s 32

G.A/=G.F /

j.g/j

s

X 

2Vreg .F /

1

X U j

 Z U .g/ .g/  .g/

V .A/

!

.gx/dx dg; (14.2.7)

where the integral on the right-hand side converges for Re s > 1. This follows from the proof of Theorem 6.2 in [15]. Example 4 of binary cubic forms was first studied by Shintani [24]. Wright [26] and Datskovsky–Wright [8] treated it adelically, but modified the action of G.F / D GL.W / on V D Sym3 W by twisting it with the inverse of the determinant. Then .g/ D det g 2 , and  D 1 for our parametrisation of unramified characters. As in the previous example, there are elements  2 Vreg .F / which have (one or three) linear factors defined over F . However, their stabilisers G do not admit rational characters over F , so G .A/=G .F / has finite volume and, indeed, Z.; s/ converges for Re s >  . Beside the zero orbit S0 , there are the singular orbits S1 and S2 of elements of the form  D 3 and  D 1 22 , resp., were , 1 ¤ 2 2 W . The orbit structure of V  is analogous. The zeta integral extends meromorphically to the complex plane and satisfies the functional equation (14.2.4). In order to describe the residues, one fixes a onedimensional subspace U0 . This gives rise to a flag of subspaces Vi D Symi

1

W
Sym4

i

U0

in V with dim Vi D i . Now S1 is the G-saturation of V1 n f0g and S2 is that of V2 n V1 . The same is true for the singular orbits SOi and subspaces VOi in V  obtained by replacing U0 with its annihilator UO 0  W  . To each Schwartz–Bruhat function  on V .A/ one associates functions i on Vi .A/=Vi 1.A/, viz. Z .xy/dy; i .x/ D Vi

1 .A/

and defines, for z 2 C n f0; 1g and i 2 f1; 2g, †i .; z/ D Z. Ki ; z/; where Z denotes the analytically continued zeta integral associated to the one-dimensional prehomogeneous vector space Vi =Vi 1 with the group GL1 and Z K .kx/dk  .x/ D K

333

Zeta functions and the trace formula

is the average over a maximal compact subgroup K  G.A/ as above. Proposition 6.1 of [8] asserts that I.; s/ D

O †0 ./ s 1

O 2/ † .; †0 ./ 3 C 1 s 6s 5

O 1/ †1 .; 32 / † .; C 2 6s 1 2s 2

†2 .; 1/ 2s

for our parametrisation of unramified characters and a measure on G.A/ specified below. The proof uses Shintani’s method of multiplying the integrand by certain functions that make the contribution from each orbit convergent and so that the original integral may be recovered. A more transparent proof using partial Fourier transforms, which exhibits the cancellations between the contributions from the orbits, is given in [19]. Although the elements of Vreg .F / which split over F into two factors do not hamper the convergence of Z.; s/ for Re s >  D 1, they do make the right-hand side of (14.2.5) divergent for s D  . This can be fixed by truncation: Theorem 14.2.6. The zeta integral for the space of binary cubic forms satisfies Z.; s/

Z O C 1 †2 .; O 1/ †0 ./ 2 j.g/js D s 1 G.A/=G.F /  .g/

X

X

U 2Vreg;U .F /

Z

Sym3 U.A/

.g.x//dx

X

.g/

2Vreg .F /

 .g/

Z

!

.gx/dx dg

V .A/

where Vreg;U D f 2 Vreg W U j g= Sym3 U . The integral on the right-hand side is convergent for Re s > 65 . Proof. We want to show that, for Re s > 1, Z X O 1/ †2 .; j.g/js  .g/ D 2s 2 G.A/=G.F /

X

U 2Vreg;U .F /

Z

Sym3 U.A/

.g.x//dx dg:

The subspaces U are the translates of U0 by elements of G.F /. The associated subspaces Sym3 U and f 2 V W U j g D U
Sym2 W are the translates of V1 and V3 , resp., and V3 \ Vreg D V3 n V2 , so the right-hand side becomes Z Z X .g.x//dx dg; j.g/js  .g/ G.A/=P .F /

2.V3.F /nV2 .F //=V1 .F /

V1 .A/

where P is the stabiliser of U0 . Since G.A/ D KP .A/, it remains to show that Z Z X Z.O 2 ; 1/ s .p.x//dx dp j.p/j  .p/ D 2s 2 V1 .A/ P .A/=P .F / 2.V3 .F /nV2 .F //=V1 .F /

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for a suitable right-invariant measure dp. If we denote the eigenvalues of p 2 P .A/ in W .A/=U0 .A/ and in U0 .A/ by a and b, resp., those in V3 .A/=V2 .A/, V2 .A/=V1 .A/ and V1 .A/ are a, b and b 2 =a, resp. Integration over N.A/=N.F /, where N is the unipotent radical of P , amounts to replacing V1 by V2 and multiplying the integrand by ja=bj. Our measure dg is twice that of [26] and corresponds to the measure da db on P .A/=N.A/, hence the right-hand side becomes Z Z X jabj2s 1jabj 0 for all (maximal) P 0  P . We denote by PT resp. OPT be the characteristic function of the set of all x 2 G.A/ such that HP .x/ TP is in C aC P resp. aP . The function X F .x; T / D . 1/dim AP =AG OPT .x/ (14.3.4) P

1

on G.A/ has compact support modulo G.F /, and if TP is sufficiently deep inside G aC P for one minimal (and hence for each) P , then F .x; T / it is the characteristic function of a set that exhausts G.A/1 as T ! 1 in the obvious sense. So it would be natural to integrate K.x; x/F .x; T /. Arthur realised that the integral is asymptotic to a polynomial that can be described explicitly. He considers the induced representation RP coming from the analogue RM of R for a Levi component M of any parabolic subgroup P defined over F . Now RP .f / is an integral operator with kernel function X Z P f .x 1 ny/dn: K .x; y/ D

2M.F /

N.A/

Arthur proved (cf. Theorem 6.1 in [2]) that Z X . 1/dim AP =AG K P .x; x/OPT .x/dx J T .f / D G.F /nG.A/1 P

(14.3.5)

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is convergent, where P runs through all parabolic subgroups over F , including G itself, and depends on T polynomially. Arthur’s trace formula (cf. [2], Theorems 19.2 and 21.6) is the equality of two expressions for J T .f /, a geometric and a spectral one. It has the general shape Z X aM . /JM . ; f / D aM ./JM .; f /dŒM; : (14.3.6) ŒM; 

The sum on the left-hand side is taken over the conjugacy classes of pairs consisting of a Levi subgroup M and an element of M.F /, while the integral on the right-hand side is taken over the conjugacy classes of pairs consisting of a Levi subgroup M and a (virtual) automorphic representation  of M.A/ \ G.A/1 . Inside the contributions from M D G to the right-hand side, one finds the trace of Rdis .f /. Thus, the trace formula can be applied to the study of the multiplicities of automorphic representations in this space. The general distributions JM .; f / come from induced representations. They are weighted characters defined with the help of intertwining operators. The general distributions JM . ; f / on the left-hand side are weighted orbital integrals over conjugacy classes. Inside the adelic points of a geometric conjugacy class, there may be infinitely many G.A/-conjugacy classes. Thus, strictly speaking, one has to fix a finite set S of places of F and consider test functions f .g/ D fS .gS /f S .g S /, where f S is the characteristic function of a maximal compact subgroup of G.AS /. This leads to a modified notion of conjugacy depending on S . For advanced applications, e.g. the classification of automorphic representations of classical groups in terms of those of GLn (cf. [3]), the trace formula has to be generalised to the twisted trace formula and undergo a number of metamorphoses (invariant form, stable form). There are potential applications which require an even deeper understanding of the terms in the trace formula, see [4].

14.4 Unipotent terms in the trace formula The global coefficients aM . / and the weight factors in JM . ; f / on the geometric side of the trace formula (14.3.6) are known only in special cases, because they are the result of an indirect argument. Now we report recent progress in their computation. A natural intermediate step in the derivation of the geometric side, previously circumvented by Arthur, is the decomposition of J T .f / into contributions JCT .f / from the (geometric) conjugacy classes C . This requires the notion of a conjugacy class induced from a conjugacy class D of a Levi component M of a parabolic subgroup P . It was proved in [20] for unipotent classes and in [13] for arbitrary classes that DN contains a dense geometric P -conjugacy class D 0 . We call D 0 the inflated class and the G-conjugacy class containing D 0 the induced class. Now we can define JCT .f /

Zeta functions and the trace formula

339

by Equation (14.3.5) with K P replaced by KCP .x; y/ D

X X Z D 2D.F /

f .x

1

ny/ dn;

N.A/

where the exterior sum runs through the geometric conjugacy classes D in M whose induced conjugacy class is C . It was conjectured in [14] and proved in [10] that the integral-sum defining JCT .f / is absolutely convergent, even for suitable noncompactly supported functions f . Arthur reduced the computation of J T .f / to that of the unipotent contribution (cf. [2], section 19), hence we consider only unipotent conjugacy classes C now. A unipotent element of G.F / is of the form D exp X for a nilpotent element X of the Lie algebra g of G over F . According to the theorem of Jacobson–Morozov, there exists another nilpotent element Y and a semisimple element H such that ŒX; Y  D H , ŒH; X  D 2X and ŒH; Y  D 2Y . Under the endomorphism Z 7! ŒH; Z, the Lie algebra decomposes into eigenspaces gn , n 2 Z. If we denote by q, u, u0 and u00 the sum of eigenspaces with n > 0, n > 0, n > 2 and n > 2, respectively, then q is a parabolic subalgebra with ideals u, u0 and u00 , and Kostant proved that they are independent of the choice of Y and H . Moreover, the normaliser Q of q in G is a parabolic subgroup over F , called the canonical subgroup of X , with unipotent radical U and normal subgroups U 0 and U 00 obtained from their analogues in g by the exponential map, and the centraliser L of H in G is a Levi component of Q. The subalgebra spanned by X , Y and H is isomorphic to sl2 .F /, and it follows from the representation theory of the latter that V D U 0 =U 00 is a regular prehomogeneous vector space under the action of L by conjugation. A general approach to the evaluation of JCT .f / for a unipotent class C , which generalises the original method of Selberg, was described in [14]. The first step is to group the contributions from the various elements of C.F / according to their canonical subgroups. The problem is that we actually have contributions from cosets

N  P , whose elements have different canonical subgroups. The trick is to replace the integral over N.A/ for in an inflated class D 0 .F / by a sum of integrals 0 0 over cosets 0 N D .A/ for a sufficiently small normal subgroup N D of P so that 0 the elements of 0 N D .F / \ D 0 .F / have the same canonical subgroup. This uses a mean-value formula of Siegel, Weil and Ono, whose applicability relies on the con0 dition that N D is also sufficiently large so that every rational function on the variety 0

N=N D relatively invariant under the stabiliser of N in P is constant. So far, no 0 case has been found in which such N D would not exist. We can now fix the canonical subgroup Q of some element of C.F / and obtain those of the other ones by conjugating with G.F /. If we combine the resulting sum over Q.F /nG.F / with the integral, we obtain under certain convergence assumptions Z Z X X T f .x 1 n0 x/ dn0 T N 0 .x/dx; JC .f / D N 0 .A/ Q.F /nG.A/1 N 0 Q

2..C \Qcan /N 0 =N 0 /.F /

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W. Hoffmann

where Qcan is the set of elements of Q whose canonical subgroup of G coincides with Q and X T N 0 D . 1/dim AP =AG OPT P

with P running through the parabolic subgroups such that is in some inflated con0 jugacy class D 0  P and N 0 D N D . According to Cor. 1 in [13], this sum is finite. The upshot is that, after introducing a simple exponential factor under the integral, T 0 the contribution JC;N 0 .f; / from each subgroup N , given by Z Z X f .x 1 n0 x/ dn0 T N 0 .x/dx; eh;HQ .x/i Q.F /nG.A/1

2..C \Qcan /N 0 =N 0 /.F /

N 0 .A/

has a chance to converge for  in the dual space of aP ˝ C with Re  in a certain chamber. Moreover, the contribution from N 0 D f1g can be expressed as a truncated zeta integral in several variables, viz. Z X hCıU=U 00 ;HL .l/i T e fV .l 1 l/T .l/ dl; JC;f1g .f; / D L.F /nL.A/\G.A/1

where Vreg D C \ U 0 =U 00 , fV .x/ D

2Vreg .F /

Z Z K

U 00 .A/

f .k

1

xuk/ dx dk

and ehıU=U 00 ;HL .l/i is the modular character for the action of L.A/ on U.A/=U 00 .A/. There are many cases in which the assumptions have been verified, and we can at least claim the following. Theorem 14.4.1. For connected reductive groups of F -rank 1 or absolute rank 2, the contribution from a unipotent geometric conjugacy class C to the geometric side of the trace formula is the value at  D 0 of X T JC;N JCT .f; / D 0 .f; /; P 0 P

where P 0 runs through the parabolic subgroups with unipotent radical N 0 , and T JC;f1g .f; / is a truncated zeta integral convergent for Re  in a chamber with apex ıU=U 0 . The case of F -rank one has been known since long (cf. [1, 12]). Here, the groups L are anisotropic modulo centre, and the only singular orbit in V is f0g. Groups of type A2 and B2 D C2 have been treated in [15], groups of type G2 in [9]. In the last two cases, for the subregular unipotent orbits, the prehomogeneous vector spaces are

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341

those of binary quadratic resp. cubic forms, cf. Examples 3 and 4. The first case not covered by the above result is the subregular unipotent orbit in the group GL4 , where we encounter the prehomogeneous vector space from Example 5. Note that for odd unipotent conjugacy classes, i. e. those with U ¤ U 0 , we have convergence at  D 0. Otherwise, this point is at the verge of the chamber of convergence, and JCT .f; 0/ is related to the constant term in the Laurent expansion of the zeta integral. The terms with N 0 ¤ f1g correspond to the principal part in that expansion. The zoo of phenomena presented in Theorems 14.2.5 and 14.2.6 is reproduced by the distribution JCT .f / with its uniform definition (14.3.5). There are conjugacy classes in general groups that can be handled with the present methods, e.g. the principal unipotent orbit. In this case, V Š g2 is a sum of root spaces, but the truncation does not respect the product structure. This leads to correction terms that can be handled using a generalisation of Arthur’s notion of .G; M /families, and an explicit formula for the weight factor in JM . ; f / was obtained (see section 5.3 in [14]). Finally, we review a different approach to the computation of JCT .f / using piecewise exponential factors. The exponents form a family  of complex linear functionals P on aP for each parabolic P compatibly with the embeddings aP 0 ! aP for P  P 0 and such that  P 1 ı Ad. / D P for 2 G.F /. Again,  is determined by P for a single minimal P . The pieces come from Arthur’s partition of G.F /nG.A/, which is constructed as follows. For each parabolic subgroup P D MN over F , we have an analogue F M .m; T M / of the function F .x; T / from (14.3.4), where TPM0 \M D TP 0 for every parabolic subgroup P 0  P of G. If we set F P .mnk/ D F M .m; T M / for m 2 M.A/, n 2 N.A/ and k 2 K, then X F P .x; T /PT .x/ D 1: P

If we insert the additional G.F /-invariant continuous factor X ehP ;HP .x/ TP i F P .x; T /PT .x/ P

P T;] under the intergral (14.3.5), we obtain a distribution J T;].f; / D C JC .f; /; C which certainly converges if Re P is nonpositive on aP for each P and which coincides with J T .f / for  D 0. At first glance, these distributions seem more complicated than JCT .f; /. In [6], however, the analogous distributions on the Lie algebra of the group GLn , with the choice P D sP for s 2 C, were expressed in terms of mean values of truncated Eisenstein series and zeta integrals for prehomogeneous vector spaces of the form End W1 


 End Wm under two assumptions on the test function f —it must be almost invariant and have small support in a suitable sense. In the proof, truncation for GLn is expressed in terms of stability. It relies on an observation by Schiffmann that, for test functions with small support, a certain condition on the slopes of all subquotients of the canonical filtration is equivalent to the condition on the slope of

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W. Hoffmann

the socle only. There is a special case of orbits in GLn , which are called regular by blocks, where the invariance condition is automatically satisfied. In this case, the zeta integrals are evaluated at a parameter in the range of convergence, see [6]. Another case that has been completely solved by the same author is that of orbits induced from the trivial orbit of Levi subgroups with two-by-two blocks, see [5]. In his case, the result can be expressed as a weighted orbital integral with a global weight factor.

References [1] J. Arthur, The Selberg trace formula for groups of F -rank one, Ann. of Math. 100 (1974), 326–385. [2] J. Arthur, An introduction to the trace formula. In Harmonic Analysis, the Trace Formula, and Shimura Varieties (J. Arthur, D. Ellwood and R. Kottwitz, eds.), Amer. Mathematical Society, Providence, RI, 2005, 1–263. [3] J. Arthur, Classifying automorphic representations. In Current Developments in Mathematics (D. Jerison, M. Kisin, T. Mrowka, R. Stanley, H.-T. Yau and S.-T. Yau, eds.), Int. Press, Somerville, MA, 2013, 1–58. [4] J. Arthur, Problems beyond endoscopy. In Representation Theory, Number Theory, and Invariant Theory (J. Cogdell, J.-L. Kim, and C.-B. Zhu, eds.), Birkhäuser, Cham, 2017, 23–45. [5] P.-H. Chaudouard, Sur la contribution unipotente dans la formule des traces d’Arthur pour les groupes généraux linéaires, Israel J. Math. 218 (2017), 175–271. [6] P.-H. Chaudouard, Sur certaines contributions unipotentes dans la formule des traces d’Arthur, Amer. J. Math. 140 (2018), 699–752. [7] P.-H. Chaudouard, Sur une variante des troncatures d’Arthur. In Geometric Aspects of the Trace Formula (W. Müller, S. Shin, and N. Templier, eds.), Springer, Cham, 2018, 88–120. [8] B. Datskovsky and D. Wright, The adelic zeta function associated to the space of binary cubic forms. II. Local theory, J. Reine Angew. Math. (Crelle) 367 (1986), 27–75. [9] T. Finis, W. Hoffmann and S. Wakatsuki, The subregular unipotent contribution to the geometric side of the Arthur trace formula for the split exceptional group G2 . To appear in Geometric Aspects of the Trace Formula. (W. Müller, Sug Woo Shin, and N. Templier, eds.) Springer, New York, 2018, arXiv:1706.00964. [10] T. Finis and E. Lapid, On the continuity of the geometric side of the trace formula, Acta Math. Vietnam 41 (2016), 425–455. [11] R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, LNM 260, Springer, New York, 1972. [12] W. Hoffmann, The non-semisimple term in the trace formula for rank one lattices, J. Reine Angew. Math. (Crelle) 379 (1987), 1–21. [13] W. Hoffmann, Induced conjugacy classes, prehomogeneous varieties, and canonical parabolic subgroups, preprint; arXiv:1206.3068. [14] W. Hoffmann, The trace formula and prehomogeneous vector spaces, In Families of Automorphic Forms and the Trace Formula. (W. Müller, Sug Woo Shin, and N. Templier, eds.), Springer, New York, 2016, 175–215.

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[15] W. Hoffmann and S. Wakatsuki, On the geometric side of the Arthur trace formula for the symplectic group of rank 2. In Mem. Amer. Math. Soc., Nr. 1244, vol. 255, 2018, American Mathematical Society, Providence, RI. [16] T. Ibukiyama and H. Saito, On zeta functions associated to symmetric matrices, II: Functional equations and special values. Nagoya Math. J. 208 (2012), 265–316. [17] J. Igusa, Some results on p-adic complex powers, Amer. J. Math. 106 (1984), 1013–1032, [18] T. Kimura, Introduction to Prehomogeneous Vector Spaces, American Mathematical Society, Providence, RI, 1998. [19] T. Kogiso, Simple calculation of the residues of the adelic zeta function associated with the space of binary cubic forms, J. Number Th. 51 (1995), 233–248. [20] G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), 41–52. [21] C. Moeglin and J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series, Cambridge University Press, Cambridge, 1995. [22] H. Saito, Convergence of the zeta functions of prehomogeneous vector spaces, Nagoya Math. J. 170 (2003), 1–31. [23] M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. 100 (1974), 131–170. [24] T. Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132–188. [25] T. Shintani, On the zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 22 (1975), 25–65. [26] D. Wright, The adelic zeta function associated to the space of binary cubic forms. I. Global theory, Math. Ann. 270 (1985), 503–534. [27] A. Yukie, On the Shintani zeta function for the space of binary quadratic forms, Math. Ann. 292 (1992), 355–374.

