Challenges For The 21st Century, Procs Of The Intl Conf On Fundamental Sciences: Mathematics And Theoretical Physics 9789812811264, 9812811265

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CHALLENGES FOR THE ST

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FUNDAMENTAL SCIENCES: 4

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H Y Chen, J Packer Jesudason, C H Lai, C H Oh, K K Phua & Eng-Chye Tan

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CHALLENGES FOR THE

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CHALLENGES FOR THE

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Editors Louis H Y Chen, J Packer Jesudason, C H Lai, C H Oh, K K Phua & Eng-Chye Tan National University of Singapore

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Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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CHALLENGES FOR THE 21s" CENTURY International Conference on Fundamental Sciences: Mathematics and Theoretical Physics Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4646-3

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M E S S A G E F R O M D I R E C T O R OF N E W T O N I N S T I T U T E

In March 1998, the Isaac Newton Institute for Mathematical Sciences was privileged to receive a visit from Deputy Prime Minister Tony Tan, who invited the Institute to collaborate with the National University of Singapore in organising a millennial conference on Mathematics and Theoretical Physics. I was happy to be involved in this exciting venture, which came to fruition exactly two years later. This volume, containing a wide range of papers presented at the Conference, reflects both the international, indeed global, nature of research in the subject, and the symbiotic manner in which mathematics can interact with theoretical physics to the mutual benefit of both.

Keith Moffatt Director, Newton

Institute

M E S S A G E F R O M D I R E C T O R OF R E S E A R C H A N D F O R M E R D E A N OF SCIENCE, NATIONAL U N I V E R S I T Y OF S I N G A P O R E

I recall that the Deputy Prime Minister, Dr Tony Tan, mooted the idea that the Faculty of Science, National University of Singapore, host two International Conferences on Fundamental Sciences — one on Mathematics & Physics and another on Chemistry & Biology — over a lunch meeting with a few of my colleagues and me, while I was Dean of Science. He also set three objectives for the Conferences: (a) scientific sessions for the academic community, (b) public lectures to enthuse the layman, and (c) lectures to students in the schools and undergraduates. This way, we would also derive maximum benefit from the eminent scientists and mathematicians brought in from afar, to share their joy of science and research with as wide an audience as possible. The strong support shown by Dr Tan — he consented to be Patron of both Conferences — gave us the courage to organize the first Conference on Mathematics & Theoretical Physics in March 2000. The Isaac Newton Institute for Mathematical Sciences, University of Cambridge, was brought in as a partner to co-organize the Conference with the Faculty of Science. There was good response from all quarters, and I would like to acknowledge the leadership shown by Prof. Keith Moffatt, Prof. Louis Chen and Dr Kok-Khoo Phua in mounting such a successful international conference. In my final months as Dean of Science, I threw the challenge to the Departments of Biological Sciences and Chemistry to organize the next Conference on Biological & Chemical Sciences. I'm pleased to hear that they have accepted the challenge and this second conference in the series will be co-organized with the University of Toronto and held in May 2001. This volume presents some of the invited papers delivered at the Conference on Mathematics & Theoretical Physics. It provides an overview of how the two disciplines interact and are interwoven, and will be of interest to a wide audience.

Soo-Ying Lee Director of Research National University of Singapore vii

FOREWORD

A turn of the century is like a change of a season that has lasted too long. It induces a nostalgia for and a romanticizing of the past followed by a longing for things unfulfilled. It is a time of stock taking for a society as it accounts for its past successes and failures. It is a time of reflection and selfassessment for a community of people bound together by the same ideals- and aspirations. The International Conference on Fundamental Sciences: Mathematics and Theoretical Physics held in Singapore from 13 to 17 March 2000 provided just such an instant to mark the new century and the new millennium and to reaffirm our conviction of the importance of the fundamental sciences, particularly in this new age of information and within a knowledge-based economic paradigm. The conference was intended to be unique and special. It had a threefold objective and was aimed at a wide spectrum of audience consisting of specialists, the general public and school students. To strive to achieve this three-fold objective, the conference consisted of scientific sessions, public lectures and lectures to schools and junior colleges. The scientific sessions served as a forum for mathematicians and theoretical physicists to interact and to reflect on the significant developments of their fields in the twentieth century and present their views on the possible development of ideas and trends in the twenty-first century. Through the public lectures, we aimed to inform the general public of some of the great discoveries in the fundamental sciences with the hope of bringing home to them the importance of these fields. The lectures for schools and junior colleges were aimed at enthusing students, especially the high-achievers, in their study of science, and at encouraging them to study the sciences in the university and perhaps go on to pursue a challenging and rewarding career in basic research. Thirty-five invited speakers gave lectures and shared their visions of the future with more than 400 participants from 27 countries. There were 3 public lectures, 31 lectures for the scientific sessions, 7 lectures to schools and 20 contributed papers in poster sessions. This volume is a collection of some of the invited papers and will serve as an overview of mathematics and theoretical physics at the turn of the 20 th century. ix

X

The conference was the result of the joint efforts of two institutions on opposite sides of the globe, namely the Faculty of Science of the National University of Singapore and the Isaac Newton Institute for Mathematical Sciences at Cambridge, UK. The Local Organising Committee would like to thank Professor Keith Moffatt and his International Scientific Committee for their advice in planning the scientific programme and selecting the invited speakers. We were honoured to have Deputy Prime Minister and Minister for Defence, Dr Tony Tan, as Patron of the conference and the guest of honour at the conference dinner, and Minister for Education and Second Minister for Defence, Admiral Teo Chee Hean, as the guest of honour at its opening ceremony. We are also thankful to the Ministry of Education, Singapore Totalisator Board, Lee Foundation, Southeast Asian Mathematical Society, Shaw Foundation, National Science and Technology Board, Singapore Airlines and World Scientific Publishing Company for their generous financial support.

Louis Chen K. K. Phua Co-chairmen Local Organising Committee

CONTENTS

Message from Director of Newton Institute

v

Message from Director of Research and Former Dean of Science, National University of Singapore Foreword

vii ix

List of Lectures

xiii

Committees

xvii

Mathematics A Review of Critical Point Theory K.-C. Chang

3

Spectrum of Convolution Dilation Operators on Weighted Lp Spaces X. Gao, S. L. Lee and Q. Sun

51

Data Tuning P. Hall

72

Gambling with the Truth: Markov Chain Monte Carlo W. S. Kendall

83

Automorphic Representations and Cohomology of Arithmetic Groups J.-S. Li and J. Schwermer

102

Applied Mathematics Meets Signal Processing S. Mallat

138

Modular Forms and Diophantine Questions K. A. Ribet

162

Chromatic Graph Theory C. Thomassen

183

XII

Glimpses of Nonlinear Partial Differential Equations in the Twentieth Century: A Priori Estimates and the Bernstein Problem N. S. Trudinger Three-Dimensional Subgroups and Unitary Representations D. A. Vogan, Jr.

196 213

Theoretical Physics How Monte Carlo Simulations can Clarify Complex Problems in Statistical Physics K. Binder

253

The Future of Particle Physics J. D. Bjorken

276

Yang-Mills Duality and the Generation Puzzle H.-M. Chan

299

The Mathematics of M-Theory R. Dijkgraaf

317

Quantum Entanglement and Secrecy A. Ekert

339

What Is Quantum Computation? A. Ekert, P. Hay den, H. Inamori and D. K. L. Oi

351

Aspects of Anomalies in Field Theory K. Fujikawa

384

Dynamical Chaos and Nonequilibrium Statistical Mechanics P. Gaspard

398

General Relativity and Quantum Mechanics J. B. Hartle

430

Integrable Integral Operators V. I. Korepin

452

Topics on Fractals in Mathematics and Physics B. B. Mandelbrot

461

Perturbative Quantum Field Theory G. Sterman

479

INTERNATIONAL CONFERENCE ON F U N D A M E N T A L SCIENCES: MATHEMATICS A N D THEORETICAL P H Y S I C S Singapore

13-17 March 2000

Lectures at Scientific Sessions "Finite Markov Chain Renaissance", David Aldous, University of California at Berkeley, USA "How Monte Carlo Simulations can Clarify Complex Problems in Statistical Physics", Kurt Binder, Johannes Gutenberg — Universitat Mainz, Germany "The Future of Particle Physics", James D Bjorken, Stanford University, USA "Advances and Open Problems in Low Dimensional Physics", Edouard Brezin, Ecole Normale Superieure, Paris, France "Yang-Mills Duality and the Generation Puzzle", Hong-Mo Chan, Rutherford & Appleton Laboratories, UK "A Review of Critical Point Theory", Kung-Ching Chang, Peking University, China "Elliptic Curves", John Coates, University of Cambridge, UK "The Mathematics of M-Theory", Robbert Dijkgraaf, University of Amsterdam, The Netherlands "Quantum Computation — from Theory to Experiments", Artur Ekert, University of Oxford, UK "Anomalies in Quantum Field Theory", Kazuo Fujikawa, University of Tokyo, Japan "Dynamical Chaos and Nonequilibrium Statistical Mechanics", Pierre Gaspard, University of Brussels, Belgium "Data Tuning", Peter Hall, Australian National University, Australia "General Relativity and Quantum Mechanics", James Hartle, University of California at Santa Barbara, USA xiii

xiv

"The Small Distance Limit in Models of Elementary Particles", Gerardus 't Hooft, University of Utrecht, The Netherlands "Birational Geometry of Algebraic Varieties", Yojiro Kawamata, University of Tokyo, Japan "Integrable Integral Operators", Vladimir E Korepin, State University of New York at Stony Brook, USA "Wavelets: A Tool for Mathematics, Science and Engineering", Seng-Luan Lee, National University of Singapore, Singapore "On the First Eigenvalue of Laplacian for Locally Symmetric Manifolds", Jian-Shu Li, Hong Kong University of Science and Technology, Hong Kong, and University of Maryland at College Park, USA "Image Processing with Sparse Approximations", Stephane Mallat, Ecole Polytechnique, France "Fractals in Physics", Benoit Mandelbrot, IBM Watson Research Center, USA "Moore's Law, Numerical Techniques, and Real-World Problems", Vladimir Rokhlin, Yale University, USA "Perspectives on Zeta Functions", Peter Sarnak, Princeton University, USA "Quantum Chaos on Graphs", Uzy Smilansky, Weizmann Institute of Science, Israel "High Energy QCD and Quantum Field Theory", George Sterman, State University of New York at Stony Brook, USA "Non-Continuum Models for Geometry, Gauge Fields, Strings,...", Dennis P. Sullivan, City University of New York, USA "Colouring Graphs", Carsten Thomassen, Technical University of Denmark, Denmark "Glimpses of Nonlinear Partial Differential Equations in the 20th Century", Neil Trudinger, Australian National University, Australia "Modern Cosmology and Astro-particle Physics", Neil Turok, University of Cambridge, UK "Dreams of a Finite Theory", Gabriele Veneziano, CERN, Switzerland "Three-Dimensional Subgroups and Unitary Representations", David Vogan, Massachusetts Institute of Technology, USA "Shadowing and Limits to Deterministic Modeling", James Yorke, University of Maryland at College Park, USA

XV

Public Lectures "Smaller than the Nucleus of an Atom", Gerardus 't Hooft, University of Utrecht, The Netherlands "Modular Forms and Diophantine Questions", Kenneth Ribet, University of California at Berkeley, USA "The Probability of Rare Events: Application in Physics, Medicine, and Genetics", David Siegmund, Stanford University, USA

Lectures to Schools "Ciphers, Quanta, and Computers", Artur Ekert, University of Oxford, UK "More than Pretty Pictures: The Continuing Usefulness of Symmetry", Roger Howe, Yale University, USA "Gambling with the Truth: Markov Chain Monte Carlo", Wilfrid Kendall, University of Warwick, UK "Squares Modulo a Prime and the Golden Theorem", Kenneth Ribet, University of California at Berkeley, USA "Physics One Hundred Years After Einstein", George Sterman, State University of New York at Stony Brook, USA "Big Bang Boom", Neil Turok, University of Cambridge, UK "The Science of Chaos", James Yorke, University of Maryland at College Park, USA

INTERNATIONAL CONFERENCE ON F U N D A M E N T A L SCIENCES: MATHEMATICS A N D THEORETICAL P H Y S I C S Singapore

13-17 March 2000

International Scientific Committee Keith Moffatt, University of Cambridge (Chairman) Sir Michael Atiyah, University of Edinburgh Edouard Brezin, Ecole Normale Superieure Louis Chen, National University of Singapore Shing-Shen Chern, University of California at Berkeley John Coates, University of Cambridge Leo Esaki, Tsukuba University Ludwig Faddeev, Steklov Institute of Mathematics at St Petersburg James Hartle, University of California at Santa Barbara Roger Howe, Yale University Yuan-Tseh Lee, Academia Sinica, Taipei Jacques-Louis Lions, College de France Shigefumi Mori, Research Institute for Mathematical Sciences, Kyoto David Olive, University of Wales, Swansea Kok Khoo Phua, World Scientific Publishing Company David Siegmund, Stanford University Chen-Ning Yang, State University of New York at Stony Brook Guang-Zhao Zhou, Chinese Academy of Sciences, Beijing xvii

xviii

Local Steering Committee Louis Chen, National University of Singapore (Co-Chairman) Shui Nee Chow, National University of Singapore Choy Heng Lai, National University of Singapore (Co-Chairman) Sing Lee, Nanyang Technological University Choo Hiap Oh, National University of Singapore Kok Khoo Phua, World Scientific Publishing Company

Local Organising Committee Belal Baaquie, National University of Singapore Kan Chen, National University of Singapore Louis Chen, National University of Singapore (Co-Chairman) Yuan Ping Feng, National University of Singapore Tailen Hsing, National University of Singapore Siew Hwa Kwa, National University of Singapore Seng Luan Lee, National University of Singapore Sing Lee, Nanyang Technological University Choo Hiap Oh, National University of Singapore Chong Kim Ong, National University of Singapore Peter Pang, National University of Singapore Kok Khoo Phua, World Scientific Publishing Company (Co-Chairman) Eng Chye Tan, National University of Singapore (Organising Secretary) Kok Choon Tan, National University of Singapore Ser Peow Tan, National University of Singapore Edward Teo, National University of Singapore