Chapter 15

Zeta functions of groups and rings—functional equations and analytic uniformity C. Voll Zeta functions are widely used tools in the study of asymptotic properties of infinite groups and rings, in particular their subobject and representation growth. We survey recent results on arithmetic and asymptotic features of such functions, focussing on various classes of subobject zeta functions, in particular submodule zeta functions associated with nilpotent algebras of endomorphisms, and representation zeta functions associated to arithmetic groups, specifically finitely generated nilpotent groups.1

15.1 Zeta functions associated to groups and rings The zeta functions considered in this text are all related to Dirichlet (series) generating functions of the form 1 X an n s ; (15.1.1) nD1

where s is a complex variable. Typically, the coefficients an are natural numbers which encode arithmetic data associated to a “global” object such as a ring or a group G (both typically infinite), for instance

(A) the numbers of subrings resp. subgroups—possibly satisfying some additional algebraic properties—of G of index n in G or (B) the finite-dimensional complex irreducible representations of G of dimension n, up to a suitable equivalence relation if necessary. A primary motivation to study zeta functions associated to such sequences .an / is to shed light on the subgroup (or subring) growth resp. representation growth of infinite groups and rings. It is well known that formal series such as (15.1.1) converge on some complex right-half P plane if and only if the coefficients an —or, equivalently, their partial sums sn D 6n a —grow at most polynomially in n. We therefore restrict our attention here to groups and rings of polynomial subgroup, subring resp. representation growth. Under this polynomiality assumption, the abscissa of convergence ˛ 2 R [ f1g of the formal series (15.1.1), viz. the infimum of all  2 R such 1 Projects

C10, C12

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C. Voll

that it converges absolutely for all s 2 C with Re.s/ >  , gives the precise degree of polynomial growth of sn : log sn ˛ D lim sup : n!1 log n Example 15.1.1. A prime example of a zeta function of the type discussed is the Dedekind zeta function K .s/ of a number field K, enumerating the ideals of finite additive index (i.e. norm) in O , the ring of integers of K: K .s/ D

X

N.a/

aGO

s

D

1 X

anG .O / n s ;

(15.1.2)

nD1

where N.a/ D jO W aj is the norm of the ideal a and anG .O / D #fa G O j N.a/ D ng. The well-known fact that the abscissa of convergence of K .s/ is equal to 1 is equivalent to the fact that the number snG .O / of ideals of O of norm at most n grows linearly with n. In fact, K .s/ allows for meromorphic continuation to the whole complex plane, with a simple pole at s D 1. It follows that snG .O /  cK n

as n ! 1

(15.1.3)

for some constant cK . The famous analytic class number formula explains how cK depends on arithmetic key invariants of the number field K. Note that, in contrast to this subtle invariant, the abscissa of convergence ˛ of K .s/ and the order of the pole arising at s D ˛ are independent of K. Ideals in the Dedekind ring O factorise—essentially uniquely—as products of prime ideals. This basic arithmetic fact is reflected analytically in the Euler product decomposition Y K .s/ D K;p .s/; (15.1.4) p2Spec.O /n.0/

where Spec.O / is the set of prime ideals of O and, for a prime ideal p of O , the Euler factor at p is defined as K;p .s/ D

1 X i D0

N.pi /

s

D

1 1 q

s

:

(15.1.5)

Here, we write q for the cardinality of the residue field O =p. In the special case K D Q, we write .s/ D K .s/ for Riemann’s zeta function. In general, our knowledge of how to characterise polynomial growth of grouptheoretically arising counting sequences .an / in terms of group- or ring-theoretic features is patchy. While finitely generated, residually finite groups with polynomial subgroup growth are precisely the virtually soluble ones of finite rank [23], the problem of classifying groups of polynomial representation growth is wide open. In contrast, rings which are finitely generated as Z-modules trivially have polynomial subring growth.

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347

The Euler product (15.1.4) illustrates a general feature: multiplicativity properties of the arithmetic functions n 7! an typically reflect arithmetic properties of the underlying counting problem and are themselves reflected in factorisations of “global” zeta functions of the form (15.1.1) as Euler products of “local” zeta functions, usually indexed by the places of some number field. In these notes, we mainly survey recent results concerning two classes of Dirichlet generating function of this form where both polynomial growth and (a strong form of) multiplicativity hold. The zeta functions therefore define analytic functions on complex right-half planes and satisfy formal Euler product decompositions. More precisely, we will discuss results concerning 1. submodule zeta functions associated to nilpotent algebras of endomorphisms, specifically local functional equations of their Euler factors, in Section 15.2, and 2. representation zeta functions associated to finitely generated nilpotent groups, specifically the uniformity of certain of their key analytic features (abscissa of convergence, meromorphic continuation, order of the leading pole) under base extension, in Section 15.3.

15.2 Submodule zeta functions of nilpotent algebras of endomorphisms Submodule zeta functions are Dirichlet generating functions of the form (15.1.1) enumerating the submodules of a module L which are invariant under an algebra E of endomorphisms of L. Let, more precisely, R be the ring of integers O of a number field K or the completion Op of such a ring at a nonzero prime ideal p of O . Let further L be a free R-module of finite rank n, say, and E be a (not necessarily unital) subalgebra of the associative algebra EndR .L/. For m 2 N, we set am . E Õ L / D # fH 6 L j H is an .E C RidL /-submodule of L with jL W H j D mg : The submodule zeta function associated to .L; E / is E ÕL .s / D

1 X

am .E Õ L/m s :

mD1

This definition may be seen as an analogue of Solomon’s zeta function; cf. [32]. Submodule zeta functions generalise ideal zeta functions, enumerating (one- or twosided) ideals of finite index in rings of finite additive rank over R. In particular, Dedekind’s zeta function K .s/ (cf. Example 15.1.1) is an example of a submodule zeta function.

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C. Voll

If, as we will assume from now, R D O , then Y E ÕL .s/ D Ep ÕLp .s/;

(15.2.1)

p2Spec.O /n.0/

where, of course, Ep WD E ˝O Op and Lp WD L ˝O Op . It follows in a straightforward manner from deep general results (cf., e.g., [12]) that each factor of the Euler product (15.2.1) is a rational function in q s (where q D q.p/ D jO W pj) and that the product has rational abscissa of convergence and some meromorphic continuation to the left of its abscissa of convergence. A Tauberian P theorem allows for explicit asymptotic formulae for the growth of the partial sums 6m a .E Õ L/ in terms of the abscissa and the order of the leading pole. While Example 15.1.1 illustrates these facts in the well-known case of Dedekind’s zeta function, the simple form of the Euler factor K;p .s/ in (15.1.5) is misleadingly atypical. Indeed, computing local factors of general submodule zeta functions is a difficult task. In the case that E is generated by a single element, the relevant submodule zeta function may be described explicitly in terms of finitely many translates of Dedekind zeta functions; cf. [26]. In general, however, this is not possible and the local submodule zeta functions are of considerable complexity. The computer algebra package Zeta (cf. [28]) produces many further examples of local submodule zeta functions; cf. also [25]. We focus in the sequel on the case that the algebra E of endomorphisms of L is nilpotent and refer to the associated submodule zeta functions also as “nilpotent”. This class of submodule zeta functions properly includes the ideal zeta functions of nilpotent Lie rings. By the Mal’cev correspondence, these are closely related to the normal zeta functions of finitely generated torsion-free nilpotent groups; cf. [16]. Numerous explicit examples of ideal zeta functions of nilpotent Lie rings are recorded in [14]. An intriguing symmetry feature satisfied by many nilpotent submodule zeta functions is a local functional equation of the form ˇ ˇ Ep ÕLp .s/ˇ D . 1/a q b cs Ep ÕLp .s/; (15.2.2) 1 q!q

satisfied by all but finitely many factors of the Euler product (15.2.1) for integers a; b; c. Here, “inverting q” is in general to be interpreted carefully in terms of the inversion of certain Frobenius eigenvalues in suitable formulae for the local factors in terms of the numbers of O =p-rational points of certain algebraic varieties associated to .L; E /. In many special cases, however, the local factors are rational functions in both q s and q, and the inversion of q may be interpreted naively in terms of these rational functions. Local functional equations of the form (15.2.2) are ubiquitous in the realm of nilpotent submodule zeta functions, but not universal. In Theorem 15.2.4, we shall describe a general, sufficient criterion for functional equations of the form (15.2.2), together with an interpretation of the data .a; b; c/.

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15.2.1 Functional equations and their breakdowns – two examples In order to show that some condition is indeed necessary for a functional equation as in (15.2.2) and to illustrate and motivate the hypotheses of Theorem 15.2.4, we first consider two examples of ideal zeta functions of nilpotent Lie rings. Example 15.2.1. Consider the Lie ring M4 of (maximal) nilpotency class 4 with presentation M4 D hz; x1 ; x2 ; x3 ; x4 j Œz; x1  D x2 ; Œz; x2  D x3 ; Œz; x3  D x4 iZ :

(15.2.3)

Here and in the sequel, we follow the convention that Lie brackets between generators which do not follow from the specified ones by antisymmetry or the Jacobi identity G are assumed to be trivial. An explicit, uniform formula for the Euler factors M .s/ 4 ;p G of the ideal zeta function M4 .s/, enumerating ideals of M4 —viz. ad.M4 /-invariant sublattices of Z 5 , the additive group underlying M4 —is given in [14, Thm. 2.37]; the formula—a rational function in p and p s —is involved, its proof is nontrivial. Inspection reveals that it satisfies the functional equation ˇ p10 14s  G .s/; (15.2.4)  G .s/ˇ 1 D M4 ;p

M4 ;p

p!p


matching the template (15.2.2) with .a; b; c/ D .5; 52 ; 5 C 4 C 3 C 2/; our chosen way of writing this data will fall into place with Theorem 15.2.4. Example 15.2.2. Consider now the Lie ring Fil4 with presentation Fil4 D hz; x1 ; x2 ; x3 ; x4 jŒz; x1  D x2 ; Œz; x2  D x3 ; Œz; x3  D x4 ; Œx1 ; x2  D x4 iZ ;

(15.2.5)

differing from (15.2.3) only in the underlined additional relation. An explicit, uniform G G formula for the Euler factors Fil .s/ of the ideal zeta function Fil .s/, enumerating 4 ;p 4 ad.Fil4 /-invariant sublattices of Z 5 , is given in [14, Thm. 2.38]; it is of similar comG plexity as the one yielding the Euler factors of M .s/. Remarkably, however, it does 4 not satisfy a functional equation of the form (15.2.2). The given presentations of M4 and Fil4 only differ in a single relation. As we shall explain, it is this additional relation that destroys a certain “homogeneity condition” (cf. Condition (15.2.3)) satisfied by M4 but not by Fil4 . Indeed, note that the fivedimensional Z-module L underlying both M4 and Fil4 may be decomposed as

L D hz; x1 i ˚ hx2 i ˚ hx3 i ˚ hx4 i ; „ ƒ‚ … „ƒ‚… „ƒ‚… „ƒ‚… DWL1

DWL2

DWL3

(15.2.6)

DWL4

with summands complementing the upper central series of M4 resp. Fil4 . The associative algebras generated by ad.M4 / and ad.Fil4 / are both generated by the two elements ad.z/ and ad.x1 /. Note, however, that only M4 satisfies the condition 8c 2 fad.z/; ad.x1 /g; 8j 2 f1; 2; 3g W

Indeed, in Fil4 we have .x2 / ad.x1 / 2 L4 .

Lj c  Lj C1 :

(15.2.7)

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15.2.2 Homogeneity implies local functional equations To explain the impact of conditions such as (15.2.7), we now return to the general setup of a free O -module L of finite rank, together with an (associative) nilpotent algebra of endomorphisms E  EndO .L/. For i 2 N0 , we define submodules Zi of L by setting Z0 D f0g and Zi C1 =Zi D CentE .L=Zi / WD fx C Zi 2 L=Zi j E .x/  Zi g for i > 0. Owing to the nilpotency of E , there exists i 2 N such that Zi D L; let c D c.L; E / be the smallest such integer. An instructive special case is that of a nilpotent O -Lie lattice together with the associative algebra E generated by ad.L/. In this case, c is just the nilpotency class of L, and .Zi /ciD0 is the upper central series of L. Only for convenience we make the assumption that the submodules Zi have successive complements in L, viz. that there exist free submodules Li which are free of finite rank and isolated in L for i 2 f1; : : : ; cg, such that, for all such i , Zi D

M

Lj :

j >c i

(This assumption is automatically satisfied if O is a principal ideal domain. In general, it is satisfied almost everywhere locally, which is all that matters in our appliL cations.) In particular, L D ciD1 Li . We also set L0 D LcC1 D f0g. The direct sum decompositions (15.2.6) are examples of such a decomposition. In contrast, the following condition does define an interesting property. Condition 15.2.3 (“Homogeneity”). The nilpotent endomorphism algebra E is generated by elements c1 ; : : : ; cd such that, for all k 2 f1; : : : ; d g and j 2 f1; : : : ; cg,

Lj ck  Lj C1 :

(15.2.8)

In (15.2.7), we had observed that M4 satisfies Condition 15.2.3 with respect to (15.2.6). In contrast, Fil4 does not; one checks easily that no other choices of decomposition or generators of ad.Fil4 / will mitigate this failure. It turns out that Condition 15.2.3 is sufficient for functional equations for almost all the Euler factors of LÕE .s/. To formulate the precise result, we set Ni D rkO .L=Zi / and n D N0 D rkO .L/. Theorem 15.2.4 ([38, Thm. 1.2]). Assume that .L; E / satisfies Condition 15.2.3. Then, for almost all p 2 Spec.O /, the following functional equation holds: P  ˇ c 1 n ˇ n . 2/ s i D0 Ni Ep ÕLp .s/ˇ Ep ÕLp .s/: D . 1/ q 1 q!q

Remark 15.2.5. Condition 15.2.3 obviously holds if E is generated by a single element. (Note that to encounter this situation one needs to leave the remit of ideal zeta function of a nilpotent Lie ring L, as ad.L/ has always at least two generators.) The

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351

paper [26] contains explicit formulae for local submodule zeta functions enumerating submodules invariant under a single endomorphism; that this endomorphism is nilpotent is an important special case, to which the general case is reduced. Likewise, Condition 15.2.3 always holds whenever E 2 D 0, viz. c 6 2. In this case, Theorem 15.2.4 is a mild generalisation of [36, Thm. C]. While the Lie ring Fil4 in Example 15.2.2 has class c D 4, there are examples of nilpotent Lie rings of class c D 3 whose Euler factors do not satisfy a functional equation of the form (15.2.2); see, for instance, [14, Thm. 1.1]. Remark 15.2.6. [38, Thm. 1.2] is more general than Theorem 15.2.4 in as much as it applies uniformly to the local submodule zeta functions associated to pairs of the form .L.O/; E .O//, where O is any finite extension of the local ring Op and L.O/ D L ˝O O and E .O/ D E ˝O O. These zeta functions enumerate E .O/-invariant submodules of the O-modules L.O/. To control the variation of the distribution of the E .Op /-invariant submodules of L.O/, viewed as Op -modules after restriction of scalars, is a much more delicate task. Local zeta functions obtained in this way from the Heisenberg (Z-) Lie lattice of strictly upper-triangular 33-matrices are studied in [29, 30]; see also [38, Sec. 5.2]. Remark 15.2.7. By appealing to a version of the model-theoretic transfer principle for p-adic integrals (cf. [7]) one may show that the local functional equations (15.2.4) hold verbatim also in characteristic p  0, viz. for local submodule zeta function associated to pairs of the form .L.Fq JT K/; E .Fq JT K//, provided q is a power of a sufficiently large prime p; cf. [38, Cor. 1.3]. The fact that finitely many residue characteristics need to be excluded is, however, essential, regardless of the characteristics of the local rings, and not just owed to the limitations of the chosen methods of proof. In the remainder of this section, we give an informal overview of the proof of Theorem 15.2.4. It combines geometric ideas with methods from p-adic integration. Let p 2 Spec.O / with uniformiser  and let Kp be the p-adic field of fractions of Op . The starting point of the proof is the trivial observation that the Ep -invariance of a lattice ƒ  Lp is really a property of the homothety class
Œƒ WD fƒ j  2 Kp g

of ƒ. The set SubModEp of homothety classes of Ep -invariant sublattices of Lp may thus be viewed as a subcomplex of the affine Bruhat–Tits building n WD .SLn .Kp //. Every homothety class Œƒ contains a unique maximal integral lattice ƒmax (i.e., ƒmax  Lp but p 1 ƒmax 6 Lp ). It clearly suffices to enumerate such maximal sublattices which are Ep -invariant: Ep ÕLp .s/ D

1

1 q

ns

X

Œƒ2SubModE

jLp W ƒmax j s :

p

Rather than study the subcomplex SubModEp directly, it proved advantageous to de-

352

C. Voll

fine a weight function m W Vn ! N0 on the whole of Vn , the vertex set of n , in such a way that 1. mjSubModE D 0; p

2. the function AG .s/ WD equation of the form

P

Œƒ2Vn

ˇ AG .s/ˇq!q

jLp W ƒmax j s q 1

D . 1/n

1

sm.Œƒ/

satisfies a functional

n

q .2/ AG .s/I

3. the function „.s/ WD Ep ÕLp .s/=AG .s/ is a “simple” rational function in q whose coefficients only depend on the rank data .Ni /ciD1 , which satisfies „.s/jq!q

1

D

q

s

Pc

1 i D0

Ni

s

,

„.s/:

The key idea in the construction of the weight function m is to define an equivalence relation  on Vn whose equivalence classes C form posets naturally isomorphic to .Z; 6/ with the property that the intersection C>0 WD C \SubModEp coincides with the “non-negative part” of C . Thus Ep ÕLp .s/ D

1

1 q

ns

X

X

C 2Vn = Œƒ2C>0

„

jLp W ƒmax j ƒ‚

DW„C

>0

.s/

s

:

…

The equivalence relation  is induced on Vn by the “staggered homothety map” ı W Lp D L1;p ˚


˚ Lc;p ! L1;p ˚


˚ Lc;p .x1 ; : : : ; xc / 7!  c

1

x1 ;  c

2


x2 ; : : : ; xc :

(15.2.9)

More precisely, we say that homothety classes Œƒ1  and Œƒ2  are -equivalent if there exists M 2 N0 such that ƒ1 ıM 2 Œƒ2 . It is easy to check that—provided c > 1, as we may assume without loss of generality—every -class C contains a (unique) homothety class Œƒ0  such that ˚ C>0 D Œƒ0 ıM  j M 2 N0 :

In other words, exactly half of C consists of Ep -invariant sublattices, viz. the “halfray” C>0 generated by Œƒ0 . For any weight function m satisfying (1), the generating function 1

1 q

AG .s/ D ns

1

1 q

ns

X

X

C 2Vn = Œƒ2C

„

jLp W ƒmax j s q ƒ‚

DW„C .s/

sm.Œƒ/

…

353

Zeta functions of groups and rings

overcounts, namely exactly along the “negative half-rays” C0 consisting of non-Ep-invariant lattices classes. The critical (and most technical) part of the proof of Theorem 15.2.4 is now to design m such that both (1) and (2) hold and, crucially, that correcting this overcounting is easy. More precisely, m is designed so that both sums „C>0 .s/ and „C a.G / ı.G /g. On the line fs 2 C j Re.s/ D a.G/g the continued zeta function is holomorphic except for a pole at s D a.G/, of order ˇ.G /.