Mathematics

A R E V I E W OF CRITICAL P O I N T T H E O R Y KUNG-CHING CHANG* Institute of Mathematics Peking Univ. Beijing 100871 China

1

Foreword

Critical point theory is a very active subject. It has been growing very rapidly since the 70's in the 20th century. Thousands of research papers tremendously enrich its contents in very diverse topics; many of the results penetrate into different branches of mathematics: nonlinear PDE and ODE, Hamiltonian systems, topology, Riemannian geometry, symplectic geometry, etc. I am not ambitious of covering everything superficially. In fact, my abilities fall short of my desires. This paper consists of two parts. In the first, I briefly review the most important milestones in the development, including some background materials. This is for the general audience. In the second, I shall focus on the methodological progress rather than the solutions of problems. The choice of topics depends on my own favorites: the minimax principle, Morse theory, nonsmooth critical point theory and the role and the failure of the PalaisSmale condition. The main progress and up-to-date information are highlighted. I hope that it will benefit newcomers to these research fields. Many important topics, e.g. the Floer homology theory, the Arnold conjecture, the Weinstein conjecture, the iV-body problem, Novikov's Morse theory, the prescribing scalar curvature problem, etc. are not covered, because of monographs, expository papers, survey articles, and special chapters of textbooks concerning these topics, which are already available. 2

A brief introduction

The calculus of variations studies the optimal shape, time, velocity, energy, volume or gain, etc., under certain conditions. It has a long history and refreshes its face according to the developments of mathematics and other sciences. Laws in astronomy, mechanics, physics, all natural sciences and technologies as well as in economic behavior obey variational principles. The 'SUPPORTED BY CNSF AND MCME 3

4

main object of the calculus of variations is to find out the solutions governed by these principles. Starting from Fermat, this is a subject in finding the minimizers of a given functional. Assume that F : Q, x MP x Rnp -» M1 is a function with suitable continuity and measurability, where fi C I " is a measurable set. Given a set 91 of p-vector functions, and a functional defined on it:

I(u) = / Jn

F(x,u(x),Du(x))dx,

where Du is the differential of u, find UQ S 91 such that J(u 0 ) = Min{I{u) | u G 9 1 } . If 91 is a space, on which the differential of the functional I makes sense, then it is well known that a necessary condition for «o to be a minimizer is that J'(uo) = 0. A point u £ 91 is called a critical point if I'(u) = 0. For the special form of I, this is a solution of the Euler-Lagrange equation:

Mx),Du{x))=Q

^k-ti^XMd

=i 2

'

* ' '"'*

{E L)

-

The classical theory of calculus of variations deals with minimizers by solving the E — L equation. Since the E — L equation is only a necessary condition for minimizers, a second order differential condition has to be derived in verifying the minimality. However, very few E-L equations can be solved explicitly. The method is limited in applications. Conversely, in his study of conformal mappings, Dirichlet introduced the following Dirichlet principle, i.e., finding a minimizing sequence {UJ} C 01: given I(UJ) —> inf{I(u) \ u € 91}, if one can show that Uj —> UQ £ 01

in some sense such that

I{Uj) -»

I(U0),

then UQ solves the (E — L) equation. This provides a direct method in solving differential equations. In this method, the positions of the minimizing sequence are not under control. As an improvement, the following Ekeland Variational Principle [100] is very fundamental, but was discovered only in the 70's.

5

Assume that / : 9T —> R 1 U {+00} is a lower semicontinuous function on a metric space (9T, d), which is bounded below. For any e > 0 let ue € 9t satisfy: I(ue) < inf I+e. Then 3vs € 91 such that I(ve) < I(ue), d(v£,u£) < 1, I(u) > I{ve) — ed(v£, u), Vu ^ t)£. A typical problem from geometry in the calculus of variations is to find geodesies between two points go, gi on a given Riemannian manifold (M,g). A path 7 on M connecting go and gi is denoted by 7 : [0,1] -¥ M with 7(1) =qui = 0,1. Let m = { 7 G C 1 _ 0 ([0,1],M) I 7 ( 0 = 9i

i = 0,1 },

and let

where || • ||2 is the scalar product induced by the metric g, or in local coordinates:

and 7 = ( 7 1 , . . . , 7 p ) , p = dimM. Then the (E - L) equation reads as the geodesic equation:

m)=° or equivalently

(!)V-E*M££.•

Geometrically, a curve satisfying the geodesic equation has the feature that tangents along the curve are parallel. Obviously there are many geodesies joining go and gi. However, besides those geodesies minimal in length, how do we reach other geodesies? Influenced by the pioneering work of G. D. Birkhoff(1917) on closed geodesies, two global methods appeared: the minimax method and Morse theory. Both are based on topological arguments. The minimax method can be seen as an outgrowth of the max-min characterization of the eigenvalues of Laplacian with Dirichlet boundary condition. It was successfully developed by Ljusternik Schnirelmann(1934) [153]. The

6

new ingredient is the topological invariant: category, by which one defines the critical values: Ck =

inf Cat(A)>k

maxiYu)

k = 1,2,- • •

u€A

where A is a subset of 9t, and Cat(A) is the category of A w.r.t the space 91. A triumph of the minimax method is the theorem of three closed geodesies, which asserts that on a closed surface with genus zero, there are at least three closed geodesies. See Ljusternik Schnirelmann(1928) [154], Ljusternik(1947)[152], see also Ballmann [19] and Klingenberg [132]. (The results for other closed surfaces were obtained by Ballmann a half century later). In 50's M. A. Krasnoselski [133] introduced the notion of genus, which is a topological index related to the group Z2, in the study of nonlinear eigenvalue problems for a class of integral equations. In the minimax argument, the genus plays a similar role as the category. An important step towards the recent developments of critical point theory is due to R. S. Palais(1966) [172]. We discuss two major contributions here: (l)the Ljusternik Schnirelmann theory is extended to infinite dimensional manifolds, in which a compactness condition, now called the PalaisSmale condition, was introduced; (2)homotopy stable families are used in replacing category in the minimax principle. In parallel work, in the 30's, M. Morse had penetrating insight into a deep relation between the critical points of a nondegenerate function and the topology of the underlying compact manifold. The Morse index of a critical point plays an important role. Although the nondegeneracy condition and the compactness requirement in his theory are not met in most variational problems, he succeeded in applying his theory in the study of closed geodesies and unstable minimal surfaces, see Morse [166], J. Milnor [165], Bott [37], Morse- Tompkins [167], and Schiffman [201]. Moreover, Morse theory was in triumph in topology during the 50's. It became a basic tool in computing the homology of compact manifolds. The work of S. Smale on the solution of the Poincare conjecture for n > 5 pushed the theory into a new peak. [206] In 50's and 60's E. Rothe [195], R. S. Palais [171], and S. Smale [207] extended the Morse theory to infinite dimensional manifolds by using the Palais-Smale condition. And later, Marino and Prodi [161] and Gromoll and Meyer [118] endeavored to weaken the restriction of nondegeneracy. All earlier results mentioned above mainly studied functionals which are bounded from below. However the functionals occurring in nonlinear PDE as well as in Hamiltonian systems may be unbounded from below and above,

7

and sometimes they are strongly indefinite, e.g.

I(u) =

^(±\S7u\2-F(u^dx

where F(u) = ±u 4 , and 0t = H&(Q,), the Sobolev space, and

I(u) = Jfy-Ju,u)-H(u)\

dt

where H is the Hamiltonian,

- ( - . ; ) is a 2n x 2n matrix, ( , ) is the inner product in R 2 n , I is the n x n identity matrix, and 71 — C 1 ( 5 1 , R 2 n ). The revival of the study of critical point theory began with the works of A. Ambrosetti and P. Rabinowitz(1974) [7] [191] and P. Rabinowitz(1974) [192]. The first paper provided a method based on the minimax principle in dealing with functional unbounded from below. The second paper gave a proof of the existence of a closed orbit of an autonomous Hamiltonian system on a starshape hypersurface in R 2 n (independent of this, see the work of Weinstein [229]). This was a breakthrough in the study of the Weinstein conjecture: Every closed connected hypersurface of contact type in a symplectic manifold carries a closed characteristic. It stimulated the study of this conjecture, which is one of the central problems in symplectic geometry. The positive answer for the hypersurface M in (M.2n,w) admitting a contact form A such that dX = U>\M is due to Viterbo [222]. Later, a vast amount of literature appeared, extending their method and applying their results to nonlinear PDE and Hamiltonian systems. About the same time, C. Conley [80] extended Morse theory to the flow on a compact space which is not assumed to be gradient-like. He defined the Conley-index for an isolated invariant set w.r.t the flow, which is the homotopy type of a pointed compact space [(iV\JVo)], where (N,NQ) is the Conleyindex pair and is determined by the flow near the invariant set. A Morse decomposition is obtained via the flow, and a Morse inequality associated with the Morse decomposition is also extended. It relates the Conley indices of these isolated invariant sets on one hand and the topology of the compact space on the other hand. See also [196]. Another impetus of to the development of the theory from the symplectic geometry is the Arnold conjecture. Given a 27r-periodic time dependent

8

Hamiltonian H on a symplectic manifold (M, w), it admits a family of vector fields Xt : w(-,Xt) — dHt, which, in turn, determines a family of symplectic diffeomorphism o by the use of Conley's theory. Then this conjecture became another central problem in the study of the symplectic geometry. Conley's theory is more general; the flow is not assumed to be gradientlike, so it is not only used for variational problems. But the Conley index is in the realm of homotopy theory, and it is difficult to compute. However, for a variational problem, classical Morse theory always provides more information than Conley's generalization. As indicated in Chang [54] [57], Conley and Zehnder's proof can be naturally derived from the Morse inequality and the minimax principle for cohomology classes, but the functional under discussion is unbounded from below; see also [52]. The next main progress is on the release of the Palais-Smale condition. In fact, most variational problems arising in geometry and physics lack this condition. Inspired by the work of Sacks and Uhlenbeck [197], the loss of compactness in many geometric problems is often related to the blowing off of bubbles. By carefully analysing the bubbles, critical point theorems are applicable to some extent. Among them, we mention few examples. Independently, H. Brezis and J. M. Coron [40] and M. Struwe [214] solved the Rellich conjecture: For any closed rectifiable arc T with fixed orientation, there is a number h(T) > 0 such that for H S (0, h(T)), there exist at least two constant mean curvature H surfaces u : Cl —>• R 3 , with u : dCl —> T, yielding the same orientation of T, a small one and a large one, where ft C M2 is the unit disk. The existence of the small surface has been known for a long time: this solution can be obtained as a local minimizer of the functional J u

() = //

\ l V u ! 2 + zHu

• (u*

A u

v) \

dxd

V-

The large solution is obtained by Mountain Pass Theorem, where the PalaisSmale Condition is verified under certain levels by comparing with bubbles.

9

T. Aubin [8] and R. Schoen [202] solved the Yamabe conjecture: For any n-dimensional compact smooth Riemannian manifold (M, g), n> 3, there exist metrics which are pointwise conformal to g, and have constant scalar curvature. This led to the attempt to find a nontrivial critical point of the functional:

/M

i ( | V g U | 2 + c n J R s U 2 ) - ^ H 2 * dVg.

/ JM

where V g is the gradient w.r.t g, c„ = 4 ^_ 2 1 \, Rg denotes the scalar curvature of g, 2* = ^2, and Vg is the volume form. A different approach was given by A. Bahri [10]. The critical Sobloev exponent causes the noncompactness of the Sobolev embedding H1 (M) —» L2 (M). New ingredients are needed to avoid the failure of the Palais-Smale Condition. One of the pioneering works in this direction is due to Brezis and Nirenberg [39]. A related, but different problem is the problem of prescribing scalar curvature (see the last section). Probably the most far reaching development of our subject in these years is the Floer theory [108] [109] [110], see also [127]. It was motivated by the Arnold conjecture mentioned above. Given a compact symplectic manifold (M, w) and a time-dependent Hamiltonian function Ht, we are going to estimate the number of contractible 27r-periodic solutions of the Hamiltonian system: u = Xji(t, u), where u>(-, XJJ) = dHt

(HS)

Since the universal covering of M is not necessarily flat, (HS) is not on K 2n ; in fact, u is a loop in M. The variational setting cannot be stated on the Sobolev space Hi{S\M), as the flat case is, because the latter is not a manifold in general. However, if then there exists an extension of the covering map 7 : R 1 —> M of u: 7 : D H-> M,

with

^\OD = 1,

where D is the unit disk with boundary dD = S1, such that ^ 7 * • w does not depend on the special choice of 7. The functional, which admits (HS) as the E — L equation, can be written as 2TT

IH(-Y) = -

[ 7*u+ [ JD

JO

Ht(i(t))dt.

The gradient flows for In are solutions of a nonlinear perturbed CauchyRiemann equation, which are related to the pseudo holomorphic curves in