There exists a constant c.G .OL // 2 R>0 such that N X

nD1


e r n G .OL /  c.G .OL //N a.G / .log N /ˇ.G /

1

as N ! 1:

(15.3.5)

Three aspects of Theorem 15.3.1 strike me as remarkable: firstly the rationality of the abscissa of convergence a.G /, secondly the feasibility of meromorphic continuation (uniformly for all L), and thirdly the fact that the invariants a.G / and ˇ.G / featuring in the asymptotic formula (15.3.5) are independent of OL , whereas the number c.G .OL // may vary rather subtly with the number field L. Remark 15.3.2. Theorem 15.3.1 is formulated in such a way as to emphasise the uniformity of certain aspects of representation growth of finitely generated nilpotent groups varying in natural families, viz. groups obtained from base extension. In fact, every abstract finitely generated nilpotent group fits into this mould, albeit only virtually so: every such group G has a finite index subgroup H D G .Z/ for a suitable unipotent group scheme G ; cf. [11, Sec. 5]. It turns out that the rational abscissa of convergence, the feasibility of partial meromorphic continuation, as well as the order of the leading pole are all commensurability invariants, whence Theorem 15.3.1 allows for a corollary on abstract finitely generated nilpotent groups; cf. [11, Cor. B]. Theorem 15.3.1 is illustrated by the following example. Example 15.3.3. Consider the group scheme H associated to the Heisenberg (Z-) Lie lattice of strictly upper-triangular 3  3-matrices. If OL is the ring of integers of a number field L, then H .OL / is the group of upper-unitriangular 3  3-matrices, a finitely generated torsion-free nilpotent group of Hirsch length 3ŒL W Q. By [33, Thm. B], the representation zeta function of H .OL / is given by the formula e irr H .OL / .s/ D

L .s 1/ D L .s/

Y

p2Spec.OL /n.0/

1 q 1 q1

s s

I

(15.3.6)

cf. Eqs. (15.1.4) and (15.1.5). By the other well-known properties of the Dedekind zeta function K .s/ recalled in Example 15.1.1, one reads off from (15.3.6) that

359

Zeta functions of groups and rings

a.H / D 2, ˇ.H / D 1, and c.H .OL // D N X

nD1

e r n .H .OL // 

1 , 2L .2/

whence

1 N2 2L .2/

as N ! 1:

(15.3.7)

Note that the special value L .2/ depends on L in a subtle way; cf. [39, Thm. 1]. In the special case that L D Q, the formula e irr H .Z/ .s/ D

1 X .s 1/ D '.n/ n s ; .s/ nD1

(where ' is Euler’s totient function) is implicit in the work [24] by Magid and Nunley, who worked out explicit representatives of the twist-isoclasses of irreducible representations of the integral Heisenberg group. The special case N X

nD1

'.n/ 

3 2 N 2

as N ! 1; 2

obtained from (15.3.7) using the well-known identity .2/ D 6 , due to Euler, is an elementary exercise in analytic number theory. For quadratic number fields, the formula (15.3.6) is due to Ezzat; cf. [15]. Numerous further examples of representation zeta functions associated to unipotent group schemes may be found in [33, Thm. B], [34], [25, Sec. 8], and [31, Ch. 6]. It is not hard to see that every rational number ˛ occurs as the abscissa of convergence of the representation zeta function of a finitely generated nilpotent group; cf. [31, Thm. 4.22]. The structure of the set of abscissae of convergence of (normal) subgroup zeta functions of finitely generated nilpotent groups is much more subtle; cf., for instance, [13, Prop. 1.1 and Qst. 1.3]. 15.3.4 Related work and directions for future research 15.3.4.1 Arithmetic subgroups of semisimple groups Let K be a number field with ring of integers O , let S be a finite set of places of K, and OS the ring of S integers of K. Let further G be an affine group scheme over OS , whose generic fibre is a connected, simply-connected semisimple algebraic group over K. One may study the representation growth of arithmetic groups such as G WD G .OS / by means of their associated zeta function G .s/; cf. (15.3.1). This is, for instance, the principal setting for [1, 2, 3]. We refer to [37, 21] for recent surveys on such representation zeta functions and comment here merely on a number of analogies between the semisimple and unipotent setups. While [3, Thm. 1.1] yields a precise general analogue of the first statement of the “unipotent” Theorem 15.3.1 in the semisimple setting, the analogue of its second statement is only known only for groups of types A1 ([19,

360

C. Voll

Thm. 7.5]) or A2 ([2, Thm. A]), where explicit formulae for almost all of the representation zeta functions of the non-archimedean completions G .Op / of G .OS / are available. In particular, it is not known whether the feasibility of (uniform) meromorphic continuation beyond the (rational) abscissa of convergence is a fluke for these “small” root systems or a general feature. In [40], M. Zordan computes the representation zeta functions enumerating continuous representations of the principal congruence subgroups SL14 .Op /; an explicit formula for the representation zeta functions of the p-adic analytic groups SL4 .Op / seems currently out of reach. 15.3.4.2 Heisenberg groups over truncated polynomial rings Theorem 15.3.1 describes the uniform variation of analytic key invariants of representation zeta functions associated to groups of rational points of certain unipotent group schemes G under base change with number rings OL . While these kinds of base extensions may be natural from a number-theoretic perspective, extensions by other, more exotic rings may look just as natural from a group theorist’s perspective. In [10], Duong studies Heisenberg groups over rings of the form O Œx=.x n/. The following result generalises (15.3.6): Theorem 15.3.4 ([10]). Let OL be the ring of integers of a number field L. For n 6 3, one has n Y L .i s 2i C 1/ e irr : .s/ D H n .OL Œx=.x // L .i s 2i C 2/ i D1

Note that the three zeta functions for n D 1; 2; 3 share the abscissa of convergence (viz. ˛ D 2) and may be continued meromorphically to the whole of C, but that the order of the pole at s D 2 is n. The simple “multiplicative” form of (15.3.4) belies the complicated “additive” computations carried out in their proofs, viz. the hands-on computations of specific p-adic integrals. Duong conjectures that (15.3.4) holds for all n. If it does, it is tempting to interpret the limit of the right hand of the identity in Theorem 15.3.4 as n ! 1 as a representation zeta function associated to the group H .OL Jx K/. There is, as yet, no general theory describing the behaviour of representation zeta functions of groups of the form G .O Œx=.x n//—let alone the completions G .OJx K/—for general unipotent group schemes G and varying O and n. A few examples for n D 2 (i.e., base change to “dual numbers”) are discussed in [27, Sec. 6.4]; cf. also [27, Qst. 7.3]. 15.3.4.3 Representation zeta functions and prehomogeneous vector spaces The paper [33] introduces three infinite families of unipotent group schemes of nilpotency class 2, all generalising the Heisenberg groups H .O /; cf. Example 15.3.3. All three families are “taylor-made” to produce finitely generated nilpotent groups whose structures encode, in natural ways, the relative invariants of certain reduced irreducible prehomogeneous vector spaces (PVSs), viz. the determinants resp. Pfaffians of generic n  n-matrices, symmetric matrices, and antisymmetric matrices. The “multiplicative” formulae for the associated representation zeta functions given

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in [33, Thm. B] have “additive” counterparts in [33, Thm. C]. Algebro-combinatorial identities of “multinomial type” such as [33, Prop. 1.5] link these two kinds of formulae, often leading to new insights on the (joint) distributions of Weyl group statistics, both old and new; cf. [33, Sec. 4], [35], and [6]. In [34], we compute the representation zeta functions of finitely generated nilpotent groups obtained as groups of O -rational points of unipotent group schemes modelled on the PVSs .SPm  GL2n ; ƒ1 ˝ ƒ1 ; V .2m/ ˝ V .2n// for n D 1 and m 2 f1; 2g, defined in [20, Ex. 2.13] (in notation differing from [20] but consistent with [18, p. 165]). For m D 2, the resulting formulae cannot be described as a finite product of translates of Dedekind zeta functions and their inverses, but still invite a Coxeter group-theoretic interpretation.

References [1] N. Avni, B. Klopsch, U. Onn and C. Voll, Representation zeta functions of compact p-adic analytic groups and arithmetic groups, Duke Math. J. 162 (2013), 111–197. [2] N. Avni, B. Klopsch, U. Onn and C. Voll, Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2 Proc. Lond. Math. Soc. 112(3) (2016), 267–350. [3] N. Avni, B. Klopsch, U. Onn and C. Voll, Arithmetic groups, base change, and representation growth, Geom. Funct. Anal. 26 (2016), 67–135. [4] M.N. Berman, Uniformity and functional equations for local zeta functions of K-split algebraic groups, Amer. J. Math. 133 (2011), 1–27. [5] M.N. Berman and B. Klopsch, A nilpotent group without local functional equations for proisomorphic subgroups, J. Group Th. 18 (2015), 489–510. [6] F. Brenti and A. Carnevale, Proof of a conjecture of Klopsch-Voll on Weyl groups of type A, Trans. Amer. Math. Soc. 369 (2017), 7531–7547. [7] R. Cluckers and F. Loeser, Constructible exponential functions, motivic Fourier transform and transfer principle, Ann. of Math., 171(2) (2010), 1011–1065. [8] J. Denef, Report on Igusa’s local zeta function, Sém. Bourbaki 43 (1990–91), 359–386. [9] J. Denef and D. Meuser, A functional equation of Igusa’s local zeta function, Amer. J. Math. 113 (1991), 1135–1152. [10] H.D. Duong, Representation growth of the Heisenberg group over OŒx=.x 3 /, J. Algebra Appl. 16 (2017), 1750077 (13 pages). [11] H.D. Duong and C. Voll, Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups, Trans. Amer. Math. Soc. 369 (2017), 6327–6349. [12] M.P.F. du Sautoy and F.J. Grunewald, Analytic Properties of zeta functions and subgroup growth, Ann. of Math., 152(2) (2000), 793–833.

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[13] M.P.F. du Sautoy and F.J. Grunewald, Zeta functions of groups and rings. In Proceedings of the International Congress of Mathematicians, Madrid, August 22–30, 2006, Vol. II, European Mathematical Society, Zürich, 2006, 131–149. [14] M.P.F. du Sautoy and L. Woodward, Zeta Functions of Groups and Rings, Springer, Berlin, 2008. [15] S. Ezzat, Counting irreducible representations of the Heisenberg group over the integers of a quadratic number field, J. Algebra 397 (2014), 609–624. [16] F.J. Grunewald, D. Segal and G.C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math. 93 (1988), 185–223. [17] E. Hrushovski, B. Martin, S. Rideau and R. Cluckers, Definable equivalence relations and zeta functions of groups, J. Eur. Math. Soc. (JEMS) 20 (2018), 2467–2537. [18] J.-I. Igusa, An Introduction to the Theory of Local Zeta Functions, AMS/IP Studies in Advanced Mathematics vol. 14, American Mathematical Society, Providence, RI, 2000. [19] A. Jaikin-Zapirain, Zeta function of representations of compact p-adic analytic groups, J. Amer. Math. Soc. 19 (2006), 91–118. [20] T. Kimura, Introduction to Prehomogeneous Vector Spaces, Translations of Mathematical Monographs 215, American Mathematical Society, Providence, RI (2003). [21] B. Klopsch, Representation growth and representation zeta functions of groups, Note Mat. 33 (2013), 107–120. [22] S. Lee and C. Voll, Enumerating graded ideals in graded rings associated to free nilpotent Lie rings, Math. Z. (2018), 1–28. [23] A. Lubotzky, A. Mann and D. Segal, Finitely generated groups of polynomial subgroup growth, Israel J. Math. 82 (1993), 363–371. [24] A.R. Magid and A. Lubotzky, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (1989). [25] T. Rossmann, Computing local zeta functions of groups, algebras, and modules, Trans. Amer. Math. Soc. 370 (2018), 4841–4879. [26] T. Rossmann, Enumerating submodules invariant under an endomorphism, Math. Ann. 368 (2016), 391–417. [27] T. Rossmann, Topological representation zeta functions of unipotent groups, J. Algebra 448 (2016), 210–237. [28] T. Rossmann, Zeta version 0.3.2, https://www.math.uni-bielefeld.de/~rossmann/Zeta/. [29] M.M. Schein and C. Voll, Normal zeta functions of the Heisenberg groups over number rings I — the unramified case, J. Lond. Math. Soc. 91 (2014), 19–46. [30] M.M. Schein and C. Voll, Normal zeta functions of the Heisenberg groups over number rings II — the non-split case, Israel J. Math. 211 (2016), 171–195. [31] R. Snocken, Zeta Functions of Groups and Rings, PhD thesis, University of Southampton (2014), https://eprints.soton.ac.uk/372833/. [32] L. Solomon, Zeta functions and integral representation theory, Adv. Math. 26 (1977), 306–326. [33] A. Stasinski and C. Voll, Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B, Amer. J. Math. 136 (2014), 501–550.

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[34] A. Stasinski and C. Voll, Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces, Forum Math. 29 (2017), 717–734. [35] A. Stasinski and C. Voll, A new statistic on the hyperoctahedral groups, Electron. J. Combin. 20 (2013), paper 50 (23 pages). [36] C. Voll, Functional equations for zeta functions of groups and rings, Ann. of Math. 172(2) (2010), 1181–1218. [37] C. Voll, Zeta functions of groups and rings—recent developments. In Groups St Andrews 2013, London Math. Soc. Lecture Note Ser. 422, Cambridge University Press, Cambridge, 2015, 469–492. [38] C. Voll, Local Functional Equations for Submodule Zeta Functions Associated to Nilpotent Algebras of Endomorphisms, Int. Math. Res. Not. IMRN (2017); https://doi.org/10.1093/imrn/rnx186. [39] D. Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions, Invent. Math. 83 (1986), 285–301. [40] M. Zordan, Representation Zeta Functions of Special Linear Groups, PhD thesis, Bielefeld University, 2016, https://pub.uni-bielefeld.de/publication/2902722.

Chapter 16

Conjectures of Brumer, Gross and Stark A. Nickel This chapter gives an introduction to generalisations of conjectures of Brumer and Stark on the annihilator of the class group of a number field. We review the relation to the equivariant Tamagawa number conjecture, the main conjecture of Iwasawa theory for totally real fields, and a conjecture of Gross on the behaviour of p-adic Artin L-functions at zero.1

Introduction Let L=K be a finite Galois extension of number fields with Galois group G. To each finite set S of places of K containing all the archimedean places, one can associate a so-called ‘Stickelberger element’ S in the centre of the group algebra CŒG. This element is constructed from values at s D 0 of the S -truncated Artin L-series attached to the complex irreducible characters of G. In particular, S is analytic in nature. By a result of Siegel [34] one knows that S always has rational coefficients. Let L and clL be the roots of unity and the class group of L, respectively. Suppose that S also contains all places of K which ramify in L=K. Then it was independently shown by Cassou-Nogues [8], Deligne–Ribet [14] and Barsky [1] that for abelian G one has AnnZŒG .L /S  ZŒG;

where AnnR .M / denotes the annihilator ideal of M regarded as a module over the ring R. In other words, the coefficients of S are almost integral. Now Brumer’s conjecture simply asserts that AnnZŒG .L /S annihilates clL . In the case K D Q Brumer’s conjecture is just Stickelberger’s theorem from the late 19th century [36]. Roughly speaking, the conjecture predicts that an analytic object gives constraints on the structure of an arithmetic object. It is this kind of conjecture which is often called a ‘Stark-type conjecture’. In fact, Harold Stark suggested the following refinement of Brumer’s conjecture. Let wL be the cardinality of L and fix a fractional ideal a in L. We will denote the action of G on a and on its class in clL by exponents on the right as usual. Then the Brumer–Stark conjecture not only predicts that awL S becomes principal, but also gives precise information about a generator of that ideal. In this survey article we explain recent generalisations of these conjectures to arbitrary, not necessarily abelian Galois extensions; these are due to the author [27] 1 Project

C7

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and, independently and in even greater generality, due to Burns [3]. A further, slightly different approach has been developed by Dejou and Roblot [13]. We discuss the relation to further conjectures in the field such as the equivariant Tamagawa number conjecture of Burns and Flach [6] and the main conjecture of equivariant Iwasawa theory which (under a suitable condition) has been proven by Ritter–Weiss [33] and, independently, by Kakde [25]. A conjecture of Gross [18] on the behaviour of p-adic Artin L-series at s D 0 also plays a pivotal role. Roughly speaking, the latter conjecture asserts that (i) the order of vanishing at s D 0 of the p-adic L-series coincides with the order of vanishing at s D 0 of a corresponding complex L-series and (ii) the special values of the p-adic and the complex L-series at s D 0 coincide up to some explicit p-adic regulator. Considerable progress has been made in the recent years by Spiess [35] and Burns [4] on part (i), and by Dasgupta, Kakde and Ventullo [12] on part (ii). We also explain their results and the relation to the equivariant Tamagawa number conjecture due to Burns [4] as well as the relation to the (non-abelian) Brumer–Stark conjecture due to Johnston and the author [23]. We provide no proofs unless they are short and we feel that it might help for a better understanding. Instead we include some examples to illustrate occurring obstacles and the underlying ideas how to overcome them. Notation and conventions All rings are assumed to have an identity element and all modules are assumed to be left modules unless otherwise stated. Unadorned tensor products will always denote tensor products over Z. We fix the following notation: R .R/ AnnR .M / Mmn .R/ K1 K clK Kc IrrF .G/

the group of units of a ring R the centre of a ring R the annihilator ideal of the R-module M the set of all m  n matrices with entries in a ring R the cyclotomic Z p -extension of the number field K the roots of unity of a field K the class group of a number field K an algebraic closure of a field K the set of F -irreducible characters of the (pro)-finite group G (with open kernel) where F is a field of characteristic 0

16.1 Preliminaries 16.1.1 Ray class groups Let L=K be a finite Galois extension of number fields with Galois group G. For each place v of K we fix a place w of L above v and write Gw and Iw for the decomposition group and inertia subgroup of L=K at w, respectively. When w is a finite place, we choose a lift w 2 Gw of the Frobenius automorphism at w; moreover, we write Pw for the associated prime ideal in L and j
jw for the corresponding absolute value. We denote the cardinality of the residue field of K at v by N.v/.

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For any set S of places of K, we write S.L/ for the set of places of L which lie above those in S . Now let S be a finite set of places of K containing the set S1 D S1 .K/ of archimedean places and let T be a second finite set of places of K such that S \ Q T D ;. We write clTL for the ray class group of L associated to the ray MTL WD w2T .L/ Pw and OL;S for the ring of S.L/-integers in L. Let OL WD OL;S1 be the ring of integers in L. Let Sf be the set of all finite places in S ; then there is a natural map ZŒSf .L/ ! clTL which sends each place w 2 Sf .L/ to the corresponding class ŒPw  2 clTL . We denote the cokernel of this map by clTL;S . When T is empty, we abbreviate cl;L;S to clL;S so that in particular clL;; D clL is the usual class group of L. Moreover, we denote the S.L/-units of L by EL;S and ˚ T define EL;S WD x 2 EL;S W x  1 mod MTL . All these modules are equipped with a natural G-action and we have an exact sequence of ZŒG-modules 

T 0 ! EL;S ! EL;S ! .OL;S =MTL / ! clTL;S ! clL;S ! 0;

where the map  lifts an element x 2 .OL;S =MTL / to x 2 OL;S and sends it to the ideal class Œ.x/ 2 clTL;S of the principal ideal .x/. 16.1.2 Equivariant Artin L-values Let S be a finite set of places of K containing S1 . Let IrrC .G/ denote the set of complex irreducible characters of G. For  2 IrrC .G/, let V be a left CŒG-module with character . We write LS .s; / for the S -truncated Artin L-function attached to  which for Re.s/ > 1 is given by the Euler product Y
1 : LS .s; / D det 1 N.v/ s w j VIw v62S

Each element in CŒG may be viewed as a complex valued function on G. The irreducible characters constitute Q a basis of the centre and we thus have a canonical isomorphism .CŒG/ ' 2IrrC .G/ C. We define the equivariant S -truncated Artin L-function to be the meromorphic .CŒG/-valued function
LS .s/ WD LS .s; / 2IrrC .G/ : If T is a second finite set of places of K such that S \ T D ;, we define Y
ıT .s; / D det 1 N.v/1 s w 1 j VIw v2T

and ıT .s/ WD ıT .s; / We set




2IrrC .G/

:

‚S;T .s/ WD ıT .s/
LS .s/] ;

where ] W CŒG ! CŒG denotes the anti-involution induced by g 7! g 1 for g 2 G. Note that LS .s/] D .LS .s; // L 2IrrC .G/ where L denotes the character contragredient

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to . The functions ‚S;T .s/ are the so-called .S; T /-modified G-equivariant Lfunctions, and we define Stickelberger elements ST .L=K/ D ST WD ‚S;T .0/ 2 .QŒG/: Note that a priori we only have ST 2 .CŒG/, but by a result of Siegel [34] we know that ST has rational coefficients. If T is empty, we abbreviate ST to S . Remark 16.1.1. Let  2 IrrC .G/ and let rS ./ be the order of vanishing of LS .s; / at s D 0. Then by [37, Chapitre I, Proposition 3.4] one has  X dimC .VGw / dimC .VG /: (16.1.1) rS ./ D v2S

Thus if either  is non-trivial and S contains an (infinite) place v such that VGw ¤ 0 or  is trivial and jS j > 1 then the -part of ST vanishes. Now suppose that S contains all ramified primes. Then if ST is non-trivial, precisely one of the following possibilities occurs: (i) K is totally real and L is totally complex, (ii) K is an imaginary quadratic field, L=K is unramified and S D S1 or (iii) L D K D Q and S D S1 .

16.2 The abelian case In this section we assume that the extension L=K is abelian. Let L denote the roots of unity in L. It was independently shown in [1, 8, 14] that one has AnnZŒG .L /S  ZŒG

(16.2.1)

whenever S contains the set Sram of all places of K that ramify in L=K. We now state Brumer’s conjecture as discussed by Tate [37]. Conjecture 16.2.1 (B.L=K; S /). Let L=K be an abelian extension of number fields with Galois group G and let S be a finite set of places of K containing the set S1 and all places of K that ramify in L=K. Then one has AnnZŒG .L /S  AnnZŒG .clL /: We will discuss relevant results on Brumer’s conjecture and its generalisations in §16.4 below. Here we only mention that for absolutely abelian number fields Brumer’s conjecture follows from Stickelberger’s theorem from the late 19th century. Theorem 16.2.2 (Stickelberger [36]). Assume that L is abelian over Q. Then, Brumer’s conjecture B.L=K; S / holds.

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Remark 16.2.3. Consider the three cases of Remark 16.1.1. In case (iii), Brumer’s conjecture (and also the Brumer–Stark conjecture below) is trivial. Case (ii) follows from the fact that each ideal of L becomes principal in the Hilbert class field of L (see [17, Remark 6.3]). Finally, case (i) can be reduced to the case that L is a CM-field (see [17, Proposition 6.4]). Therefore, we shall henceforth assume that L=K is a CM-extension, that is, L is a CM-field, K is totally real and complex conjugation induces a unique automorphism j in G. Let wL be the cardinality of L . Then clearly wL 2 AnnZŒG .L / and so Brumer’s conjecture asserts that awL S is a principal ideal for every fractional ideal a in L. The Brumer–Stark conjecture now gives precise information on a generator of this ideal. To make this precise, we need the following definition. Definition 16.2.4. Let L be a CM-field. Then,  2 L is called an anti-unit if  1Cj D 1. Conjecture 16.2.5 (BS.L=K; S /). Let L=K be an abelian CM-extension with Galois group G and let S be a finite set of places of K containing the set S1 and all places of K that ramify in L=K. Then, for every non-zero fractional ideal a in L, there is an anti-unit  2 L such that 1. awL S D ./.

p 2. The extension L. wL /=K is abelian.

Here, the first part almost follows from B.L=K; S /, and Harold Stark originally suggested that the second condition might hold. The conjecture was first stated in published form by Tate [37]. Remark 16.2.6. An anti-unit  2 L that fulfils (1) is determined up to a root of unity. In order to generalise these conjectures to arbitrary Galois CM-extensions, we need reformulations of both conjectures. For this, we introduce the following terminology. Let S and T be two finite sets of places of K. We then say that Hyp.S; T / is satisfied if the following holds:  S contains all the archimedean places of K and all places which ramify in L=K, i.e. S  Sram [ S1 .