10

(M, w). ^From the a point of view of Smale, the set Tt of all bounded solutions of the gradient flows, consisting of all critical points, together with their connecting orbits, carries all the topological data. In general, Wl is not a manifold. By the bubbling-off analysis, it is proved that SDT is compact. On 971, there is a natural M-action defining a gradient-like flow. After an approximation procedure, 97? contains the cohomology structure of M. This is a rough description of the ideas behind Floer's proof of the first part of the Arnold fixed-point conjecture for W\W2(M) — 0. As for the nondegenerate case, because of the nondegeneracy, 97t has a structure which looks like that of a Morse-Smale system on a smooth, compact manifold. For any two nondegenerate contractible orbits u, v, let 9Jt(«, v) denote the submanifold, which consists of those flow lines connecting u and v. It is proved that if £ e Wl(u, v), i.e., £ is a connecting flow line, with £(t) —» u or v as t —> =F°°, then dimVJl(u,v) = IndexF(£), where F is a Fredholm operator which is the linearization of the gradient like operator. If the first Chern class [C{\ vanishes on ^ ( M ) , i.e., Js2 u*ci = 0 Vu : S2 H> M, then lndF(u) = i{u) — i(v), where i is the Maslov index, by which the relative Morse index between these two nondegenerate critical points is defined. A chain complex generated by the finitely many nondegenerate critical points was constructed by the analytic gluing method. The homology of this complex is called Floer homology. By showing that Floer homology was a model of singular homology of the underlying manifold M, Floer was able to show the second part of the Arnold conjecture under the assumptions: W |TT2(M) = [CI]TT2(M) = 0- Many people made efforts to avoid the assumption W ITT2(M) = 0, see Hofer and Salamon [128], where Novikov's Morse theory for closed 1-forms is applied. Finally, the nondegenerate case was completely solved by G. Liu and G. Tian [144] and K. Fukaya and K. Ono [114], in which the virtual moduli cycle plays a crucial role. However, the degenerate case without the assumption w|7r2(M) = 0 is still open. H. Hofer [126] proved Weinstein's conjecture for a wide and significant class of 3-manifolds; G. Liu and G. Tian [145] used Gromov-Witten invariants for proving certain cases of the conjecture. Although critical point theory has been developing rapidly and has been applied successfully in solving many famous open problems, there is still an immense amount of virgin soil waiting to be upturned. In particular, the lack of compactness is a common problem in nonlinear analysis; so far we know very little in this case. Many challenging problems, e.g, nonlinear wave equations (small divisors), the supercritical exponent semilinear elliptic problems, multiple high dimensional harmonic maps, etc., are far from being understood.

11

3 3.1

Minimax Methods Linking

In dealing with critical points of a functional which is unbounded from above and below, the direct method is invalid. A. Ambrosetti and P. Rabinowitz [7] proposed that if the functional possesses a mountain pass geometry, then the minimax method works. Namely, let 0? be a complete Banach-Finsler manifold, assume that there exists an open set O C 01 and two points po e O and p\ £ O such that max{/(po), J(pi)} = b1oot6[0,l]

Then one may choose a sequence un C 0T such that I(un) -> c,I'(un)

->• 0,and d(u n ,7„[0,1]) -*• 0.

Furthermore, if c = 60, then d(un, dO) —> 0.

12

A sequence {un} satisfying the above four limits is called a (PS)ao,c sequence along {7 n }Since a (PS)QO,C sequence along {7 n } is more restricted than a {PS)C sequence, a weaker type (PS) condition is sufficient to ensure the existence of a critical point. This enlarges the area of applications of critical point theorems. These kinds of ideas have been developed in various aspects by N. Ghoussoub, see [116], As a direct application, they improved the Mountain Pass Lemma as follows: if 60 = bi and (PS)b0 holds, then there exists a critical point u0 G dO with I(UQ) = bo = C. We point out that the (PS)C condition is crucial in the proof of the existence of a critical point. There is a counter-example due to H. Brezis and L. Nirenberg [42]: on K2, one defines

I(x,y)=x2

+

(l-x)3y2;

obviously, / possesses a Mountain Pass Geometry, (0,0) is the only critical point, but is not a mountain pass point. Mountain pass geometry is a special case of linking, which is essentially a notion of intersection of sets, see Ambrosetti and Rabinowitz [7], Benci and Rabinowitz [35] and W. M. Ni [169]. Let D be a ^-topological ball in a complete Banach Finsler manifold 9t, and let S be a subset of OT. We say that_aD and S link (homotopically), if dD n S = 0 and • c

and

I'(un) -> 9,

where c = inf max I oip{u),

and

r = {

oo n g D

{un} being a (PS)C sequence and d(un,(pn(D)) Furthermore, if c = mfsi,

->0.

then d(un, S) -» 0.

13

In fact, the linking geometry, which includes the Mountain Pass geometry as a special case, creates homotopy stable families with boundary conditions. In contrast with Palais's earlier work, it is the boundary condition which ensures the applicability of the minimax methods to functions unbounded from below. On one hand, the requirement in the definition of linking that D be a topological ball, is too strong in applications. M. Schechter and K. Tintarev [200] say that A links B for two subsets A, B of a reflexive Banach space 91, if (1) A n 5 = 0 (2) VV> G $, 3t G (0,1] such that ip(t, A) D B ^ 0, where $ = { ip g C([0,1] x OX, OT) I possesses the following properties: a) Vi G [0,1), r/>(t, •) is a homeomorphism of 91 onto itself, and ip(t, - ) _ 1 is continuous on [0,1] x 91, 6)V(0,-) = id, c) 3«o € 91 such that i()(l,u) = uoVu G OX and ^ ( i , u) —• uo as £ —>• 1 uniformly on bounded subsets of 91}. The advantage of the definition of Schechter and Tintarev can be seen from the following statement: if 91\A is path connected, then aA links B" implies "B links A". Since the positions of A and B are not symmetric in defining the minimax values, this provides a possibility of obtaining two critical values, and consequently, two distinct critical points. On the other hand, the restriction that dimD < oo in the definition of linking does not fit for strongly indefinite variational problems, e.g. for the Hamiltonian systems and a class of semilinear elliptic systems. V. Benci and P. Rabinowitz(1979) [35] eliminated the restriction as follows. Let 0T be a real Hilbert space, 91 = X © X1-, where X is a linear subspace, and let Px and Px± be the orthogonal projections onto X and X1respectively. Let r = {

{B,B') be a be Gmap. Fix a set A of G-spaces. Define: A-Cat{f)

:=inf { m G N U {+00} | 3 closed F0, -Fi,... ,Fm C 01 s.t. (i)97' C F0 and 3 a G-homotopy pt). It is an index, and coincides with the cohomology index for representation spheres, but differs in general; see the work of T. Bartsch [22]. All the indices discussed above do not depend upon the given function / . However, as we have mentioned at the end of the last section, if we know more about the distribution of the values of I, there is more room to use minimax methods. Pseudo-index is a notion related to a given index i and a given function / . Let J be an interval (a, b), —00 < a < b < +00. Define Tj(I) = { T] I G—homeomorphism satisfying v\i-i(W\J)

= id|j-i(Ri\J)

a n d

-%(u))
• N U {+00} satisfies: (2a)A cB=>i*(A)
i*(4)-i*(B)

(2c)i*(^p))>i*(A)

\/A,B G £*. VAGE*,BGS.

VAGS*,V7 ? Gr J (7).

Under the same assumptions of the Multiplicity Theorem, let c*k=

inf

supJ(u)

Vfc = l , 2 , . . .

i'(A)>ku€A

and ii c = c*k = ••• = c*k+m, then i(ATc) > m + 1. The notion of the pseudo-index first appeared in the work of Ambrosetti and Rabinowitz [7], and then it was developed by V. Benci [31], Benci and Rabinowitz [35] and K. C. Chang [59]. Special pseudo-indices have been constructed according to the behavior of I. Applications can be found in these papers. An index theory is said to satisfy the d-dimension property if 3 an integer d > 0 such that i(Vdk n dB^O))

=k

Vdfc-dimensional subspaces Vdk G £ such that Vdk n Fix G = {6}, where B\ (0) is the unit ball with center 9 of the Banach space 71. In dealing with strongly indefinite functionals on a Banach space OT with a G action, Y. Q. Li [149] introduced the notion of limit index i°° w.r.t an index i as follows. Let Y and Z be G-invariant closed subspace of 01 such that 01 = Y © Z, where Z = U^ = 1 2j, and Zj is a dn^-dimensional G-invariant subspace with Z\ C Z2 C • • •. Wl G E, let Aj = A D (Y © Zj). If i satisfies the d-dimensional property, then we define i°° : S - > Z U { - o o , + o o } , t°°(i4)= lim"

(i(Aj)-rij).

j-yco

The Multiplicity Theorem is modified in the same way as it is in the pseudo-index theory, but the pseudo-index i* is replaced by the limit index i°°, and the (PS)C condition should be replaced by the following (PS)* condition with respect to {Y © Zn}: Set / „ = I\Y®Z„- For every sequence {un} satisfying un G Y © Zn, I(un) —>• c and I'n(un) ->• 0 as n -> 00 possesses a subsequence, which strongly converges to a critical point of I.

18

On the other hand, for a G-invariant function J, one may derive the existence of many critical points by linking geometry via various Borsuk- Ulam type theorems directly, see the work of Bartsch [23] and Bartsch, Clapp, and Puppe [25]. 3.3

Morse theory for junctionals with isolated critical points

• Local Theory The Morse index characterizes the local feature of a nondegenerate critical point. But, in nonlinear analysis, nondegeneracy is not known a priori, and might not even hold. Without loss of generality (in the estimations of the number of the critical points), we may assume that the critical points of a C 1 -functional are isolated. By the work of E. Rothe [195], critical groups can be introduced to replace the Morse index. Given a topological pair (A, B) and an Abelian group G, we denote by Hq(A, B; G) the q-th. relative singular homology group of the pair with coefficient group G. Let yi be a C 1 -complete Banach-Finsler manifold, and I be a C 1 function defined on it. Let u be an isolated critical point of I, and let c = I(u) and Ic = {u G OT | I(u) < c}. Cq(I, u) = Hq{Ic n O, (Ic \ {u}) n O; G) is called the qth critical group of / at u, q = 0,1, 2 , . . . , where O is an isolated neighborhood of u. These groups are well defined, i.e., they do not depend on the special choice of O. The critical groups enjoy the homology invariance and other important properties, which make them computable for many types of isolated critical points, cf [54]. Parallel to the work of E. Rothe, Gromoll and Meyer [118] defined a pair (W,W-) for an isolated critical point, where W is a closed isolated neighbourhood of u, and W- is in some sense the exit set of W w.r.t a pseudo-gradient flow -q of / . Chang revealed the relationship between the pair and the critical groups [54]. In case when VI is a Hilbert space, I e C2 (91) and I' is a compact vector field. Under the (PS) condition, one has the formula oo

ind(/',u) = ^ ( - l ) 9 r a n k C Q ( / , w ) , 9=0

where ind is the Leray-Schauder index of / ' at u. Critical groups have also been extended to dynamically isolated critical sets. A critical subset S is said to be dynamically isolated if there exists a closed neighborhood O of S and regular values a < (3 of I satisfying O C I-1 [a, j3] and Cl(d) n l n J " 1 [a-/3] = s> w h e r e K i s t h e critical set, O =

19 U{ r](t, O) | t G R 1 } and 77 is a pseudo-gradient flow of I. The critical groups for S are defined to be Cq(I,S)

= H*(I(, n 0+,Ia n 0+;G),

q = 0,1,2,...

where 0 + = U{r7(£, O) \ t > 0 } , and # * stands for the relative singular cohomology groups; see the work of Chang and Ghoussoub [62]. The notion of Gromoll-Meyer pair (GM pair, in short) of a dynamically isolated critical set has also been extended. For a C 1 functional Z o n a complete Banach-Finsler manifold 01, a pair (W, W_) of subsets is called a GM pair for a dynamically isolated critical set S w.r.t a pseudo-gradient flow r), if (i)W is a closed neighborhood of S, which is convex along flow lines, and satisfying W D K = S, and J is bounded below on W, (n)W- is an exit set for W w.r.t 77, (iii)W_ is closed and is a union of a finite number of submanifolds which are transversal to the flow 77. Again, one has Cq(I,S)=H«(W,W-;G)

Vq.

Moreover, under a natural condition on W, the GM pair (W, W-) is a Conley index pair for any isolating neighborhood O of the invariant hull of the dynamically isolated critical set S. The relationship between the Laray-Schauder degree and the Conley index is given by 00

deg(r,N,0)

= 53(-l)*rankff«(W, N0)

for any Conley index pair (N,N0)

for an isolating neighborhood O on which

9 $ I'[dN). A similar but different approach can be seen in the work of Benci [29]. Equivariant Morse theory for isolated critical orbits was studied by Z. Q. Wang [225], Hingston [123] and K. C. Chang [58]. However, for a strongly indefinite functional, the Morse index is infinity, and all critical groups are trivial. Szulkin [216] and Kryszewski and Szulkin [135] refined the definition of critical groups with the aid of infinite dimensional cohomology theory due to Geba and Granas [115]. Assume that VI is a real Hilbert space, and that there is a filtration (sJln)^'=1 of 01, let (dn)'^L1 be a sequence of non negative integers, and let £ = { O l n , ^ } . For a closed pair (A, B) of 01, we define the g-th £ -cohomology group of (A, B)

20

with coefficient field F by

Hj(A,B) = [(H«+d"(Anmn,Bnmn))~=1 where the RHS is the asymptotic group defined by oo

'(Hi+d"(An,Bn))~=1}

= l[Ho+d»(An,Bn)

®~=1H«+d"(An,Bn)

I

n=l

in which An =AD%l,Bn

= Bnmn,

Vn.

After suitable modifications of the definition of the Gromoll-Meyer pair (W, W-), the critical groups for an isolated critical point u of a functional I G Cx(0T) satisfying the (PS)* condition w.r.t (y) = k

21

if 7 € Vk- The number k is called the Conley-Zehnder index of 7. If 7(i) is the fundamental solution matrix of the linear Hamiltonian system: - JZ{t) =

B{t)Z{t)

where B(t) is a nondegenerate loop in the symmetric matrix space, i.e., it does not have Floquet multiplier 1, then the Maslov index of B, i(B) = k if j(j) = k. If B(t) is degenerate, i.e., ifit has a Floquet multiplier 1, Y. M. Long [157] extended the definition of Maslov index as a pair (i-(B),n(B)): i-(B)

n(-B)=dimker(7(l)-l), — lim i(C) where C is nondegenerate. C-s-B

Let us state the relationship between the Maslov index and the relative Morse index for strongly indefinite functionals. In an abstract setting, we write (HS) as I{u) = \(Au,u)

+ G{u)

in

m =

Hi(S\R2n)