 S \ T D ;.

T  EL;S is torsionfree. T is torsionfree whenever T contains at least two Remark 16.2.7. Note that EL;S places of different residue characteristic or at least one place of sufficiently large norm.

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Lemma 16.2.8. Let S be a finite set of places of K containing Sram [ S1 . Then, the elements 1 N.v/w 1 , where v runs through all the finite places of K such that the sets S and Tv WD fvg satisfy Hyp.S; Tv /, generate AnnZŒG .L /. Moreover, if we restrict to totally decomposed primes v, the greatest common divisor of the integers 1 N.v/ equals wL . Proof. This is an obvious reformulation of [37, Ch. IV, Lemma 1.1].



Corollary 16.2.9. Let L=K be an abelian extension of number fields and let S be a finite set of places of K containing the set S1 and all places of K that ramify in L=K. Then, Brumer’s conjecture B.L=K; S / holds if and only if ST 2 AnnZŒG .clL / for all finite sets T of L such that Hyp.S; T / is satisfied. Proof. We have ST D ıT .0/S and ıT .0/ 2 AnnZŒG .L / whenever Hyp.S; T / is  satisfied. As ıTv .0/ D 1 N.v/w 1 , the result follows from Lemma 16.2.8. Let NL=K W L ! K  be the field-theoretic norm map. For  2 L we define S WD fv finite place of K W v j NL=K ./g:

Proposition 16.2.10. Let S be a finite set of places of K containing Sram [ S1 . Then, the Brumer–Stark conjecture BS.L=K; S / is equivalent to the following claim: For every non-zero fractional ideal a of L there is an anti-unit  2 L such that awL
S D ./

and, for each finite set T of primes of K such that Hyp.S [ S ; T / is satisfied, there is an T 2 EST such that (16.2.2)  ıT .0/ D TwL : Proof. This is [37, Ch. IV, Prop. 1.2].



Remark 16.2.11. In many cases, one can omit the condition that  is an anti-unit: Suppose that the order of a in the class group is odd. Then, we may write a D b2 .u/ for some u 2 L . Now assume that bwL S is principal and generated by some ˇ 2 L such that (16.2.2) holds with  replaced with ˇ. As .1 j /S D 2S , we then have awL S D ./, where  WD uwL S
ˇ 1 j is an appropriate anti-unit.

16.3 The general case 16.3.1 How to generalise to non-abelian extensions? Now let L=K be an arbitrary Galois CM-extension with Galois group G. How can we formulate Brumer’s conjecture and the Brumer–Stark conjecture for this more general situation?

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We still have the Stickelberger elements ST at our disposal. In view of Corollary 16.2.9 one is tempted to conjecture that ST still annihilates the class group whenever Hyp.S; T / is satisfied. Although the Stickelberger elements always belong to the centre of QŒG, it is, however, in general not true that ST has integral coefficients. Thus ST does in general not even act on the class group! 3 Example 16.3.1. This example p ispdue to Nomura [32, §6]. Let ˛ be a root of x 3; 4001; ˛/. Then, L=Q is a Galois CM-extension 11x C 7 and set L D Q. with Galois group G isomorphic to Z=2Z  S3 , where j generates the first factor and S3 denotes the symmetric group on 3 letters. We write

S3 D h;  j  3 D  2 D 1;   

1

D

1

i:

Then Sram D f3; 4001g and for S D Sram [ S1 and T D f7g one has ST D

1 .1 3

j / 3410

1774. C  2 / C 44. C   C  2  /

which visibly does not belong to ZŒG.




The idea is to replace the centre of ZŒG by a larger ring I .G/ such that ST always belongs to I .G/ and such that I .G/ D ZŒG when G is abelian. In order to achieve annihilators, one then has to multiply by a certain ‘denominator ideal’ H.G/. We next introduce these purely algebraic objects. 16.3.2 Denominator ideals and the integrality ring Let R be a Noetherian integrally closed domain with field of fractions E. Let A be a finite-dimensional separable E-algebra and let A be an R-order in A. Our main examples are group rings A D RŒG and A D EŒG, where R and E are either Z and Q or Z p and Q p for a prime p, respectively. The reduced norm map nr D nrA W A ! .A/ is defined component-wise on the Wedderburn decomposition of A and extends to matrix rings over A (see [10, §7D]). We choose a maximal R-order M such that A  M  A. Following [20, §3.6], for every matrix H 2 Mnn .A/ there is a generalised adjoint matrix H  2 Mnn .M/ such that H  H D HH  D nr.H /
1nn (note that the conventions in [20, §3.6] slightly differ from those in [26]). If HQ 2 Mnn .A/ is a second matrix, then .H HQ / D HQ  H  . We define

H.A/ WD fx 2 .A/ j xH  2 Mnn .A/ 8H 2 Mnn .A/ 8n 2 Ng; I .A/ WD hnr.H / j H 2 Mnn .A/; n 2 Ni .A/ : One can show that these are R-lattices satisfying

H.A/
I .A/ D H.A/  .A/  I .A/  .M/:

(16.3.1)

Hence H.A/ is an ideal in the commutative R-order I .A/. We will refer to H.A/ and I .A/ as the denominator ideal and the integrality ring of the R-order A, respectively.

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Remark 16.3.2. The integrality ring is the smallest subring of .A/ that contains .A/ and the image of the reduced norm of all matrices with entries in A. The denominator ideal measures the failure of the generalised adjoint matrices having coefficients in A. If p is a prime and G is a finite group, we set

I .G/ WD I .ZŒG/; H.G/ WD H.ZŒG/;

Ip .G/ WD I .Z p ŒG/; Hp .G/ WD H.Z p ŒG/:

The first claim of the following result is a special case of [20, Prop. 4.4]. The second claim then follows easily from (16.3.1). Proposition 16.3.3. Let p be prime and G be a finite group. Then, one has Hp .G/ D .Z p ŒG/ if and only if p does not divide the order of the commutator subgroup of G. Moreover, in this case, we have Ip .G/ D .Z p ŒG/. Let A D RŒG, where R is either Z or Z p for a prime p. As above let M be a maximal order containing A. The central conductor of M over A is defined to be F .A/ WD fx 2 .M/jxM  Ag and is explicitly given by (cf. [10, Thm. 27.13])

F .A/ D where D

1

M jGj D .1/ 

1

.E./=E/;

(16.3.2)

.E./=E/ denotes the inverse different of the extension E./ WD E..g/ W g 2 G/ over E D Quot.R/

and the sum runs over all irreducible characters of G modulo Galois action. It is clear from the definition that we always have F .A/  H.A/. As above we set

F .G/ WD F .ZŒG/;

Fp .G/ WD F .Z p ŒG/:

Example 16.3.4. Let p and ` be primes with ` odd. We compute the denominator ideals of Z p ŒD2` , where D2` denotes the dihedral group of order 2`. In the case ` D 3, one has D6 ' S3 , the symmetric group on three letters. We let Mp .D2` / be a maximal Z p -order containing Z p ŒD2` . Then ( .Z p ŒD2` /; if p ¤ `, Hp .D2` / D Fp .D2` /; if p D `; ( .Z p ŒD2` /; if p ¤ `, Ip .D2` / D .Mp .D2` // if p D `. In fact, in the case that p ¤ `, the result follows from Proposition 16.3.3. In the case p D `, the result is established in [20, Ex. 6].

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373

Example 16.3.5. Let p be a prime and let q D `n be a prime power. We consider the group Aff.q/ D Fq Ì F q of affine transformations on Fq , the finite field with q elements. Let Mp .Aff.q// be a maximal Z p -order such that Z p ŒAff.q/  Mp .Aff.q//  Q p ŒAff.q/. Then by [21, Prop. 6.7] we have ( .Z p ŒAff.q//; if p ¤ `, Hp .Aff.q// D Fp .Aff.q//; if p D ` ¤ 2; ( .Z p ŒAff.q//; if p ¤ `, Ip .Aff.q// D .Mp .Aff.q///; if p D ` ¤ 2. If p D ` D 2, then we have containments 2H2 .Aff.q//  F2 .Aff.q//  H2 .Aff.q//;

2.M2 .Aff.q///  I2 .Aff.q//  .M2 .Aff.q///: Note that the commutator subgroup of Aff.q/ is Fq so that the case p ¤ ` again follows from Proposition 16.3.3. Example 16.3.6. Let S4 be the symmetric group on 4 letters. If p is an odd prime, then Ip .S4 / D .Mp .S4 // and Hp .S4 / D Fp .S4 /. However, if p D 2 we have

F2 .S4 / ¨ H2 .S4 / ¨ .Z 2 ŒS4 /I .Z 2 ŒS4 / ¨ I2 .S4 / ¨ .M2 .S4 //: This follows from [21, Prop. 6.8]. 16.3.3 Integrality conjectures Assume that L=K is a Galois CM-extension with Galois group G. Let S be a finite set of places of K containing Sram [S1 . We choose a maximal order M.G/ such that ZŒG  M.G/  QŒG. If T is a finite set of places of K such that Hyp.S; T / is satisfied, then ıT .0/ D nr.1 N.v/w 1/ belongs to .M.G//. We define AS to be the .ZŒG/-submodule of .M.G// generated by the elements ıT .0/, where T runs through the finite sets of places of K such that Hyp.S; T / is satisfied. Conjecture 16.3.7. Let S be a finite set of places of K containing Sram [ S1 . Then, we have an inclusion AS S  I .G/: As the integrality ring I .G/ is always contained in the centre of the maximal order M.G/, we may also state the following considerably weaker conjecture. Conjecture 16.3.8. Let S be a finite set of places of K containing Sram [ S1 . Then we have an inclusion AS S  .M.G//:

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Remark 16.3.9. If p is a prime, we let Mp .G/ WD Z p ˝ M.G/ which is a maximal Z p -order in Q p ŒG. As we have \ \ I .G/ D .QŒG/ \ Ip .G/ and .M.G// D .QŒG/ \ .Mp .G// p

p

the integrality conjectures of this section naturally decompose into local conjectures at each prime p. Example 16.3.10. Consider the Galois CM-extension L=Q from Example 16.3.1. As before let S D Sram [ S1 and T D f7g. We have seen that in this case ST does not lie in .Z 3 ŒG/. However, one has 
 71 1 13 37 ST D nr .1 j / C  11 2 C 19 C   C  2  2 I3 .G/: 2 2 2 2 Theorem 16.3.11. Both Conjecture 16.3.7 and Conjecture 16.3.8 hold when L=K is abelian.

Proof. Lemma 16.2.8 implies that AS D AnnZŒG .L /. Then, the result follows from (16.2.1) and the fact that I .G/ D ZŒG in this case.  Recall that a finite group is called monomial if each of its irreducible characters is induced by a linear character of a subgroup. The class of monomial groups includes all nilpotent groups [10, Thm. 11.3] and, more generally, all supersoluble groups [38, Ch. 2, Cor. 3.5]. Theorem 16.3.12. Let L=K be a Galois extension with Galois group G ' H  C , where H is monomial and C is abelian. Let S be a finite set of places of K containing Sram [ S1 . Then, we have an inclusion AS S  .M.H //ŒC : In particular, Conjecture 16.3.8 is true for monomial extensions. Proof. This is due to the author [30, Thm. 1.2]. The proof heavily relies on the abelian case and functoriality of Artin L-functions.  For non-abelian extensions, unconditional results on Conjecture 16.3.7 are rather sparse. Here we only mention the following special case of [30, Cor. 5.12]. Corollary 16.3.13. Let ` be an odd prime. Let L=K be a Galois CM-extension with Galois group isomorphic to D4` , the dihedral group of order 4`. Then, Conjecture 16.3.7 holds. Proof. We first note that D4` ' D2`  C2 with C2 WD Z=2Z and that dihedral groups are monomial. Taking Example 16.3.4 into account, we see that the p-part of

Conjectures of Brumer, Gross and Stark

375

Conjecture 16.3.7 directly follows from Theorem 16.3.12 if p is odd. Now consider the case p D 2. Let N be the commutator subgroup of D2` so that D2` =N ' C2 . It follows from [21, Proposition 2.13] that the group ring Z 2 ŒD2`  is ‘N -hybrid’ meaning that it decomposes into a direct product of Z 2 ŒD2` =N  ' Z 2 ŒC2  and some maximal order (see Definition 16.4.23 below). As

I2 .D4` / D .Z 2 ŒD4` / D .Z 2 ŒD2` /ŒC2 ; the result follows by combining Theorems 16.3.12 and 16.3.11.



We put !L WD nr.jL j/ D nr.wL /. Proposition 16.3.14. Let S be a finite set of places of K containing Sram [ S1 . 1. Suppose that Conjecture 16.3.7 holds. Then, !L S 2 I .G/: 2. Suppose that Conjecture 16.3.8 holds. Then, !L S 2 .M.G//: Proof. It suffices to show that !L S 2 Ap .G/ for each prime p, where Ap .G/ D Ip .G/ in case (1) and Ap .G/ D .Mp .G// in case (2). By Lemma 16.2.8, there is a totally decomposed place v0 of K (in fact infinitely many places) such that jL j D .1 N.v0 //
c, where c is a unit in Z p , and such that Hyp.S; T0 / is satisfied with T0 WD fv0 g. As nr.c/ belongs to Ap .G/, we have !L S D nr.c/ST0 2 Ap .G/ 

as desired.

16.3.4 The non-abelian Brumer and Brumer–Stark conjectures The following conjecture was first formulated in [27] and is a non-abelian generalisation of Brumer’s Conjecture 16.2.1. Conjecture 16.3.15 (B.L=K; S /). Let S be a finite set of places of K containing Sram [ S1 . Then, for each x 2 H.G/, we have x
AS S  Ann .ZŒG/ .clL /: Remark 16.3.16. Suppose that G is abelian. As we have already observed, Lemma 16.2.8 implies that AS D AnnZŒG .L / in this case. Since we have H.G/ D ZŒG, Conjecture 16.3.15 recovers Brumer’s Conjecture 16.2.1. Remark 16.3.17. When Conjecture 16.3.7 holds, then x
AS S is at least contained in .ZŒG/ for each x 2 H.G/.

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Remark 16.3.18. If M is a finitely generated Z-module and p is a prime, we define its p-part to be M.p/ WD Z p ˝ M . Replacing the class group clL by clL .p/ for each prime p, Conjecture B.L=K; S / naturally decomposes into local conjectures B.L=K; S; p/. It is then possible to replace H.G/ by Hp .G/ (see [27, Lem. 1.4]). Moreover, if p does not divide the order of the commutator subgroup of G then Hp .G/ D Ip .G/ D .Z p ŒG/ by Proposition 16.3.3 and so granting the hypotheses on S the statement of the local conjecture simplifies to AS S  Ann .Z p ŒG/ .clL .p//: Remark 16.3.19. Burns [3] has also formulated a conjecture which generalises many refined Stark conjectures to the non-abelian situation. In particular, it implies Conjecture 16.3.15 (see [3, Prop. 3.5.1]). A further approach to non-abelian Brumer and Brumer–Stark conjectures is due to Dejou and Roblot [13]. Recall that !L WD nr.jL j/. The following is a non-abelian generalisation of the Brumer–Stark Conjecture 16.2.5. Conjecture 16.3.20 (BS.L=K; S /). Let S be a finite set of places of K containing Sram [ S1 . Then for each x 2 H.G/ we have x
!L
S 2 .ZŒG/. Moreover, for each non-zero fractional ideal a of L, there is an anti-unit  D .x; a; S / 2 L such that ax
!L
S D ./ and for each finite set T of primes of K such that Hyp.S [ S ; T / is satisfied there is T an T 2 EL;S such that   z
ıT .0/ D Tz
!L

for each z 2 H.G/.

Remark 16.3.21. If G is abelian, we have

I .G/ D H.G/ D ZŒG

and

!L D jL j D wL :

Hence it suffices to treat the case x D z D 1 in this situation. Then, Proposition 16.2.10 implies that Conjecture 16.3.20 generalises Conjecture 16.2.5. Remark 16.3.22. Suppose that Conjecture 16.3.7 holds. Then, !L S 2 I .G/ by Proposition 16.3.14 and thus x
!L
S 2 .ZŒG/ for each x 2 H.G/. Remark 16.3.23. When we restrict to ideals whose classes in clL have p-power order, we again obtain local conjectures BS.L=K; S; p/ for each prime p. 16.3.5 The weak Brumer and Brumer–Stark conjectures Since H.G/ always contains the central conductor F .G/, we can state the following weaker versions of Conjectures B.L=K; S / and BS.L=K; S /.

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377

Conjecture 16.3.24 (Bw .L=K; S /). Let S be a finite set of places of K containing Sram [ S1 . Then, for each x 2 F .G/ we have x
AS S  Ann .ZŒG/ .clL /: Conjecture 16.3.25 (BSw .L=K; S /). Let S be a finite set of places of K containing Sram [ S1 . Then for each x 2 F .G/ we have x
!L
S 2 .ZŒG/. Moreover, for each non-zero fractional ideal a of L, there is an anti-unit  D .x; a; S / 2 L such that ax
!L
S D ./ and, for each finite set T of primes of K such that Hyp.S [ S ; T / holds, there is an T 2 EST such that  z
ıT .0/ D Tz
!L for each z 2 F .G/.

Remark 16.3.26. Suppose that Conjecture 16.3.8 holds. Then both x
AS S and x
!L
S lie in .ZŒG/ (for the latter use Proposition 16.3.14). Remark 16.3.27. We may again formulate local conjectures Bw .L=K; S; p/ and BSw .L=K; S; p/ for each prime p. 16.3.6 Implications between the conjectures We first discuss dependence on the set S . Lemma 16.3.28. Let S and S 0 be two finite sets of places of K such that S contains Sram [ S1 . If S  S 0 , one has B.L=K; S / Bw .L=K; S / BS.L=K; S / BSw .L=K; S /

H) H) H) H)

B.L=K; S 0 / Bw .L=K; S 0 / BS.L=K; S 0 / BSw .L=K; S 0 /:

Proof. We only give the proof in the case of Brumer’s conjecture; the other cases followQsimilarly. So assume that B.L=K; S / holds. We have S 0 D nr./S , where  D v2S 0 nS .1 w 1 / 2 ZŒG. If x lies in H.G/, so does xQ WD x
nr./. Hence we see that xAS 0 S 0  xA Q S S belongs to .ZŒG/ and annihilates clL .  The relation between the Brumer–Stark conjecture and Brumer’s conjecture is slightly more subtle. Lemma 16.3.29. Let S be a finite set of places of K containing Sram [ S1 . Then, BS.L=K; S / BSw .L=K; S /

H) H)

B.L=K; S / Bw .L=K; S /

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Proof. We give the proof for the strong conjectures. Let a be a non-zero fractional ideal of L and let x 2 H.G/. Then ax
!L S D ./ and ./z
ıT .0/ D .T /z
!L for all z 2 H.G/. Hence T ax
z
!L
S D ./z
ıT .0/ D .T /z
!L : (16.3.3)

Since !L 2 .QG/ , we find N 2 N such that N
!L 1 2 .ZŒG/. Moreover, jGj
.ZŒG/  F .G/  H.G/ such that we may choose z D jGj
N
!L 1 . However, the group of fractional ideals has no Z-torsion such that equation (16.3.3) implies that x
ST belongs to .ZŒG/ (take a to be a totally decomposed prime) and T ax
S D .T /.  16.3.7 A strong Brumer–Stark property Definition 16.3.30. Let S be a finite set of places of K containing Sram [ S1 . We say that L=K satisfies the strong Brumer–Stark property SBS.L=K; S / if

H.G/


1 T
  Ann .ZŒG/ .clTL / 2 S

for all finite sets T of K such that Hyp.S; T / holds. Remark 16.3.31. It is clear that SBS.L=K; S / holds if and only if 1 T
  Ann .Z p ŒG/ .clTL .p// 2 S for all primes p. It is then easy to see that the strong Brumer–Stark property S tBS.L=K; S; p/ as discussed in [27] for all p implies our strong Brumer–Stark property SBS.L=K; S /.

Hp .G/


Remark 16.3.32. It follows as in the proof of Lemma 16.3.28 that SBS.L=K; S / implies SBS.L=K; S 0 / whenever S  S 0 . Proposition 16.3.33. Let S be a finite set of places of K containing Sram [S1 . Then, SBS.L=K; S / implies BS.L=K; S /. Proof. The proof is similar to [27, Prop. 3.9].



Remark 16.3.34. In order to prove BS.L=K; S / and B.L=K; S /, it thus suffices to show that, for every prime p, we have:

Hp .G/


1 T
  Ann .Z p ŒG/ .clTL .p// 2 S

for all finite sets T of K such that Hyp.S; T / holds. We refer to this property as SBS.L=K; S; p/. This is in fact not much stronger than BS.L=K; S; p/. Nomura [32, Proposition 4.2] showed that for odd p in fact SBS.L=K; S; p/ and BS.L=K; S; p/ are equivalent.

Conjectures of Brumer, Gross and Stark

379

Remark 16.3.35. Replacing the denominator ideal H.G/ by the central conductor F .G/ one can formulate a weaker variant SBSw .L=K; S / such that SBSw .L=K; S / implies BSw .L=K; S / (we refrain from calling this the “weak strong Brumer–Stark property” for obvious reasons).