Li

where A is a bounded self adjoint operator on 9t defined by {Au,u)=

Jo

(-Ju,u)dt

V«GC1(S'1,E2n),

and r2lT

G(u) = - / H(t,u(t))dt. Jo Let B(t) = V\H(t, u(t)) be the loop in the symmetric matrix space. Its action is also a compact symmetric operator on 91. Let {Pn} be a sequence of finite dimensional orthogonal projections, which strongly converges to the identity and commutes with the operator A + B. If (A + B) is invertible, then i{B) = lim {m(Pn(A + B)Pn) - m(Pn{A + P)Pn)} n—>oo

where m is the Morse index and P is the orthogonal projection onto the kernel of A For any B, we have n(B) =dimker(A + B), i.{B)=i(B

+ PB),

22

where Pg is the orthogonal projection to kev(A + B). See the works of K. C. Chang, J. Q. Liu and M. J. Liu [69], G. Fei and Q. Qiu [106] and S. Li and J. Q. Liu [142]. In his study of closed geodesies, R. Bott [38] established an iteration formula for the Morse index, by which closed geodesies can be distinguished geometrically. This work was extended to periodic orbits for convex Hamiltonian systems by I. Ekeland [99]. In a series of papers, Y. M. Long extended it further for the Maslov-type index of symplectic paths, in which the Hamiltonian is not assumed to be convex (see for instance [155]). These works were motivated by the minimal period problem [156]. • Global Theory Deformation is used throughout Morse theory. It is constructed by the flow, based on the pseudo-gradient vector field and the Palais-Smale condition. As a direct consequence, we have the following existence theorem. Principle I (Nontrivial Interval Theorem) Let 91 be a C1 complete Banach-Finsler manifold, and let I be C 1 and satisfy the (PS) condition in I~l[a, b], — oo < a < b < oo. If Ia is not a strong deformation retract of lb, then there must be a critical value c € (a, b). Returning to the Mountain Pass Lemma, it was observed by Tian [221] that if c is a mountain pass value, then \/e > 0, the two points po and p\ are path connected in the level set Ic+e> but not in Ic-e- These imply that 3u € K such that I(u) = c and Ci(I,u) ^ 0, if I has only isolated critical points. Hofer [125] defined an isolated critical point u to be of mountain pass type if for any small open neighborhood O of u, the set Ic n 0\{u} is not path connected, where c = I(u). Under the assumptions that 91 is a Hilbert space, I G C 2 , I"(u) is a Fredholm operator of index 0 and dimker7"(w) = 1 if 0 e a(I"(u)), the following equivalence was proved: (l)u is of mountain pass type, (2)Ci(J,u)^0and (3)Cg(I,u) = SqlG. See [54], [125] and [24]. The idea carries over to high dimensional linking. Since the homology groups and the homotopy groups coincide only in the first order, and the critical groups are homological, we come to a definition [29] [56] of homological link. Let D be a fc-topological ball in a complete Banach-Finsler manifold 91, and let S C 91 be a subset. We say that dD and S homologically link, if dD n S = 0 and \T\ n 5 ^ 0 V singular k chain r with dr = 3D, where

23

\T\ is the support of r. In contrast, we call the definition of linking given previously homotopical linking. The relation between these two notions of linking is as follows: suppose that dD and S homotopically link. Assume that (1) S ("1 D = single point, (2) 5 is a path-connected orientable submanifold with codimension k, (3) 3 a tubular neighborhood N of S such that N C\ D is homeomorphic to D, (4) there is no nontrivial relative singular chain r G Ck(M\S,dD) with dr = dD. Then dD and S homologically link. Thus by the same argument as given in an article of Tian [221], one has a critical point u, such that I(u) = c* where c* is the minimax value w.r.t the relative homology class [T),8T = dD, and Ck(I,u) ^ 0. See the works of J. Q. Liu [146] [147], C. Viterbo [223] and Chang [58]. The Morse index estimation for saddle points was obtained earlier by Lazer and Solimini [137] and one can refer to the work of Perera and Schechter for some more recent variants [182]. The estimation of the critical groups is useful in getting multiple critical points. Extending a result due to Schechter and Tintarev [200], Perera [179] proved the existence of a pair of critical points ui and u2 with J(ui) < a < I(u2)

and

Ck^(I,u{)

^ 0,Ck(I,u2)

^ 0,

provided I\dD < a < I\s, rank(i fc _i) > oo

as

j - > oo,

such that -wj(oo)

->XT

in

W^D.R'),

as j —>• oo, where oo is the north pole at S . Furthermore, in the work of Qing and Tian [186], it was proved that Xoo and {w;}™ are connected together without any necks. Since the limiting bubble tree depends on the subsequence {£,}, there a natural question about its uniqueness.

37

Recently K. C. Chang and J. Q. Liu studied the heat flow for minimal surfaces with Plateau boundary condition [63] [64]. Again this has been applied to multiplicity results on the coboundary minimal surfaces. It is expected that the information on bubble trees will be useful in the study of minimal surfaces and harmonic maps. References [1] Abbondandolo A., A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces, Topol. Math. Nonlinear Anal. 9, 325-382 (1997). [2] Adimurthi, Mancini G., The Neumann problem for elliptic equations with critical nonlinearity, Nonlinear Analysis in honor of G. Prodi, S.N.S. Pisa, 9-25 (1991). [3] Ahmedou M. O., El Mehdi K. O., Computation of topology at infinity for Yamabe-type problems on annulir-domains, Part I, Part II, Duke Math. J. 94, 215-229, 231-255 (1998). [4] Ahmedou M. 0., El Mehdi K. O., On an elliptic problem with critical nonlinearity in expanding annuli, J. Funct. Anal. 163, 29-62 (1999). [5] Alama S., Li Y. Y., On "multibump" bound states for certain semilinear elliptic equations, Indiana Univ. Math. J. 4 1 , 983- (1993). [6] Ambrosetti A., Critical points and Nonlinear variational problems, Bulletin Soc. mathematique de France torn 120 (1992), Memoire. [7] Ambrosetti A., Rabinowitz P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14, 349-381 (1973). [8] Aubin T., Equations differentiales non lineaires et problems de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55, 269-296 (1976). [9] Bahri A., Critical points at infinity in some variational problems, Pitman Research Notes in Math. v. 182, Longman, London (1989). [10] Bahri A., Another proof of the Yamabe conjecture for locally conformally flat manifolds, Nonlinear Anal. TMA 20, 1261-1278 (1993). [11] Bahri A., The scalar curvature problem on sphere of dimension larger or equal than 7, preprint (1994). [12] Bahri A., Brezis H., Elliptic differential equations involving the Sobolev critical exponent on manifolds, preprint (1994). [13] Bahri A., Coron J. M., On a nonlinear elliptic equation involving the Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 4 1 , 253-294 (1988). [14] Bahri A., Coron J. M., The scalar-curvature problem on the standard

38

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S P E C T R U M OF CONVOLUTION DILATION OPERATORS O N W E I G H T E D Lp SPACES X I A O J I E G A O , S. L. L E E A N D Q I Y U S U N Department of Mathematics National University of Singapore 10 Kent Ridge Road, Singapore 119260, Republic of matgxj, matleesl, [email protected]

Singapore

The convolution dilation operators Wc,af(x)