16.4 Relations to further conjectures and results 16.4.1 The relation to the equivariant Tamagawa number conjecture We only give a vague description of the statement of the equivariant Tamagawa number conjecture (ETNC) for the relevant Tate motive as formulated by Burns and Flach [6]. Let L=K be a finite Galois extension of number fields with Galois group G. We regard h0 .Spec.L// as a motive defined over K and with coefficients in the semisimple algebra QŒG. Let A be a Z-order such that ZŒG  A  QŒG. The ETNC for the pair .h0 .Spec.L//; A/ asserts that a certain canonical element T .L=K; A/ of the relative algebraic K-group K0 .A; R/ vanishes. This element incorporates the leading coefficients of the Artin L-functions attached to the irreducible characters of G and certain cohomological Euler characteristics. We note that the ETNC for the pair .h0 .Spec.L//; ZŒG/, the Lifted Root Number Conjecture of Gruenberg, Ritter and Weiss [19], the vanishing of the element T .L=K; 0/ defined in [2, §2.1], and the ‘leading term conjecture at s D 0’ of [5] are all equivalent (see [2, Theorems 2.3.3 and 2.4.1] and [5, Remarks 4.3 and 4.5]). Moreover, Burns and Flach [7, §3, Cor. 1] show that the ETNC for the pair .h0 .Spec.L//; M.G//, where M.G/ is a maximal order, is equivalent to the strong Stark conjecture (as formulated by Chinburg [9, Conj. 2.2]) for L=K. If L=K is a Galois CM-extension, the ETNC (over the maximal order, and in general away from its 2-primary part) naturally decomposes into a plus and a minus part. The following result is due to the author [27, Thm. 3.5]. Theorem 16.4.1. Let L=K be a Galois CM-extension of number fields with Galois group G. Let M.G/ be a maximal order in QŒG containing ZŒG. Then, the minus part of the ETNC for the pair .h0 .Spec.L//; M.G// implies BSw .L=K; S / and Bw .L=K; S / for all finite sets S of places of K containing S1 [ Sram . There is also a prime-by-prime version of Theorem 16.4.1 (see [27, Thm. 4.1]). Combined with [28, Cor. 2] this leads to the following unconditional result (see also [27, Cor. 4.2]). We denote the maximal totally real subfield of L by LC and let L0 be the Galois closure of L over Q. For a natural number n we let n be a primitive n-th root of unity. Theorem 16.4.2. Let p be an odd prime. Assume that no prime of LC above p splits in L whenever L0  .L0 /C .p /. Then, BSw .L=K; S; p/ and Bw .L=K; S; p/ are true for every set S of places of K containing Sram [ S1 .

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Remark 16.4.3. We stress that for a given extension L=K the hypotheses on p in Theorem 16.4.2 are fulfilled by all primes that do not ramify in L0 . In particular, the hypotheses are satisfied by all but finitely many primes. Theorem 16.4.4. Let L=K be a Galois CM-extension of number fields with Galois group G. Let p be an odd prime. Then, the minus p-part of the ETNC for the pair .h0 .Spec.L//; ZŒG/ implies BS.L=K; S; p/ and B.L=K; S; p/ for all finite sets S of places of K containing S1 [ Sram . Proof. When L .p/ is a cohomologically trivial G-module, this is due to Greither [16] (if G is abelian) and to the author [27, Thm. 5.1]; this includes the cases L .p/ D 1 and p − jGj. The general case follows from recent work of Burns [4, Proof of Cor. 3.11 (iii)].  Remark 16.4.5. The proofs of Theorem 16.4.4 actually show that a refinement of SBS.L=K; S; p/ holds and then use an argument similar to Proposition 16.3.33. There are meanwhile quite a few cases where the ETNC has been verified for certain non-abelian extensions. Here we only mention the following result of Johnston and the author [21, Thm. 4.6]. Theorem 16.4.6. Let L=Q be a Galois extension with Galois group G ' Aff.q/, where q D `n is a prime power. Then, the ETNC for the pair .h0 .Spec.L//; M.G// holds and the p-part of the ETNC for the pair .h0 .Spec.L//; ZŒG/ holds for every prime p ¤ `. Remark 16.4.7. An extension L=Q as in Theorem 16.4.6 never happens to be a CM-extension. However, Burns’ conjecture on the annihilation of class groups [3] predicts non-trivial annihilators for any Galois extension of number fields. Theorem 16.4.6 can then be combined with Example 16.3.5 to show that Burns’ conjecture holds in this case (up to a factor 2 if ` D 2). This is [21, Thm. 7.6]. Remark 16.4.8. The `-part of the ETNC in the situation of Theorem 16.4.6 is considered in recent work with Henri Johnston [24]. Suppose in addition that L is totally real and Leopoldt’s conjecture holds for L at `. Then, the ETNC for the pair .h0 .Spec.L//.1/; ZŒG/ holds. Moreover, the ETNC for the pair .h0 .Spec.L//; ZŒG/ holds if ` is at most tamely ramified (see [24, Cor. 10.6]). For the proof one has to verify the ‘`-adic Stark conjecture at s D 1’ for L=Q which might be seen as an analogue at s D 1 of Gross’ conjecture 16.4.15 below. 16.4.2 p-adic Artin L-functions Let p be an odd prime and let K be a totally real field. Let L=K be a Galois extension of K such that L is totally real and contains the cyclotomic Z p -extension K1 of K and ŒL W K1  is finite. We put G WD Gal.L=K/ and €K WD Gal.K1 =K/ ' Z p such that G ' H Ì €, where H WD Gal.L=K1/ and

Conjectures of Brumer, Gross and Stark

381

€ D G =H ' €K . Thus L=K is a one-dimensional p-adic Lie extension. We choose a topological generator K of €K . We write cyc for the p-adic cyclotomic character cyc W Gal.L.p /=K/ ! Z  p; defined by  ./ D  cyc ./ for any  2 Gal.L.p /=K/ and any p-power root of unity . Let ! and  denote the composition of cyc with the projections onto the first and second factors of the canonical decomposition Z  p D Q p  .1 C pZ p /, respectively; thus ! is the Teichmüller character. We note that  factors through €K and put u WD . K /. Fix a character 2 IrrCp .G / and let S be a finite set of places of K containing all archimedean places and all places that ramify in L=K. Note that S in particular contains the set Sp of all p-adic places. Each topological generator K of €K permits the definition of a power series G ;S .T / 2 Q cp ˝Q p Quot.Z p ŒŒT / by starting out from the Deligne–Ribet power series for one-dimensional characters of open subgroups of G (see [8, 14, 1]) and then extending to the general case by using Brauer induction (see [15]). One then has an equality Lp;S .1

s; / D

s ;S .u H .us

G

1/ 1/

;

where Lp;S .s; / W Z p ! Cp denotes the ‘S -truncated p-adic Artin L-function’ (a p-adic meromorphic function) attached to constructed by Greenberg [15], and where, for irreducible , one has ( . K /.1 C T / 1; if H  ker , H .T / D 1; otherwise. 16.4.3 The interpolation property and Gross’ conjecture Let p be an odd prime and choose a field isomorphism  W C ' Cp . For a character 2 IrrCp .G / we put  WD  1 ı 2 IrrC .G /. If is a linear character and r > 1 is an integer then for every choice of field isomorphism  W C ' Cp one has the interpolation property Lp;S .1

r; / D  .LS .1

r; . !

r 

/ // :

(16.4.1)

This can be extended to characters of arbitrary degree provided that r > 2 (see [15, §4]). However, the argument fails in the case r D 1. Nevertheless, it seems plausible to conjecture the following. Conjecture 16.4.9. For each

2 IrrCp .G /, one has

Lp;S .0; / D  LS .0; . !


// :

1 

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A. Nickel

Remark 16.4.10. As both sides in (16.4.9) are well-behaved with respect to direct sum, inflation and induction of characters, it is easy to see that Conjecture 16.4.9 holds when is a monomial character (also see the discussion in [18, §2]). In fact, Conjecture 16.4.9 is a special case of a Conjecture of Gross [18] which we now recall. Let  2 IrrC .Gal.L.p /=K// be a non-trivial character and let rS ./ be the order of vanishing of the S -truncated Artin L-function LS .s; / at s D 0. We write LS .0; / for the leading coefficient in the Laurent series expansion of LS .s; / at s D 0. Now let 2 IrrCp .G / and choose a field isomorphism  W C ' Cp . Then, formula (16.1.1) shows that rS . / WD rS .. !

1 

//

does in fact not depend on the choice of . The first part of Gross’ conjecture [18] concerns the order of vanishing of p-adic Artin L-functions and asserts the following. Conjecture 16.4.11. Let 2 IrrCp .G /. Then, the order of vanishing of the S truncated p-adic Artin L-function Lp;S .s; / at s D 0 equals rS . /. Remark 16.4.12. Suppose that is linear. If rS . / vanishes, Conjecture 16.4.11 holds by (16.4.1). The conjecture is also known when rS . / D 1 (see [18, Prop. 2.13]). Theorem 16.4.13. Let 2 IrrCp .G /. Then, the order of vanishing of the S -truncated p-adic Artin L-function Lp;S .s; / at s D 0 is at least rS . /. Proof. For being a linear character, this has been proved by Spiess [35] using Shintani cocycles (his approach actually allows p to be equal to 2). The general case has recently been settled by Burns [4, Thm. 3.1].  The rest of this subsection is mainly devoted to (the second part of) Gross’ conjecture. This may be skipped by the reader who is only interested in the Brumer and Brumer–Stark conjectures. Fix a character 2 IrrCp .G / and choose a Galois CM-extension L over K such that ! 1 factors through G WD Gal.L=K/. We denote the kernel of the natural augmentation map ZŒS.L/ ! Z that maps each w 2 S.L/ to 1 by XL;S . The usual Dirichlet map L;S W R ˝ EL;S

! R ˝ XL;S X 1 ˝  7! log jjw w w2S.L/

is an isomorphism of RŒG-modules. For each place w of L, Gross [18, §1] defines a p-adic absolute value k
kw;p W L ! Z  p

383

Conjectures of Brumer, Gross and Stark

as the composite map ab  L ,! L w ! Gal.Lw =Lw / ! Z p I

here, Lab w denotes the maximal abelian extension of Lw , the first arrow is the natural inclusion, the second arrow is the reciprocity map of local class field theory and the last map is the p-adic cyclotomic character. We define a homomorphism of Z p ŒGmodules p;L;S W Z p ˝ EL;S

! Z p ˝ XL;S X 1 ˝  7! logkkw;p w: w2S.L/

1 Now, choose a field isomorphism  W C ' Cp . Then, L;S and p;L;S induce an endomorphism 1 .Cp ˝Z p p;L;S / ı .Cp ˝ L;S / W Cp ˝ XL;S ! Cp ˝ XL;S :

We define a p-adic regulator ./

1 Rp;S . / WD detCp ..Cp ˝Z p p;L;S /ı.Cp ˝ L;S / j HomCp ŒG .V

!

1

; Cp ˝XL;S //:

Proposition 16.4.14. Fix 2 IrrCp .G / and choose a field isomorphism  W C ' Cp . ./ Then Conjecture 16.4.11 holds for if and only if Rp;S . / ¤ 0. 

Proof. This follows from [4, Thm. 3.1 (iii)]. Lrp;S .0;

r

For an integer r, let / be the coefficient of s in the power series expansion of Lp;S .s; / at s D 0. We can now state the second part of Gross’ conjecture. Conjecture 16.4.15. Fix Then, one has

2 IrrCp .G / and choose a field isomorphism  W C ' Cp .

./ S. / . /
 LS .0; . ! .0; / D Rp;S Lrp;S


// :

1 

Remark 16.4.16. By means of Proposition 16.4.14, it is clear that Conjecture 16.4.15 is only interesting when Conjecture 16.4.11 holds. The following recent result due to Dasgupta, Kakde and Ventullo [12] generalises the approach developed in [11]. Theorem 16.4.17. Conjecture 16.4.15 holds for linear characters. Corollary 16.4.18. Suppose that Conjecture 16.4.11 holds for all Then, Conjecture 16.4.15 is also true for all 2 IrrCp .G /. Proof. This follows from Theorem 16.4.17 by Brauer induction.

2 IrrCp .G /. 

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A. Nickel

16.4.4 Conditional Results We can now state the following result which has been proved by Johnston and the author [23, Thm. 5.2 and Cor. 5.4]. We refer to the ‘equivariant Iwasawa main conjecture’ (EIMC) for totally real fields (see [25] or [33], for instance). Theorem 16.4.19. Let L=K be a Galois CM-extension with Galois group G and let p be an odd prime. Suppose that the EIMC for the extension L.p /C 1 =K holds. Suppose further that Conjecture 16.4.9 holds for all irreducible characters of C Gal.L.p /C 1 =K/ which factor through Gal.L =K/. Then, SBS.L=K; S; p/ and thus BS.L=K; S; p/ and B.L=K; S; p/ are true for every finite set S of places of K containing Sp [ Sram [ S1 . Remark 16.4.20. We write p .F / for the p-adic -invariant attached to the cyclotomic Z p -extension of a number field F (see [22, Rem. 4.3] for details). When p .L.p /C / vanishes, then the EIMC has been proved independently by Kakde [25] and Ritter and Weiss [33]. Without assuming the vanishing of -invariants considerable progress has been made in [22]. This includes the case p − jGj. Remark 16.4.21. Suppose that L=K is abelian. Then, Conjecture 16.4.9 holds by (16.4.1). Under the somewhat stronger condition that  vanishes, Theorem 16.4.19 has been shown by Greither and Popescu [17] by an entirely different method. This method has been generalised to arbitrary Galois CM-extensions by the author in [29]. In order to get rid of the p-adic places one has to assume the full strength of Gross’ conjecture. Theorem 16.4.22. Let L=K be a Galois CM-extension with Galois group G and let p be an odd prime. Suppose that p .LC / vanishes or that p − jGj. Suppose further that Gross’ Conjecture 16.4.11 holds for all 2 IrrCp .Gal.LC 1 =K//. Then, the minus p-part of the ETNC for the pair h0 .Spec.L/; ZŒG/ holds. In particular, both BS.L=K; S; p/ and B.L=K; S; p/ are true for all finite sets S of places of K containing Sram [ S1 . Proof. This is [4, Cors. 3.8 and 3.11]. Note that the last part follows from Theorem 16.4.4.  16.4.5 Unconditional results We now discuss certain cases where the Brumer– Stark conjecture holds unconditionally. Let p be aP prime and let G be a finite group. For a normal subgroup N E G, let eN D jN j 1 2N  be the associated central trace idempotent in the group algebra Q p ŒG. Definition 16.4.23. Let N E G. We say that the p-adic group ring Z p ŒG is N hybrid if (i) eN 2 Z p ŒG (i.e. p − jN j) and (ii) Z p ŒG.1 eN / is a maximal Z p -order in Q p ŒG.1 eN /.

Conjectures of Brumer, Gross and Stark

385

Theorem 16.4.24. Let L=K be a finite Galois CM-extension of number fields. Let N be a normal subgroup of G WD Gal.LC =K/ and let F D .LC /N . Let p be an odd prime and let P be a Sylow p-subgroup of G WD Gal.F=K/ ' G=N . Suppose that Z p ŒG is N -hybrid, G is monomial, and F P =Q is abelian. Let S be a finite set of places of K such that Sp [ Sram .L=K/ [ S1  S . Then, both BS.L=K; S; p/ and B.L=K; S; p/ are true. Proof. It follows from the theory of hybrid Iwasawa algebras [22] that the relevant case of the EIMC holds. We also recall that Conjecture 16.4.9 holds for monomial characters. Then the result follows from Theorem 16.4.19. See [23, Thm. 10.5] for details.  We recall that a Frobenius group is a finite group G with a proper nontrivial subgroup V such that V \ gVg 1 D f1g for all g 2 G V , in which case V is called a Frobenius complement. A Frobenius group G contains a unique normal subgroup U , known as the Frobenius kernel, such that G is a semidirect product U Ì V . Corollary 16.4.25. Let L=K be a finite Galois CM-extension of number fields and let G D Gal.LC =K/. Suppose that G D U Ì V is a Frobenius group with Frobenius kernel U and abelian Frobenius complement V . Suppose further that .LC /U =Q is abelian (in particular, this is the case when K D Q). Let p be an odd prime and let S be a finite set of places of K such that Sp [ Sram .L=K/ [ S1  S . Suppose that either p − jU j or U is a p-group (in particular, this is the case if U is an `-group for any prime `.) Then, both BS.L=K; S; p/ and B.L=K; S; p/ are true. Proof. This is [23, Cor. 10.7]. We recall the proof for convenience. First note that G is monomial by [23, Lemma 9.7] since V is abelian. Suppose that p − jU j. Let N D U and F D .LC /N . Then Z p ŒG is N -hybrid by [23, Prop. 9.4]. Hence, the desired result follows from Theorem 16.4.24 in this case since F=Q is abelian, which forces F P =Q to be abelian. Suppose that U is a p-group. Taking N D f1g and F D LC , we apply Theorem 16.4.24 with G D G and P D U to obtain the desired result.  Example 16.4.26. In particular, U is an `-group in Corollary 16.4.25 when G is one of the following Frobenius groups (for a natural number n we denote the cyclic group of order n by Cn ):  G ' Aff.q/ D Fq Ì F q , where q is a prime power and Aff.q/ is the group of affine transformations on Fq ,  G ' C` Ì Cq , where q < ` are distinct primes such that q j .` acts on C` via an embedding Cq ,! Aut.C` /,

1/ and Cq

 G is isomorphic to any of the Frobenius groups constructed in [22, Ex. 2.11]. Note that in particular Aff.3/ ' S3 is the symmetric group on 3 letters (which is the smallest non-abelian group) and Aff.4/ ' A4 is the alternating group on 4 letters.

386

A. Nickel

In certain situations, we can also remove the condition that Sp  S . To illustrate this, we conclude with the following two results (the first is [23, Thm. 10.10], whereas the second is due to Nomura [31, 32]). Theorem 16.4.27. Let L=Q be a finite Galois CM-extension of the rationals. Suppose that Gal.L=Q/ ' hj i  G, where G D Gal.LC =Q/ D N Ì V is a Frobenius group with Frobenius kernel N and abelian Frobenius complement V . Suppose further that N is an `-group for some prime `. Then, both BS.L=Q; S; p/ and B.L=Q; S; p/ are true for every odd prime p and every finite set S of places of Q such that Sram .L=Q/ [ S1  S . Theorem 16.4.28. Let ` be an odd prime. Let L=K be a Galois CM-extension with Galois group G ' D4` . Let p be a prime and suppose that p does not split in Q.` /. Then, BS.L=K; S; p/ and B.L=K; S; p/ both hold for every finite set S of places of K such that Sram [ S1  S .

References [1] D. Barsky, Fonctions zeta p-adiques d’une classe de rayon des corps de nombres totalement réels. In Groupe d’Etude d’Analyse Ultramétrique (5e année: 1977/78) (Y. Amice, D. Barsky, and P. Robba, eds.), Secrétariat Math., Paris, 1978, Exp. No. 16, pp. 1–23. [2] D. Burns, Equivariant Tamagawa numbers and Galois module theory. I, Compos. Math. 129 (2001), 203–237. [3] D. Burns, On derivatives of Artin L-series, Invent. Math. 186 (2011), 291–371. [4] D. Burns, On derivatives of p-adic L-series at s D 0, J. Reine Angew. Math. (Crelle) (to appear). [5] D. Burns and M. Breuning, Leading terms of Artin L-functions at s D 0 and s D 1, Compos. Math. 143 (2007), 1427–1464. [6] D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570. [7] D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients. II, Amer. J. Math. 125 (2003), 475–512. [8] P. Cassou-Nogues, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques, Invent. Math. 51 (1979), 29–59. [9] T. Chinburg, On the Galois structure of algebraic integers and S-units, Invent. Math. 74 (1983), 321–349. [10] C.W. Curtis and I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. 1, Wiley, New York, 1981. [11] S. Dasgupta, H. Darmon and R. Pollack, Hilbert modular forms and the Gross–Stark conjecture, Ann. of Math. 174 (2011), 439–484. [12] S. Dasgupta, M. Kakde and K. Ventullo, On the Gross–Stark conjecture, Ann. of Math. 188 (2018), 833–870.

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[13] G. Dejou and X.-F. Roblot, A Brumer–Stark conjecture for non-abelian extensions, J. Number Th. 142 (2014), 51–88. [14] P. Deligne and K.A. Ribet, Values of abelian L-functions at negative integers over totally real fields, Invent. Math. 59 (1980), 227–286. [15] R. Greenberg, On p-adic Artin L-functions, Nagoya Math. J. 89 (1983), 77–87. [16] C. Greither, Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture, Compos. Math. 143 (2007), 1399–1426. [17] C. Greither and C.D. Popescu, An equivariant main conjecture in Iwasawa theory and applications, J. Algebraic Geom. 24 (2015), 629–692. [18] B.H. Gross, p-adic L-series at s D 0, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 979–994. [19] K.W. Gruenberg, J. Ritter and A. Weiss, A local approach to Chinburg’s root number conjecture, Proc. London Math. Soc. 79 (1999), 47–80. [20] H. Johnston and A. Nickel, Noncommutative Fitting invariants and improved annihilation results, J. Lond. Math. Soc. 88 (2013), 137–160. [21] H. Johnston and A. Nickel, On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results, Trans. Amer. Math. Soc. 368 (2016), 6539–6574. [22] H. Johnston and A. Nickel, Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture, Amer. J. Math. 140 (2018), 245–276. [23] H. Johnston and A. Nickel, On the non-abelian Brumer–Stark conjecture and the equivariant Isawana main conjecture, Math. Z. (to appear). [24] H. Johnston and A. Nickel, On the p-adic Stark conjecture at s D 1 and applications, preprint, arXiv:1703.06803. [25] M. Kakde, The main conjecture of Iwasawa theory for totally real fields, Invent. Math. 193 (2013), 539–626. [26] A. Nickel, Non-commutative Fitting invariants and annihilation of class groups, J. Algebra 323 (2010), 2756–2778 [27] A. Nickel, On non-abelian Stark-type conjectures, Ann. Inst. Fourier 61 (2011), 2577–2608. [28] A. Nickel, On the equivariant Tamagawa number conjecture in tame CM-extensions, Math. Z. 268 (2011), 1–35. [29] A. Nickel, Equivariant Iwasawa theory and non-abelian Stark-type conjectures, Proc. Lond. Math. Soc. 106 (2013), 1223–1247. [30] A. Nickel, Integrality of Stickelberger elements and the equivariant Tamagawa number conjecture, J. Reine Angew. Math. (Crelle) 719 (2016), 101–132. [31] J. Nomura, On non-abelian Brumer and Brumer–Stark conjecture for monomial CMextensions, Int. J. Number Th. 10 (2014), 817–848. [32] J. Nomura, The 2-part of the non-abelian Brumer–Stark conjecture for extensions with group D4p and numerical examples of the conjecture. In Algebraic Number Theory and Related Topics 2012 (A. Shiho, T. Ochiai and N. Otsubo, eds.), Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, 33–53. [33] J. Ritter and A. Weiss, On the “main conjecture” of equivariant Iwasawa theory, J. Amer. Math. Soc. 24 (2011), 1015–1050.

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[34] C.L. Siegel, Über die Fourierschen Koeffizienten von Modulformen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1970 (1970), 15–56. [35] M. Spiess, Shintani cocycles and the order of vanishing of p-adic Hecke L-series at s D 0, Math. Ann. 359 (2014), 239–265. [36] L. Stickelberger, Ueber eine Verallgemeinerung der Kreistheilung, Math. Ann. 37 (1890), 321–367. [37] J. Tate, Les conjectures de Stark sur les fonctions L d’Artin en s D 0, Lecture notes edited by D. Bernardi and N. Schappacher, Progress in Mathematics 47, Birkhäuser Boston, Boston, 1984. [38] M. Weinstein (ed.), Between Nilpotent and Solvable, Polygonal Publ. House, Passaic, 1982.

Chapter 17

Displays and p-divisible groups T. Zink The theory of displays is a Dieudonné theory for formal p-divisible groups which is an equivalence of categories over an arbitrary p-adic ring. Over a more restricted class of rings one obtains a classification of all p-divisible groups. We explain basic ideas and some recent results of this theory. The last paragraph ameliorates the discussion of isogenies of displays found in the literature.1

17.1 Introduction Let p be a fixed prime number. Let A be an abelian variety over a field k of charactristic p. The formal group of A is a very interesting and subtle invariant of A. This is in contrast to the case of characteristic 0 where the formal group is determined by dim A. Dieudonné gave a classification of the formal groups over k entirely in terms of linear algebra. This classification was extended to cases where k is replaced by a p-adic ring R. In the generalisation one has to replace the formal group of A by a finer invariant, the p-divisible group. To avoid confusion, it must be said that this is not an abstract group but an inductive limit of commutative finite flat group schemes. Many of the most important results in arithmetic proved in the last years depend on a profound konwledge of p-divisible groups. Ever since the introduction of this notion by J. Tate in 1967, very surprising results on these groups were proved continuously. Some of the main contributors are A. Grothendieck, W. Messing, B. Mazur, P. Berthelot, J.-M. Fontaine, Ch. Breuil, M. Kisin, F. Oort, A.J. de Jong, A. Vasiu, L. Fargues, P. Scholze, J. Weinstein. The classification of p-divisible groups is by objects of linear algebra over a padic ring R which are called displays. In the case of a perfect field R, they coincide with the classical Dieudonné modules. The notion was developed to study the bad reduction of Shimura varieties. Some important recent results on p-divisible groups have been explained in terms of displays. We review here some linear algebra of displays which is buried in the literature and we add some supplements. This is essential in order to apply the main theorems of the classification. This article complements the overview [16]. We mention the duality theory for displays only briefly. For a concise review of these results we refer to [5], Chapter 3. In the papers [7] and [8] it is proved that the crystalline cohomology 1 Project

B5

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is often endowed with a display structure. This is applied to deformation theory of algebraic varieties. Further applications of the theory can be found in the papers [19, 17, 5, 22].

17.2 Frames We make the following conventions. A ring will be a commutative ring with unit element unless otherwise stated. Let S be a ring and let  W S ! S be a ring endomorphism. If M is an S -module, we write M ./ D S ˝;S M; where the right S -module structure on S is modified by restriction of scalars via  . If  W M ! N is a  -linear homomorphism, we define its linearisation  ] W M ./ ! N by  ] .s ˝ m/ D s.m/. We call  a  -linear epimorphism if ' ] is surjective, and we call  a  -linear isomorphism if  ] is bijective. Frames were introduced in [21] and [11]. Let F=Qp be a finite field extension and O D OF its ring of integers. We fix a prime element  2 O. Let  D O=O be the residue class field and q D pf D ]. Definition 17.2.1. An O-frame F D .S; I; R; ; / P consists of the following data: S is an O-algebra, I  S is an ideal, R D S=I ,  W S ! S is an endomorphism of O-algebras, P W I ! S is a  -linear map of S -modules. We require: (i) The ideal I and the prime number p are contained in the Jacobson radical of S. (ii) For each s 2 S , we have  .s/  s q mod S . (iii) The set .I P / generates S as an S -module. Lemma 17.2.2. For each O-frame F , there is a unique element  2 S such that  .a/ D  .a/ P for a 2 I: We have  2 I C S and in particular  belongs to the radical of S . Proof. The uniqueness is immediate from condition (iii). Moreover, by (iii) we may write: X 1 D si P .ai / (17.2.1) i

for suitable elements si 2 S and ai 2 I . Then, the element X  D si  .ai / i

satisfies the requirement. This follows if we multiply (17.2.1) by  .a/. Finally, by condition (ii), we have  .ai / 2 I C S . Therefore, the same holds for  . 

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Definition 17.2.3. A morphism of O-frames ˛ W F D .S; I; R; ; / P ! F 0 D .S 0 ; I 0 ; R0 ;  0 ; P 0 / is a ring homomorphism ˛ W S ! S 0 such that ˛.I /  I 0 and such that P 0 .˛.a// D ˛..a// P for a 2 I: The last equation implies that  0 .˛.s// D ˛. .s//;

s 2 S:

Indeed, by Equation (17.2.1), 1 D We multiply (17.2.1) with  .s/:

X i

˛.si /P 0 .˛.ai //:

 .s/ D

X i

si .a P i s/:

Then we obtain by the definition X ˛. .s// D ˛.si /P 0 .˛.ai s// i

D

X i

˛.si /P 0 .˛.ai // 0 .˛.s// D  0 .˛.s//;

as required. Example 17.2.4. Let R be a p-adic ring. We denote by W .R/ the ring p-typical Witt vectors [2]. It is endowed with the Frobenius endomorphism F and with the Verschiebung V . The ring W .R/ is again a p-adic ring [20]. We define a Z p -frame

W .R/ D .W .R/; V W .R/; R; ; /; P where  W W .R/ ! W .R/ is the Frobenius endomorphism  ./ D F, and P .V/ D , for  2 W .R/. We call this the Witt frame. Example 17.2.5. Let R be an O-algebra. Then the Witt vectors WO .R/ are defined [1]. As a set, WO .R/ Š RN . There is a unique O-algebra structure which is functorial in R and such that the maps wn W WO .R/ ! R given by n

wn .x0 ; x1 ; x2 ; : : :/ D x0q C x1q

n 1

C  2 x2q

n 2

C : : : C n

1 q xn 1

C  n xn

are O-algebra homomorphisms. For x 2 R, the element Œx WD .x; 0; 0; 0; : : :/ is called the Teichmüller representative.

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There are maps F; V W WO .R/ ! WO .R/: V

.x0 ; x1 ; x2 ; : : :/ D .0; x0 ; x1 ; x2 ; : : :/;

F

.x0 ; x1 ; x2 ; : : :/ D Œx0q C.x1 ; x2 ; : : :/:

F is a ring homomorphism, V is additive and the following relations hold, F V D ; V. F/ D .V/;

;  2 WO .R/:

As before, we define an O-frame

WO .R/ D .WO .R/; V WO .R/; R; ; /; P where  ./ D relative to O.

F

 and . P V/ D , for  2 WO .R/. We call this the Witt frame

Example 17.2.6. Let R ! k be an O-algebra homomorphism. We assume that k is a perfect field of characteristic p > 3 and that the kernel a is a nilpotent ideal. Then, we have an exact sequence 0 ! WO .a/ ! WO .R/ ! WO .k/ ! 0: There is a unique Frobenius equivariant section ı W WO .k/ ! WO .R/. We denote by WO O .a/  WO .a/ the subset of all Witt vectors which have only finitely many nonzero components. This is a subring. We denote by WO O .R/  WO .R/ the subring generated by WO O .a/ and the image of ı. This subring is invariant under F and V . As before, we obtain a O-frame O O .R/ D .WO .R/; V WO .R/; R; ; /; W P O O where  D F , and P D V 1 as before. We call this the small Witt frame relative to O. In the case O D Z p we have WO .R/ D W .R/ and we find the Z p -frame of Example 17.2.4. Example 17.2.7. Let R; S be p-adic O-algebras. We consider a surjective O-algebra homomorphism S ! R such that the kernel a is endowed with divided powers n relative to O ([1], 1.2.2). This means that expressions .aq = n / for a 2 a are defined such that the following properties hold. We define the divided Witt polynomials for .a0 ; a1 ; a2 ; : : :/ 2 WO .a/: n

w0n .a0 ; a1 ; a2 ; : : :/

n 1

aq aq D 0n C 1n  

1

C:::C

anq 1 C an : 

Taken together, the w0n define an isomorphism of O-modules WO .a/ Š aN :

(17.2.2)

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The image of a Witt vector on the right hand side is called its logarithmic coordinates Œu0 ; u1 ; : : :, ui 2 a. We have F

Œu0 ; u1 ; u2 ; : : : D Œu1 ; u2 ; u3 ; : : :;

V

Œu0 ; u1 ; u2 ; : : : D Œ0; u0 ; u1 ; : : ::

By (17.2.2), aN becomes a WOF .S /-module: Œu0 ; u1 ; u2 ; : : : D Œw0 ./u0 ; w1 ./u1; w2 ./u2 ; : : :;

 2 WOF .S /

We denote by JO .S / the kernel of the composite map WO .S / ! WO .R/

w0

We have JO .S / D IO .S / C WO .a/. We extend V

! R: 1

(17.2.3)

W IO .S / ! WO .S / to a map

P W JO .S / ! WO .S /

(17.2.4)

by setting .Œu P 0 ; u1 ; u2 ; : : :/ D Œu1 ; u2 ; u3 ; : : :;

ui 2 a:

With respect to the maps (17.2.3), (17.2.4) we obtain a O-frame

WO .S=R/ D .WO .S /; JO .S /; R; ; /; P

(17.2.5)

where  D F is the Frobenius endomorphism of WO .S /. We call this the relative Witt frame. Let R ! k and S ! k as in Example 17.2.6. Let ˛W S ! R be a surjective O-algebra homomorphism compatible with the augmentations to k and such that the kernel of ˛ is endowed with divided powers. Then we can define the small relative Witt frame O .S=R/ D .WO .S /; JO .S /; R; ; /: W P O O O

17.3 Displays Definition 17.3.1. Let F be a O-frame. An F -display (or window) P D .P; Q; F; FP / consists of the following data: A finitely generated projective S -module P , a submodule Q  P , and two  linear maps F W P ! P; FP W Q ! P: The following conditions are required. (i) IP  Q:

(ii) The factor module P =Q is a finitely generated projective R-module.

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(iii) If a 2 I and x 2 P , we have the relation FP .ax/ D .a/F P x: (iv) FP .Q/ generates P as an S -module. (v) The projective R-module P =Q lifts to a finitely generated projective S -module. More precisely, the last condition says that there is a finitely generated projective S -module T such that P =Q ' R ˝S T . We call rankWO .R/ P the height of P and rankR P =Q the dimension of P . Morphisms of F -displays are defined in the obvious way. Remarks: Let  2 S be the element in Lemma 1. Then, we have F .y/ D  FP .y/;

for y 2 Q:

Indeed, we multiply Equation (17.2.1) with F .y/ and obtain X X F .y/ D si FP .ai y/ D si  .ai /FP .y/: i

i

It follows from the axioms that there exist S -submodules L and T of P such that P D T ˚ L;

Q D IT ˚ L

(17.3.1)

Indeed, choose a finitely generated projective S -module T which lifts P =Q. Then, we find an S -module homomorphism which makes the following diagram commutative, /T P❉ ❉❉ ❉❉ ❉❉ ❉"  P =Q The homomorphism P ! T is surjective by the lemma of Nakayama. Let L be the kernel. Taking a splitting of the exact sequence 0 !L !P !T

! 0;

we obtain the decomposition (17.3.1). We will call (17.3.1) a normal decompositon of P . The  -linear homomorphism F ˚ FP W T ˚ L ! P is then a  -linear isomorphism. Indeed, consider its linearisation .F ˚ FP /] W .T ˚ L/./ ! P:

(17.3.2)

Displays and p-divisible groups

395

Since we have projective modules of the same rank on both sides, it is enough to show that (17.3.2) is surjective. By the axioms, the elements of the form FP .at C l/;

for a 2 I; t 2 T; l 2 L

generate the S -module P . Since FP .at C l/ D .a/F P .t/ C FP .l/, the surjectivity of (17.3.2) follows. Assume conversely that we are given a finitely generated projective S -module P , a decomposition P D T ˚ L, and an isomorphism ˆ W P ./ ! P:

We may write ˆ

t0 l0

!!

D



A B C D



! t0 : l0

Here, A W T ./ ! T , B W L./ ! T , C W T ./ ! L, B W L./ S -linear homomorphisms and t 0 2 T ./ , l 0 2 L./ . We use the notation  and P also for the following maps:  WT

(17.3.3) ! L are

! S ˝;S T;

t 7 ! 1 ˝ t; where T can be any S -module, and P W I ˝S T

! S ˝;S T;

a ˝ t 7 ! P .a/ ˝ t: The F -display obtained from ˆ is then defined as follows: We set Q D I T ˚ L;

P D T ˚ L:

Let t 2 T , l 2 L and y 2 I T D I ˝S T be elements. We define the maps F and FP as follows: !!    y A B P .y/ P F D ; C D  .l/ l (17.3.4) !!    t A B  .t/ F D : C D  .l/ l Let T be a free module with basis e1 ; : : : ; ed and L a free module with basis ed C1 ; : : : ed Cc , then, T ./ has the basis 1 ˝ e1 ; : : : ; 1 ˝ ed and L./ has the basis 1 ˝ ed C1 ; : : : ; 1 ˝ ed Cc . Now the linear map A W T ./ ! T may be simply regarded as a d  d -matrix with coefficients in S . In this sense, we may regard   A B (17.3.5) C D as an element of GLd Cc .S /.

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Let ˛ W F ! F 0 be a morphism of O-frames as in Definition 17.2.3. We now define the base change functor ˛ W .F -displays/ ! .F 0 -displays/:

(17.3.6)

Let P D .P; Q; F; FP / be an F -display. We define P 0 D S 0 ˝S P;


Q0 D Ker S 0 ˝S P ! R0 ˝R P =Q :

There are unique  -linear maps F 0 W P 0 ! P 0 and FP 0 W Q0 ! P 0 such that F 0 .s 0 ˝ x/ D  0 .s 0 / ˝ F .x/; s 0 2 S 0 ; x 2 P; 0 0 0 0 0 0 0 0 FP .s ˝ y/ D  .s / ˝ FP .y/; FP .a ˝ x/ D P .a / ˝ F .x/; a0 2 I 0 ; y 2 Q: One shows easily that ˛ P D .P 0 ; Q0 ; F 0 ; FP 0 / is an F 0 -display. In the case where P is given by a matrix (17.3.5), we can simply apply the homomorphism S ! S 0 to the coefficients of this matrix. Then, we obtain the matrix of the base change .P 0 ; Q0 ; F 0 ; FP 0 /. Now we describe morphisms of displays. Let ˛ W P ! P 0 be a morphism of F -displays. We fix normal decompositions P 0 D T 0 ˚ L0 :

P D T ˚ L;

Then the display structure on P resp. P 0 is given by a matrix     0 A B A B0 : resp. C 0 D0 C D We also write ˛ in matrix form   X Y W T ˚ L ! T 0 ˚ L0 ; U Z

(17.3.7)

(17.3.8)

where X W T ! T 0 , U W T ! L0 , Z W L ! L0 , Y W L ! I T 0 . The image of Y is in I T 0 because ˛.Q/  Q0 . We will denote by  .X / W T ./ ! T 0./ the map obtained from X by base change and similarly  .U / and  .Z/. We denote by P .Y / W L./ ! T 0./ the linearisation of the  -linear homomorphism P

L ! I ˝ T 0 ! T 0./ : The matrix (17.3.8) defines a homomorphism iff the following diagram is commutative ˛ IT ˚ L ! I T 0 ˚ L0 ? ? ? ? P0 yF FP y T ˚L

˛

! T 0 ˚ L0 :

Displays and p-divisible groups

We may write this as an equation of matrices,       0  .X / .Y P / X Y A B A B0 : D   .U /  .Z/ C 0 D0 U Z C D

397

(17.3.9)

This diagram expresses the condition that ˛ commutes with FP . But then ˛ also commutes with F because of the equation FP .ax/ D .a/F P .x/ for a 2 I , x 2 P and the axiom that the elements P .a/ generate the unit ideal.

17.4 Classification of p-divisible groups Theorem 17.4.1 ([12]). Let S ! R be as in Example 17.2.7. Let (p-div=R) be the category of p-divisible groups over R. Then, there is a natural (with respect to S ! R) functor .p-div=R/ ! .WZ p .S=R/-displays/ In the case S D R we obtain a functor .p-div=R/ ! .W .R/-displays/ This functor commutes with base change with respect to a ring homomorphism R ! R0 which is defined for p-divisible groups and for displays with respect to the morphism of frames W .R/ ! W .R0 / (17.3.6). Theorem 17.4.2 (Gabber, [12]). Let R be a perfect ring of characteristic p. Then the last functor is an equivalence of categories. The proof by Gabber is not published. Theorem 17.4.3. We assume that p > 3. Let O D Z p and let R Example 17.2.6. Then, we have an equivalence of categories:

! k as in

O Z .R/-displays/: BT W .p-div=R/ ! .W p The functor BT has an explicit description and commutes with the duality theory which exists for both categories [10]. Let R be a complete regular local ring with perfect residue class field k of characteristic p > 0. We consider a ring homomorphism   Sd WD W .k/ ŒT1 ; : : : ; Td  ! R; (17.4.1)

which maps the indeterminates T1 ; : : : ; Td to a regular system of parameters of R. We find a power series E 2 Sd with constant term p 2 W .k/ which generates the

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T. Zink

kernel of (17.4.1). Let  be the standard Frobenius on Sd , i.e.  .Ti / D Tip . We define .Es/ P D  .s/ and obtain a Z p -frame

B D .Sd ; ESd ; R; ; /: P Theorem 17.4.4 ([11]). Let p > 3. Then, the category of p-divisible groups over the complete regular local ring R is equivalent to the category of B -displays. This theorem was conjectured by Breuil in the case dim R D 1. The conjecture was proved by Kisin [4]. It was proved by Vasiu and Zink [18] if E is an Eisenstein polynomial. The theorem implies a classification of finite flat group schemes of order a power of p over R in the same way as explained after Lemma 17.6.5. Definition 17.4.5. Let R be an O-algebra. Let .X; / be a p-divisible group X over R with an action  W O ! End X:

We call the action  strict if the action of O on Lie X induced by  coincides with the action via O ! R on the R-module Lie X . In this case, we call .X; / a p-divisible O-module. Theorem 17.4.6 ([1]). We assume that p > 3. Let R ! k be a homomorphism of O-algebras as in Example 17.2.6. Then, we have an equivalence of categories, O .R/-displays/ BT W .p-divisible O-modules=R/ ! .W O

which is compatible with base change R ! R0 .

Let X be a smooth and projective scheme over a ring R. One may ask whether i there is a display structure on the crystalline cohomology Hcris .X=W .R//. If X=R is an abelian variety, one can use Theorem 17.4.1 to define a display structure on 1 Hcris .X=W .R//. There is a category of higher W .R/-displays [7, 14]. It is an exact tensor category which contains the W .R/-displays as a full subcategory. Examples of i schemes X such that Hcris .X=R/ admits a display structure are given in [7, 8, 3, 14]. This is for instance the case if X is a smooth complete intersection in P m R relative to R. The display structure sometimes determines the deformation theory of X completely [8]. The theory of higher displays is based on the relative de Rham–Witt complex [6].

17.5 The nilpotency condition We will now discuss the nilpotency condition for F -displays. Proposition 17.5.1. Let P D .P; Q; F; FP / be an F -display. There is a unique homomorphism of S -modules, V ] W P ! P ./

399

Displays and p-divisible groups

such that V ] .FP .y// D 1 ˝ y;

V ] .F x/ D  ˝ x;

y 2 Q; x 2 P:

If we choose a normal decomposition P D T ˚L, we can describe V ] as follows. We consider the inverse matrix of (17.3.3)   AM BM W T ˚ L ! T ./ ˚ L./ : CM DM Then the map V ] is given by V

]

t l

!

D



 AM  BM CM DM

! t : l



(17.5.1)

It follows from (17.3.4) that F ] ı V ] D  idP ;

V ] ı F ] D  idP ./ :

Definition 17.5.2. Let P DD .P; Q; F; FP / be an F -display. We call P nilpotent if there is a natural number N such that the iterate of V ] , V N ] W P ! P .

N/

;

is zero modulo the ideal I C S . We may write this condition in matrix form. We set T0 D R=R ˝S T , L0 D R=R ˝S L, P0 D R=R ˝S P . If we tensor V ] with R=R, we obtain a map ]

V0 W T0 ˚ L0 ! R=R ˝Frob;R=R T0 ˚ R=R ˝Frob;R=R L0 ; which is given by the image of the matrix (17.5.1) under the morphism S ! R=R,    AM0  BM 0 : CM 0 DM 0 The first row of this matrix is zero. Therefore, the nilpotency condition says that, for large e, ./ . e 1 / ı : : : ı DM 0 ı DM 0 D 0: (17.5.2) DM 0

Here, DM 0./ D R=R ˝Frob;R=R DM 0 . Assume that the S -modules T and L are free and that we have chosen a basis in each of them. Then the matrix DM 0./ is obtained from DM 0 by raising all entries to the p-th power. The condition is equivalent to saying that the map e 1 DM . / ı : : : ı DM ./ ı DM (17.5.3)

400

T. Zink e

has image in .I C S /L. / . If p is nilpotent in R, we can iterate (17.5.3) until the image is zero modulo I . Therefore, for  nilpotent in R, we may reformulate the nilpotency condition as follows: e (NP) For large numbers e, the image of the map (17.5.3) is contained in IL. / . Equivalently we can say: (NP)’ For large numbers e, the map V e] of Definition 17.5.2 is zero modulo I . Proposition 17.5.3 ([11]). Let ˛ W F D .S; I; R; ; / P ! F 0 D .S 0 ; I 0 ; R0 ;  0 ; P 0 / be a morphism of frames which induces an isomorphism R ! R0 and a surjection S ! S 0 . We assume that p is nilpotent in R. Let c be the kernel of S ! S 0 . We assume that each finitely generated projective S 0 -module lifts to a finitely generated projective S -module. If  .c/ D 0, the base change functor is an equivalence of the category of nilpotent F -displays with the category of nilpotent F 0 -displays. If P .c/ D 0, the base change functor is an equivalence of the category of F displays with the category of F 0 -displays. We note that c  I . Therefore, P is defined on c. Consider an F -display .P1 ; Q1 ; F1 ; FP1 / and let .P; Q; F; FP / be the F 0 -display obtained by a base change. Let P D T ˚ L be a normal decomposition of P . One checks easily that there is a normal decomposition P1 D T1 ˚ L1 of P1 which lifts the normal decomposition of P . The proposition is a consequence of the following lemma. Lemma 17.5.4. Let P D .P; Q; F; FP / be a nilpotent F 0 -display with a normal decomposition P D T ˚ L. Assume that Pi D .Pi ; Qi ; Fi ; FPi / with i D 1; 2 are F -displays which lift P . Assume further that normal decompositions Pi D Ti ˚ Li are given which lift the given decomposition P D T ˚ L. Then, the morphism idP lifts uniquely to a morphism of F -displays P1 ! P2 . Proof. Since c is contained in the radical of S we may find isomorphisms of S modules T1 Š T2 resp. L1 Š L2 which lift the identity of T respectively L. We assume for simplicity that these modules are free, and we fix isomorphisms T1 Š S d and L1 Š S c and obtain induced isomorphisms T2 Š S d and L2 Š S c . If ˛ W P1 ! P2 is a morphism which lifts the identity, it is given by a matrix of the form     X Y Ed 0 C : U Z 0 Ec Here, the second matrix has coefficients in c, matrices. By (17.3.9) we have to find X , Y , following equation is satisfied,       A1 B1 X Y A1 B1 A2 C D C1 D1 C2 U Z C1 D1

and Ed respectively Ec are the unit U , Z with entries in c such that the   B2 A C 2 D2 C2

B2 D2



  .X / .Y P /   .U /  .Z/

401

Displays and p-divisible groups

By our assumption, we have  .X / D 0,  .U / D 0,  .Z/ D 0. Therefore, the previous equation becomes        X Y A1 B1 A2 A1 B2 B1 0 A2 .Y P / D C : U Z C1 D1 C2 C1 D2 B1 0 C2 .Y P / If we multiply the last equation from the right by 

A1 C1

B1 D1



1

D



AM1 CM 1

 BM 1 ; DM 1

we obtain an equation of the form       0 A2 .Y P / AM1 X Y A B C D 0 C2 .Y P / U Z C D CM 1

 BM 1 : DM 1

The first matrix on the right hand side has coefficients in c. To solve this equation for X; U; Y; Z is the same as solving the following equation for Y : Y

A2 .Y P /DM 1 D B :

(17.5.4)

We consider the assignment Y 7! A2 .Y P /DM 1 as an endomorphism of the abelian group Md;c .c/, because .c/ P  c. Existence and uniqueness follows if we show that this operator is nilpotent. If we iterate e-times U we obtain: A2
 .A2 /
 e

1

.A2 /
P e .Y /
 e

1

.DM 1 /
: : :
 .DM 1 /
DM 1 :

(17.5.5)

By the nilpotency condition we may assume that M D  e 2 .DM 1 /
: : :
 .DM 1 /
DM 1 has coefficients in the ideal I . We set N D P e 1 .Y /. This is a matrix with coefficients in c and is therefore annihilated by  . We have .N P / .M / D P .NM / D  .N /.M P / D 0: But this shows that (17.5.5) is equal to 0, which means that U is nilpotent. We remark that, in the case P .c/ D 0, Equation (17.5.4) is trivial. Therefore, we need no nilpotency condition to solve it. This proves the last part of the proposition.  The following theorem was first proved in [20] for O D Z p and an additional assumption on R. The additional assumptions were removed in [9]. Theorem 17.5.5 ([1]). Let R be an O-algebra such that  is nilpotent in R. Then, there is an equivalence of categories BT W .formal p-divisible O-modules=R/ ! .nilpotent WO .R/-displays/:

402

T. Zink

Let S ! R be a morphism of O-algebras as in Example 17.2.7 for frames and such that  is nilpotent in R. Proposition 17.5.3 implies that the base change functor BT W .nilpotent WO .S=R/-displays/ ! .nilpotent WO .R/-displays/ is an equivalence of categories. Let P be a nilpotent WO .R/-display. It is obtained by a base change from a WO .S=R/-display PQ which is canonically defined. We set DP .S / D PQ =IO .S /PQ : This is a crystal on the category of morphism S ! R as above. Starting with a formal p-divisible O-module X , Grothendieck and Messing [15] have associated to X a crystal DX via the theory of the universal extension. It is proved in [1] that for P D BT X we have a canonical isomorphism DX .S / Š DP .S /: This is a main ingredient when proving Theorem 17.5.5.

17.6 Isogenies Let F be a frame as above. Let ˛ W P ! P 0 be a morphism of F -displays of the same height. We assume that P and P 0 are free S -modules. If we choose a basis in each of these modules, then det ˛ 2 S is defined. Up to a unit, it is independent of the choice of the basis. Lemma 17.6.1. Let ˛ W P ! P 0 be a morphism of F -displays. Assume that the modules P and P 0 are free of the same rank h, and that P =Q and P 0 =Q0 are free of the same rank d . Then, we have:  .det ˛/ D  det ˛: for some unit  2 S .

Proof. The morphism ˛ may be expressed by a matrix. In the notation of (17.3.9), we have the identity      .X / .Y P /  .X /  .Y P / det D det (17.6.1)   .U /  .Z/  .U /  .Z/ Indeed, let S be any ring and  2 S . Then, we have for any block matrix with quadratic blocks on the diagonal that     A11 A12 A11 A12 : D det det A21 A22 A21 A22 Indeed, this follows by reduction to a universal case where  is not a zero divisor.

Displays and p-divisible groups

403

The right hand side of (17.6.1) is  .det ˛/ because  .Y P / D  .Y /. Therefore, the claim follows from (17.3.9).  Proposition 17.6.2. Let R be an O-algebra such that  is nilpotent in R. We assume that Spec R is connected. Let P D .P; Q; F; FP / and P 0 D .P 0 ; Q0 ; F 0 ; FP 0 / be WO .R/-displays. We assume that P and P 0 are free WO .R/-modules and that P =Q and P 0 =Q0 are free R-modules. (Locally on Spec R this is fulfilled.) Let ˛ W P ! P 0 be a morphism of displays such that det ˛ ¤ 0. Then, there is u 2 Z >0 and a unit  2 WO .R/ such that det ˛ D  u : If det ˛ ¤ 0, we call ˛ an isogeny of displays. The integer u is called the height of the isogeny ˛. Proof. We begin with the case R D 0. We set  D det ˛. By the last proposition, we find F  D 
 for some  2 WO .R/: (17.6.2) We write  D

Vt

 where w0 ./ ¤ 0. We claim that .17:6:2/ implies F

Ft

 D


:

(17.6.3)

To verify this, we may assume that t > 0. We obtain FVt

 D 

Vt

D

Vt

t

. F /:

Since R D 0, the operators F and V acting on W .R/ commute. Therefore, we deduce (17.6.3) t Let w0 ./ D x and w0 . F / D e 2 R . We apply w0 to Equation (17.6.3) and obtain x q D ex: (17.6.4) Since the product

x.x q

1

e/ D 0

has relatively prime factors, it follows that D.x/ [ D.x q

D.x/ \ D.x

1

q 1

e/ D Spec R

e/ D ¿:

Hence by connectedness either D.x/ D Spec R or D.x/ D ¿. In the first case, x is nilpotent. But then we find x D 0, by iterating Equation (17.6.4). This is a contradiction to our choices. Therefore, D.x/ D ¿ and x is a unit. Then,  is a unit too. We find t t Ft  D F V  D  t :

404

T. Zink t

But by (17.6.2), F  may be expressed as the product of  with a unit. This proves the result in the case R D 0. We consider an epimorphism of rings R ! S such that the kernel a satisfies a D a2 D 0. It suffices to show that the assertion of the proposition holds for R if it holds for the base change det ˛S over S . By assumption, we have det ˛S D  u N ; for some unit N 2 W .S /. We choose a lift  2 W .R/ of N . We obtain det ˛

 u  2 W .a/:

If we apply the Frobenius of W .R/, we obtain F

det ˛ D  u F :

But, on the other hand, there is a unit  2 W .R/ such that F

det ˛ D  det ˛: 

The last two equations give the result.

Definition 17.6.3. Let R be a connected O-algebra such that  is nilpotent in R. Let ˛ W P ! P 0 be a morphism of WO .R/-displays of the same height. The morphism ˛ is called an isogeny if locally on Spec R we have det ˛ D  u ;

u 2 Z >0 ;  2 WO .R/

for some unit  2 WO .R/. The non-negative number u is defined locally on Spec R, but if Spec R is connected, it is well defined globally. We call u the height of the isogeny. Proposition 17.6.4. Let R be an O-algebra such that  is nilpotent in R and such that the ideal of nilpotent elements of R is nilpotent. Let ˛ W P ! P 0 be an isogeny of WO .R/-displays of height u. Then, there exists locally on Spec R a morphism of WO .R/-displays ˇ W P 0 ! P , such that ˇ ı ˛ D  u idP ;

˛ ı ˇ D  u idP 0 :

Proof. We choose normal decompositions P D T ˚ L;

P 0 D T 0 ˚ L0 :

Since the question is local, we may assume that all these WOF .R/-modules are free, T Š WOF .R/d Š T 0 , L Š WOF .R/c Š L0 . We write h D d C c for the height of the displays. We represent ˛ by a block matrix, !!  !  t t X VY ˛ D : (17.6.5) U Z ` `

405

Displays and p-divisible groups

We note that X is a quadratic d d matrix and Z is a quadratic c c matrix. We set   X VY M˛ D : (17.6.6) U Z For an arbitrary block matrix with entries in a ring of Witt vectors WOF .R/ of the type above, we define  F  X Y s : M˛ D  FU FZ

We will denote by Ed the unit matrix of size d  d . If  is invertible in R, we have the relation     1Ed 0 1Ed 0 s F : M˛ M˛ D 0 Ec 0  1 Ec By reduction to a universal case, we conclude that det sM˛ D det FM˛ D

F

det M˛ ;

and that M˛ 7! sM˛ is a homomorphism of matrix algebras. Let M 2 M.h  h; S / be a matrix with entries in a commutative ring S . Then, we denote by adM the adjoint matrix. We have M adM D adMM D .det M /Eh ;

ad

.M1 M2 / D adM2 adM1 :

If A 2 M.h  h; S / is an invertible matrix, we have ad

.AMA

1

/ D A adMA

1

:

This again is proved by reduction to a universal case. Moreover, for a matrix M˛ as above, the adjoint matrix (17.6.6). We have ad F

. M˛ / D

F ad

. M˛ /;

ad

M˛ is again of type

ad s

. M˛ / D s .adM˛ /:

The first equation is clear because F is a ring homomorphism and the second equation is a consequence of the first one. We write FP W IR T ˚ L ! T ˚ L in matrix form !! !   V t t A B FP D : F` C D ` The quadratic block matrix on the right hand side will be denoted by ˆ. In the same way, we define with respect to P 0 the matrix   0 A B0 : ˆ0 D C 0 D0

406

T. Zink

The matrix M˛ defines a morphism of displays iff   F    0 X A B A B0 X VY D C 0 D0 C D U Z F U In our notation, this reads

 Y : F Z

M˛ ˆ D ˆ0 s M˛ :

(17.6.7)

 u  det ˆ D .det ˆ0 / F  u :

(17.6.8)

u

There is a unit  2 WOF .R/ such that det M˛ D  . Now, we obtain from (17.6.7) We take the adjoint of Equation (17.6.7), ˆ adM˛ D s.adM˛ / adˆ0 :

ad

Since ˆ and ˆ0 are invertible, we conclude from the last equation .det ˆ/ adM˛ ˆ0 D .det ˆ0 /ˆ s.adM˛ /: Let us assume that the ring R is reduced. Then,  is not a zero divisor in WOF .R/. In this case, we obtain from Equation (17.6.8)  det ˆ D

F

 det ˆ0 :

(17.6.9)

If we insert this in the latter equation, we obtain 

M˛ ˆ0 D ˆ F

1 ad

Therefore, the matrix N˛ D 

1 s ad

. M˛ /:

1 ad

M˛ defines the desired morphism ˇ W P0 ! P:

Finally, we consider the case where R is not reduced. We consider the morphism R ! Rred . Let 0 2 WOF .Rred / be the image of  2 WOF .R/. Then, 0 is a solution of Equation (17.6.9) in the ring WOF .Rred /. By Lemma 17.6.5 below there is a unique lifting 1 2 WOF .R/ of 0 such that (17.6.9) holds with  replaced by 1 . As above, we see that NQ ˛ WD 1 1 adM˛ defines a morphism ˇ W P 0 ! P . We obtain M˛ ı NQ ˛ D NQ ˛ ı M˛ D 1 1  u Eh :

(17.6.10)

We denote by a the kernel of R ! Rred . We may write 1 1  D 1 C , where  2 WOF .a/. Then, the matrix .1 C / u Eh defines by (17.6.10) an endomorphism of P (resp. P 0 ) for the coordinates we have chosen. Since this matrix commutes with ˆ (resp. ˆ0 ), the matrix equation for a morphism (compare (17.6.9)) reads in our case .1 C F / u Eh D .1 C / u Eh : Since all entries of the Witt vector  WD  u are nilpotent, the equation  D F  implies that  D 0. Therefore, the right hand side of (17.6.10) is  u Eh . The proof of the proposition is finished by the following lemma. 

Displays and p-divisible groups

407

Lemma 17.6.5. Let R ! S be an epimorphism of OF -algebras such that  is nilpotent in R. We assume that the kernel c is nilpotent. Let u 2 WO .R/ and denote F its image in WO .S / by u. N Let N 2 WO .S / be a solution of the equation F

F

uN F N D N : Then, there is a unique  2 WOF .R/ which lifts N such that u F  D : We omit the easy proof. Let R be an Artinian local ring whose residue class field has characteristic p. Let G be a finite flat local group scheme over R of order a power of p. We call this simply a finite flat local p-group. By Raynaud, there is an exact sequence 0 ! G ! X 0 ! X 1 ! 0; where X 0 and X 1 are formal p-divisible groups. One obtains a fully faithful functor from the category of finite flat local group schemes to the derived category of bounded complexes of formal p-divisible groups [13]. By Theorem 17.5.5, the last category is equivalent to the derived category D b .nilDsp/ of bounded complexes of nilpotent W .R/-displays. The resulting functor .finite flat local p-groups=R/ ! D b .nilDsp/ has as essential image the isogenies P 0 ! P 1 . This is a general scheme to classify O O .R/finite flat group schemes. If the residue class field of R is perfect we can use W displays to classify all finite flat p-groups not only local ones.

References [1] T. Ahsendorf, C. Cheng and T. Zink, O-displays and -divisible formal O-modules, J. Algebra 457 (2016), 129–193. [2] N. Bourbaki, Algébre Commutative, Chapitre 9, Masson 1983. [3] O. Gregory and A. Langer, Higher displays arising from filtered de Rham–Witt complexes, preprint, 2016. [4] M. Kisin, Crystalline representations and F -crystals, Algebraic geometry and number theory, Progr. Math. 253 (2006), 459–496. [5] S. Kudla, M. Rapoport and T. Zink, On Cherednik uniformization, in preparation. [6] A. Langer and T. Zink, De Rham–Witt cohomology for a proper and smooth morphism, J. Inst. Math. Jussieu 3 (2) (2004), 231–314. [7] A. Langer and T. Zink, De Rham–Witt cohomology and displays, Docum. Math. 12 (2007), 147–191.

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[8] A. Langer and T. Zink, Grothendieck–Messing deformation theory for varieties of K3-type, TJM 1 (2019), 455–517. [9] E. Lau, Displays and formal p-divisible groups. Invent. Math. 171 (2008), 617–628. [10] E. Lau, A duality theorem for Dieudonné displays, Annales ENS 4e série t.42 (2009), 241– 259. [11] E. Lau, Frames and finite group schemes over complete regular local rings. Docum. Math. 15 (2010), 545–569. [12] E. Lau, Smoothness of the truncated display functor, J. Amer. Math. Soc. 26 (2013), 129–165. [13] E. Lau, Relations between Dieudonné displays and crystalline Dieudonné theory. Algebra Number Theory 8 (2014), 2201–2262. [14] E. Lau, Higher frames and G-displays, preprint, 2016. [15] W. Messing, The Crystals Associated to Barsotti–Tate Groups, LNM 264, Springer, Berlin, 1972. [16] W. Messing, Travaux de Zink. Séminaire Bourbaki 2005/2006, exp. 964, Astérisque 311 (2007), 341–364. [17] M. Rapoport and T. Zink, On the Drinfeld moduli problem of p-divisible groups, Cambridge J. Math. 5 (2017), 229–279. [18] A. Vasiu and T. Zink, Breuil’s classification of p-divisible groups over regular rings of arbitrary dimension. In Proceedings of Algebraic and Arithmetic Structures of Moduli Spaces (I. Nakamura and L. Weng, eds.), Advanced Studies in Pure Mathematics 58 (2010), 461–479. [19] A. Vasiu and T. Zink, Purity results for p-divisible groups and abelian schemes over regular bases of mixed characteristic, Docum. Math. 15 (2010), 571–599. [20] T. Zink, The display of a formal p-divisible group. Astérisque 278 (2002), 127–248. [21] T. Zink, A Dieudonné theory for p-divisible groups, Class field theory — its centenary and prospect, Adv. Stud. Pure Math 30 (2001), 139–160. [22] T. Zink, De Jong–Oort purity for p-divisible groups. In Algebra, Arithmetic, Geometry (Y. Tschinkel, and Y. Zarhin, eds.), Birkhäuser, Basel, 2009, 693–701.

List of contributors Michael Baake, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Viorel Barbu, Octav Mayer Institute of Mathematics, Romanian Academy, Bd. CAROL I, nr. 11, 700506 Ia¸si, Romania, email: [email protected] Barbara Baumeister, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Wolf-Jürgen Beyn, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Vladimir I. Bogachev, Department of Mechanics and Mathematics, Moscow State University, GSP-1, 1 Leninskiye Gory, 119991 Moscow, Russia, email: [email protected] Kai-Uwe Bux, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Martin Callies, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld Franz Gähler, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Barbara Gentz, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Friedrich Götze, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Alexander Grigor’yan, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Andriy Haydys, Institut für Mathematik, Albert-Ludwigs-Universität Freiburg, ErnstZermelo-Str. 1, 79104 Freiburg, email: [email protected] Sebastian Herr, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Werner Hoffmann, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected]

410

List of contributors

Christian Huck, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Moritz Kassmann, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Dawid Kielak, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Yuri Kondratiev, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Holger Kösters, Institut für Mathematik, Universität Rostock, Ulmenstr. 69, Haus 3, 18057, Bielefeld, email: [email protected] Oleksandr Kutoviy, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Henning Krause, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Andreas Nickel, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Str. 9, 45127, Essen, email: [email protected] Denny Otten, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Michael Röckner, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Stanislav V. Shaposhnikov, Department of Mechanics and Mathematics, Moscow State University, GSP-1, 1 Leninskiye Gory, 119991 Moscow, Russia, email: [email protected] Greg Stevenson, School of Mathematics & Statistics, University of Glasgow, University Place, G12 8QQ, Glasgow, United Kingdom, email: [email protected] Pavlo Tkachov, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Christopher Voll, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected] Peter Zeiner, Department of Mathematics, Xiamen University Malaysia, Jalan Sunsuria, Bandar Sunsuria, 43900 Sepang, Selangor, Malaysia, email: [email protected] Thomas Zink, Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, email: [email protected]

Index ˛-stable measure, 191 nondegenerate, 191 ˛-stable process, 184 abscissa of convergence, 330, 345, 358 absolute order, 250, 260 activation energy, 119 adeles, 326 Allen–Cahn stochastic partial differential equation, 118, 120 analytic number theory, 165, 359 anti-unit, 369, 370 arithmetic subgroup, 302, 334, 355, 359 Arrhenius Law, 112 Artin L-function p-adic, 381 complex, 367 equivariant, 367 Asplund’s theorem, 44 asymptotic approximation, 19, 21–23 asymptotic expansion, 22, 113 asymptotic freeness, 4, 11 asymptotics exponential, 111, 112, 118 subexponential, 112, 113, 116, 120, 122 atomic space, 168 autocorrelation, 199–201, 203, 208 automorphic form, 334, 335 automorphic representation, 335 BDLP model, 88, 92 bifurcation, 113, 119, 120 pitchfork, 113, 114, 116, 119 symmetry breaking, 115, 117 birth-and-death process, 84 boundary conditions Dirichlet, 172 Neumann, 118–120 periodic, 119, 173 braid

diagram, 310 strand diagram, 249, 310 braid group, 247 Artin presentation of, 248 braided Thompson group, 311 Brownian motion, 40, 55, 63, 107, 117, 120, 188 Bruhat–Tits building, 351 Brumer’s conjecture abelian, 368, 370 non-abelian, 375 weak, 377 Brumer–Stark conjecture abelian, 369, 370 non-abelian, 376 weak, 377 Brumer–Stark property strong, 378, 380 weak, 379 canard solution, 125 capacity, 65 generalised, 65 Newtonian, 116 Catalan number, 2, 238 generalised Catalan number, 261 Cauchy problem, 31–33, 35, 36, 38, 130, 131, 133, 144, 145 central conductor, 372, 377, 379 central limit theorem, 4, 19, 20, 241 entropic, 22 free, 4, 21, 23 centraliser, 147, 257, 339 chain complex, 69 character, 325 cyclotomic, 381 monomial, 382 Teichmüller, 381 unramified, 326 characteristic boundary, 107 circular law, 9

412

Index

class group, 368, 375 ray class group, 367, 378 classifying space, 299 CM-extension, 369 co-moving frame equation, 145 coincidence rotation, 214 coincidence site lattice, see lattice combinatorial Morse theory, 303 cube complex, 303 descending link, 304 level set, 303 Morse function, 303 piecewise Euclidean complex, 303 piecewise spherical complex, 303 sublevel complex, 303 sublevel set, 303 commensurable, 334 comparison principle, 94 computer algebra package Zeta, 348 configuration space, 77, 247 congruence subgroup, 335, 355 conjecture Charney’s, 257 Oppenheim’s, 18 smashing, 280 connected sum, 58 connectivity properties d -spherical, 319 l-connected, 301 essentially l-connected, 302 essentially homotopically trivial, 302 continuity equation, 41 contour method, 129, 148, 153–155 coordinate system adapted, 174 conjugate, 223 special holomorphic, 223 core, 210 correlation function, 15, 78 Coulomb potential, 178 coupling strength, 113 Coxeter element, 260 quasi-Coxeter element, 266

Coxeter group, 256, 286, 361 Coxeter system, 256 dual, 260 cristalline cohomology, 398 curvature constant, 228 of a metric space, 257 principal, 164 Ricci, 58 cut and project, 198, 199 cycling, 122, 125 cylindric functions, 78 decoupling conjecture, 165 Dedekind zeta function, see zeta function special value, 359 Deligne–Ribet power series, 381 denominator ideal, 371–373 Dieudonné theory, 389 differential equation ordinary, 107 partial, 110, 129, 159 partial Allen–Cahn, 118 stochastic, 44, 107, 109 differential gradient system, 41 diffraction spectrum, see spectrum diffusion, 40, 63, 96, 107 exit, 110 non-reversible, 121 reversible, 122 diffusion matrix, 31 digraph, 68 Dini continuity, 31 Diophantine approximation, 16, 17 Dirac comb, 199, 201, 202 Dirac equation cubic, 159, 173 Soler model, 173 Thirring model, 173 Dirac–Klein–Gordon system, 159, 177 Dirichlet form, 63, 66, 116 generating function, see generating function map, 382

Index

problem, 110, 187 series, 213, 214, 216, 326, 345 dispersion, 166 kernel, 92 set, 148, 151 dispersive, 159 display, 393 distribution, 240 of algebraic numbers, 16 of eigenvalues of random matrices, see spectral distribution of irrational quadratic forms, 18 of zeros of random polynomials, 16 dual Matsumoto property, 262 dynamical system, 107, 129, 172, 197, 198, 202 continuous-time, 107 critical manifold, 125 measure-theoretic, 203 normal form, 115 slow manifold, 125 slow–fast, 125 randomly perturbed, 125 topological, 201, 211 dynamical spectrum, see spectrum dynamics global, 125 gradient, 107, 112, 113, 117 local, 125

413

equivariant Tamagawa number conjecture, 379, 384 Euler characteristic, 231, 239, 306, 379 form, 267 product, 215, 327, 347, 356, 357 totient function, 359 Euler–Mascheroni constant, 217 Evans function, 148 evolution equation, 75, 129 exceptional sequence, 270, 271 exit problem, 110 exponential asymptotics, 111, 112, 118 exponential sum, 165, 167

factor, 204 factor map, 204, 205, 208 factorisation, 216, 251, 347 T -reduced, 261 filiform Lie ring, see Lie ring finite-time extinction, 41 finiteness properties, 300 Brown’s criterion, 301 higher generation by subgroups, 252 type Fm , 300 first exit location, 109, 111, 112, 122 time, 109, 111, 112, 122 first passage time, 122 eigenfunction, 146, 147, 149, 154, 166, first-hitting time, 112 167, 212 first-passage time, 123, 125 elliptic Harnack inequality, 65 Fisher information, 23 elliptic law, 10 Fokker–Planck equation, 81, 108 ellipticity Fokker–Planck–Kolmogorov equation, uniform, 109 29, 31, 34 energy form, 184 parabolic, 29 energy functional, 118, 161 folded node singularity, 125 energy space, 161 Fourier entropic limit theorem, 5, 22 coefficient, 164 equilibrium extension, 163 asymptotically stable, 111, 112 restriction, 162 distribution, 108 series, 119, 165 metastable, 107 equivariance, 129, 271, 328, 392 symbol, 151

414

Index

transform, 68, 162, 184, 199, 200, 203, 209, 240, 328, 329 fractals, 65 fractional Laplace operator, 184 fractional derivatives, 68 fractional Laplace operator, 183 free additive convolution, 244 free convolution, 4, 20, 21 free multiplicative convolution, 243 free nilpotent group, see nilpotent group free probability, 4, 11, 20, 21, 23, 241 freeness, 4, 241 freezing method, 129, 133–136, 139, 141, 149, 152 functional equation for Igusa’s local zeta function, 354 for zeta integrals, 329 global, 329 local, 348 Galerkin approximation, 119 Gaussian measure, 35 Gelfand triple, 44, 46, 47 generalised adjoint matrix, 371 generalised Cartan lattice, 269 generating function, 213–215, 246 Dirichlet, 345 geodesic, 167, 250, 306, 323 trapped, 166 geometric dimension, 300 Ginibre matrix, 14 Ginzburg–Landau equation quintic-cubic, 138 Girko–Ginibre matrix, 8, 9 global maximum principle, 183 Green function, 57, 58 Gross’ conjecture first part, 382 second part, 383 Gross–Pitaevskii hierarchy, 173 Grothendieck group, 267 group algebraic, 324, 334, 355 dihedral, 372, 374, 386 Frobenius, 385

monomial, 374, 385 nilpotent, 348, 355, 374 of affine transformations, 373, 380, 385 symmetric, 371, 373, 385 group scheme, 389 unipotent, 356 growth polynomial representation, 355 polynomial subgroup, 345 polynomial subring, 345 subgroup, 345 subring, 345 GUE matrix, 12 Gumbel distribution, 123 Hölder regularity, 188, 189, 191 Haar measure, 68, 199, 201, 325, 326, 354 Hall basis, 353 Hamiltonian PDE, 132 Hamiltonian structure, 172 Hardy–Littlewood–Sobolev, 164 harmonic analysis, 159 Harnack’s inequality, 39 elliptic, 65 parabolic, 64 Hausdorff dimension, 64 series, 356 heat semigroup, 67 heat kernel, 58, 60, 67, 68 heavy tail, 101 Hecke operator, 336 height function, 16, 302, 330 height lattice, 206 higher generation by subgroups, 308 m-generating, 308 coset complex, 308 Hilbert space, 39, 67, 134, 185, 208, 336 holomorphic cubic form, 223 holomorphic prepotential, 223 homogeneity condition, 350

Index

homology theory on graphs, 68 hull continuous, 211 discrete, 210 Hurwitz action, 262 hybrid group ring, 375, 384 icosian ring, 216 idele norm, 326 Igusa’s local zeta function, 354 independence, 241 infinite-dimensional diffusion, 34 infinitesimal generator, 109 inflation, 204 inflation multiplier, 206 integrable solution, 32 integrality conjecture, 373 weak, 373 integrality ring, 371–373 integration p-adic, 351 fractional, 164 interaction strong, 136 weak, 136 interpolation, 164 property, 381 invariant density, 108 measure, 108, 325 positively, 110 unitarily, 12 Iwahori-Hecke algebra, 264 Iwasawa main conjecture, 384 jump kernel, 63 process, 66, 184, 190 Künneth formula, 70 Kazdan–Warner equation, 226 kinetic equation, 91, 92 Kolmogorov backward equation, 81 forward equation, 81, 108

415

operator, 29 Korteweg–de Vries equation, 152, 179 Kramers’ Law, 113, 118, 119 Kronecker factor, 205, 212 L-function Artin, 367 automorphic, 327 Lévy process, 67, 184, 188, 190 Lagrangian immersion, 224 Laplace–Beltrami operator, 55, 165 large deviations, 111, 122, 125 action functional, 111 principle, 111, 112, 118 quasipotential, 111, 122 rate function, 111, 114, 123 theory, 112 Wentzell–Freidlin theory, 111 lattice, 165, 199, 213, 216, 235 coincidence, 198, 213 coincidence site lattice, 215 enumeration, 197 join, 236 meet, 236 non-crossing partition lattice, 235 planar, 214 point problem, 17 rational, 217 similar, 198, 213 well-rounded, 216 Lebesgue–Poisson measure, 79 Li–Yau estimate, 59 Lie algebra, 130, 335, 355 bracket, 146 group, 129 lattice, 350, 356 ring, 348 filiform, 349 maximal class, 349 Littlewood–Paley theory, 166 local boundedness of solution densities, 31

416

Index

local functional equation, see functional equation local semi-circle law, 7 long range potential, 61 Lorentz covariance, 173 Lorenz gauge, 178 Lyapunov exponent, 123 Lyapunov function, 32, 39, 152 -invariant, 384 Mal’cev correspondence, 348 manifold compact, 165 critical, 125 Kähler, 221 Riemannian, 55, 165 slow, 125 Zoll, 167 Markov chain, 123, 125 generator, 85 kernel, 125 operator, 67 process, 66, 81, 107 martingale problem, 38 mathematical physics, 34, 173 maximal class Lie ring, see Lie ring maximal compact subgroup, 330 maximal equicontinuous factor, 202, 204, 205 measure ergodic, 198, 212 Hausdorff, 163 positive definite, 200 probability, 2, 31, 78, 198, 202, 240 pure point, 200–202, 208 Radon, 63, 77 spectral, 208 translation bounded, 199, 202 meromorphic continuation, 328, 330, 348, 358 metastable equilibrium, 107 metric of constant negative curvature, 228 Poincaré, 229

special Kähler, 221 Mikado braid, 263 mixed-mode oscillations, 125 mixed spectrum, 203 model set, 197 regular, 198 weak, 198, 201, 203 Morris–Lecar model, 122 mutual local derivability, 204 negative eigenvalues, 56 nilpotent group, 348, 355, 374 noise, 44, 107 isotropic, 112 noise-induced exit, 110 phenomena, 107 transition, 117 non-crossing partition, 2 non-Gaussian solution, 35 non-parabolic, 59 nonlinear diffusion theory, 41 nonlinear dispersive equations, 159 well-posedness, 160 normal form, 115, 310 null structure, 174, 175 number of integer lattice points, 165 numerical scheme, 179 observables, 78 operator integrodifferential, 183 local, 183 nonlocal, 184, 186, 187 orbital integral, 336 weighted, 325 order of vanishing, 368, 382 orthogonality, 134, 167 orthoscheme metric, 257 p-adic distance, 66–68 integration, see integration Lie extension, 381 p-divisible groups, 389

Index

parabolic, 55, 59 equation, 132 Harnack inequality, 64 parabolic-hyperbolic equation, 132 @-invariant path, 68 partition, 235 block, 235 non-crossing, 2, 236, 286 Kreweras complement, 236 non-crossing partition in a Coxeter system, 261 refinement, 235 pathwise uniqueness, 49 periodic orbit stable, 122 unstable, 121, 122 phase condition, 134 fixed, 134 pitchfork bifurcation, 113, 114, 116, 119 Poincaré inequality, 6, 65 Poincaré map random, 123, 125 Poisson measure, 78 Poisson problem, 110 polynomial growth, see growth polynomial representation growth, see growth polynomial subgroup growth, see growth polynomial subring growth, see growth poset, 235 Möbius function, 238 non-crossing partition lattice, 235 order complex, 239 positively invariant, 110 potential theory, 113, 116 prehomogeneous vector space, 324, 360 regular, 328 probability solution, 32–37 products of random matrices, 10, 14 profinite completion, 355 projection boundary conditions, 153, 154 pseudo-differential operator, 38 quadratic form, 216 quantum field theory, 173 quantum many-body system, 173

417

quasi-observables, 80 quasipotential, 111, 112, 122 quiver, 267, 276 R-transform of Voiculescu, 245 random matrix, 1, 240 random perturbation, 107 rate of convergence, 6, 21–23 reconstruction equation, 134 reduced norm, 324, 371 reductive group, 334 regularity class, 148 critical, 160 subcritical, 160 supercritical, 160 regulator, 383 relative entropy, 22 relative equilibrium, 131 relative invariant, 324, 360 relaxation time, 112 representation rigid, see rigid representation zeta function, see zeta function residence time, 122 resonance, 122, 177, 179 rigid, 355 root lattice, 213, 214, 216 root system, 259 crystallographic, 260 rotating wave, 129, 133, 135, 137, 139, 147–149 round sphere, 167

S -lattice, 216 S -transform of Voiculescu, 246 saddle non-quadratic, 115, 120 quadratic, 115 quartic, 115, 116 scaling, 160 factor, 206 Vlasov, 88 scattering, 174, 177 Schrödinger equation

418

Index

energy, 161 energy-critical, 161 linear, 161 nonlinear, 142, 159 Schrödinger operator, 56, 60 Schwartz distribution, 44 second order equation, 152 evolution system, 129 operator, 38 PDAE, 140 system, 141 semi-circle law, 2 semi-linear elliptic inequalities, 55 semi-relativistic Hartree, 178 Shannon entropy, 5, 22 shift space, 210, 211 Sierpinski gasket, 64 silver mean, 206 singular drift, 61 Sobolev embedding theorem, 32, 132, 133, 148 space, 41, 130, 159 Sobolev–Slobodeckij space, 184, 185 solitary waves, 142 solution densities, 38 spectral distribution, 1, 9 global, 5, 8, 12 local, 5, 12 spectral gap, 125, 145 spectral measure, 20, 208, 245 spectral projector, 166 spectrum, 132, 145, 166, 198 diffraction, 202, 203, 209, 212 dynamical, 198, 201, 209 essential, 145, 147–149 point, 145, 149, 155 pure point, 202, 203, 205, 211 spherical harmonics, 167 spinor, 173 stabiliser, 131, 301, 325, 330 stability orbital, 144, 145 with asymptotic phase, 144, 145

stable-like bound, 64 stationary phase, 164 solution, 118 state, 118, 119 Stickelberger element, 368, 370, 371, 373–379 Stickelberger’s theorem, 368 Stieltjes transform, 2 stochastic Allen–Cahn equation, 118, 120 completeness, 66 differential equation, 44, 107, 109 infinite-dimensional equation, 42 partial differential equation, 34 Allen–Cahn, 118, 120 parabolic, 119 porous media equation, 44 reaction diffusion equation, 49 resonance, 122 total variation flow, 29 variational inequality, 42 Strichartz estimates, 164, 166 strong solution, 50, 130 strong Stark conjecture, 379 sub-Gaussian bound, 64 sub-Poissonian, 83 subcategory coreflective, 268, 277 localising, 277 thick, 268, 277 subgroup growth, see growth subring growth, see growth substitution, 204 constant-length, 203 random, 211 rule, 210 sums of squares, 167 Tate motive, 379 Tauberian theorem, 348 theorem classical central limit theorem, 241 free central limit theorem, 243 Hall’s theorem, 239

Index

nerve theorem, 253 topological combinatorics arc matching complex, 306 chess board complex, 305 matching, 305 matching complex, 305 topology local, 210 vague, 77, 201 torus, 167 flat, 167 irrational, 167, 172 split, 336 total variation based image restoration model, 41 total variation flow, 41 transition kernel, 123 noise-induced, 107, 117 optimal path, 112, 114, 118, 123 probability density, 108 rate, 119 state, 118, 119 time, 115, 116, 118, 120 travelling wave, 96, 132, 135, 139, 140, 147–150, 154 truncation parameter, 331, 337 twist-equivalence, 355 twist-isoclass, 356

419

Vlasov equation, 91, 92 Vlasov scaling, 88 Voiculescu’s free entropy, 5, 23 walk dimension, 64 weak solution, 50, 82, 152, 187, 188 weak turbulence, 172 Wentzell–Freidlin theory, 111 Weyl group, 267, 269, 286 Weyl group statistics, 361 Wiener process, 44 cylindrical, 118 Wigner matrix, 1 Wigner measure, 243 Witt vectors, 391 Yamada–Watanabe theorem, 50 Yang–Mills equation, 178 Yosida approximation, 43 Yukawa potential, 178

Zakharov system, 159, 178 Zakharov–Kuznetsov equation, 179 zeta function, 345 associated with a prehomogeneous vector space, 326 Dedekind, 214–216, 346 functional equation, 329, 348 graded subobject, 354 ideal, 347 normal, 348 ultra-metric space, 66, 67 proisomorphic, 355 unipotent flow, 17, 19 representation, 355 unipotent group scheme, see group Riemann, 16, 214, 215 scheme Solomon’s, 347 uniqueness of semigroups, 38 zeta integral uniqueness of solutions, 38, 160 associated with a prehomogeneous universal distribution, 2, 5, 9, 13–15 vector space, 325, 326 upper rate function, 63 functional equation, 329 variational principle, 111, 112, 116 local, 327 variational solution, 41, 185 meromorphic continuation, 330 visible lattice points, 198 truncated, 331

Michael Baake, Friedrich Götze and Werner Hoffmann, Editors

This book is a collection of survey articles about spectral structures and the application of topological methods bridging different mathematical disciplines, from pure to applied. The topics are based on work done in the Collaborative Research Centre (SFB) 701. Notable examples are non-crossing partitions, which connect representation theory, braid groups, non-commutative probability as well as spectral distributions of random matrices. The local distributions of such spectra are universal, also representing the local distribution of zeros of L-functions in number theory. An overarching method is the use of zeta functions in the asymptotic counting of sublattices, group representations etc. Further examples connecting probability, analysis, dynamical systems and geometry are generating operators of deterministic or stochastic processes, stochastic differential equations, and fractals, relating them to the local geometry of such spaces and the convergence to stable and semi-stable states.

ISBN 978-3-03719-197-2

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SCR Baake et al. | Egyptienne F | Pantone 116, 287 | RB 30 mm

Spectral Structures and Topological Methods in Mathematics

Spectral Structures and Topological Methods in Mathematics

Michael Baake, Friedrich Götze and Werner Hoffmann, Editors

Series of Congress Reports

Series of Congress Reports

Spectral Structures and Topological Methods in Mathematics Michael Baake Friedrich Götze Werner Hoffmann Editors