= a

c(ax - y)f(y)dy,

f € LP(R),

where a > 1, for special integrable kernels c arise in many diverse disciplines, such as wavelet analysis, geometric modelling, dynamical systems and the mathematical theory of quasicrystals. Their spectrum and eigenfunctions play an important role in applications. The object here is to give a complete understanding of the spectrum of the operator for any compactly supported integrable kernel c in the setting of the weighted Lp space, L£,(R) that comprises functions / for which fw belong to LP(R). We prove that under an oscillation condition on w, WCi 1 and a compactly supported function c in i 1 ( R ) with | R c(x)dx = 1, the convolution dilation operator Wc,a • i p ( R ) -> LP(R) denned by Wc,af(x)

:= a f c(ax - y)f(y)dy,

f € L P (R),

(1.1)

is a continuous analogue of the transfer operator (also known as Ruelle operator) that arises in a number of diverse fields, such as wavelet analysis 3 ' 15 ' 20 , geometric modeling 4 ' 8 ' 12 ' 21 , dynamical systems 16,17 ' 18 and the mathematical theory of quasicrystals 1 ' 2 . Its spectrum and eigenfunctions play an important role in applications. It is easy to check that Wc,a is a bounded operator on 51

52

Lp(R) for any 1 < p < oo. For any K > 0, denote by L p ([-A",if]) the space of all Lp-functions with support in the interval [-K, K], and by LQ([-K, K\) the space of all functions / e LP([-K,K]) with JRf(x)dx = 0. Let K0 be the smallest positive number that satisfies supp(c) C [—(a—l)Ko, (a— 1)KQ}. Then it can be checked that for any K > K0, the spaces LP([—K,K\) and LQ([—K,K]) are invariant subspaces of WCtCt. An eigenfunction of WC)Q with eigenvalue 1 is a solution of the convolution dilation equation cj) = a

c(a • -y)(y)dy.

(1.2)

JR

The simplest convolution dilation equation is one with kernel c = | x ( - i , i ] a n d dilation a = 2, and it was studied by Kebaya and Iri 14 and Rvachev 19 independently. Recent interests in convolution dilation equations are associated with nonstationary multiresolution and wavelets 5 ' 9 , nonstationary subdivision processes 6 ' 7 , and invariant densities for model sets and quasicrystals 1 ' 2 . It is known that (1.2) has a unique compactly supported solution (f> normalized so that J R 4>{x)dx = 1 and the solution is infinitely differentiable and supported in [—i^oi-^o]13For a Banach space X and an operator T on X, we shall denote the resolvent set, spectrum, the set of all eigenvalues, and the spectral radius of T on X by P(T,X), cx(T,X),E(T,X) and p(T,X) respectively. Then E(T, X) c a{T, X) = C\P(T, X)

(1.3)

and p(T,X)

= sup{|A| : A 6 a(T,X)}

= lim ||T"|| 1 / n .

(1-4)

n—KXI

Note that if 0 6 LP([-K,K]) is the solution of (1.2), then ^ k \ the fc-th derivative of (j>, is the eigenfunction of the restricted operator WCiQ|i,P([_x,K]) with eigenvalue a~k for any K > K0. This follows by taking derivatives of both sides of (1.2). Set E 0 : = { Q - f e : A: = 0 , 1 , . . . } . Then any A G So is an eigenvalue of the operator Wc,a on the Banach space LP([—K,K]), and any A € £ o \ { l } is an eigenvalue of the operator WCiQ on LQ([—K,K]). Moreover, the operator Wc>a is a compact operator on Lp([-K,K}) and on L%([-K,K\) for any K > K013. Therefore, the following result about spectrum of the restricted operator WCta\LP^_x,K]) a n d Wc,a\L>([-K,K])

follows13.

53

Theorem 1.1 Let a > 1, 1 < p < oo, c be a compactly supported function in L 1 (R) with J R c(x)dx = 1, and let K > K0. Then Wc,a is a compact operator onLp([-K,K}) andLpa{[-K,K)). Moreover E(Wc,a,LP(l-K,K})) o-(Wc,a,L?(l-K,K}))

= X0,

(1-5)

= {0}Ui;0,

(1.6)

= l,

(1.7)

= a-1.

(1.8)

p(Wc KQ. Therefore, one would expect that the spectrum of the operator WCta on the entire space L P (R) to be the same as in (1.6). However, this is not the case, and it turns out that the spectrum of the operator Wc>a on L P (R) is the closed disc with radius a 1 _ 1 / p , and that all the nonzero complex numbers with absolute value strictly less than one are eigenvalues of Wc 1, c be a compactly supported function in Z/1(R) with J R c(x)dx — 1, and w be a weight function satisfying (1.9). Then WCtCt is bounded on ££,(R) if and only if w satisfies (1.10). For any K > 0, let Lpw([-K, K}) be the space of all Lpw(R) functions with support in the interval [-K, K). Observe that for any weight w that satisfies (1.9), the norm || • \\PtW in Lpw([-K, K\) is equivalent to || • ||p in Lp{{-K, K]). Therefore, part of the results of Theorem 1.1 can be stated as follows. Theorem 1.3 Let a, p, c, K and Wcoo w{ax) then w satisfies (1.10), and we have

(1.14)

Theorem 1.4 Let l 0. Then E(Wc,a,Lpw(R))D{\€C

0 < |A| < r} U S 0 ,

a(W c , Q ,LP(R)) = { A G C |A| < ra1'1^}

U S0,

1 1 p

P ( ^ , a , F J R ) ) = { A e C |A| > r a - / } \ E 0 , p(WCta,Lpw(R))

= max(l, r a 1 " 1 ^ ) .

Consider the weight ws(x) = (1 4- | i | ) s , where s £ R. Then lim|x|_).00 ws(x)/ws(ax) = a~s and LP(R) = LP, 0 (R). Therefore, by taking w — ws in Theorem 1.4, we have

55

Corollary 1.5 Let p,a,c 3

and Wc>a be as in Theorem 1.4, and let ws(x) =

(l + |x|) , s £ R . Then £(Wc,a,L^(R))D{AeC

0 < |A| < a " s } u S 0 ,

a(Wc,a,Lla(R))

=

{\€C

lAl^a-'+^/PjUEo,

P(Wc,a,Lls(K))

=

{\GC

|A| > a - + 1 - 1 / " } \ S 0 )

p(W c , Q) LS,.(R)) = m a x ( l , Q - + 1 - 1 / " ) . Next, we shall show that if (1.9) is satisfied then lim ^ = 0 (1.15) |x|->oo w{ax) is a necessary and sufficient condition for WCta to be a compact operator on L^(R). This characterization of Wc 1, andc be a compactly supported function in L 1 (R) with J-Rc(x)dx = 1, and suppose that w is a weight function that satisfies (1.9). Then WCta is a compact operator on L^,(R) if and only if w satisfies (1.15). Furthermore, if (1.15) holds, then £(W c , Q ,L£(R)) = £o

(1.16)

and £7(WC,Q,JLP(R))

= {0}US0.

(1.17)

Now, take positive numbers A and 7 with 7 < 1. Since lim|x|_+00 ex^~a~'^x^ = 0, by setting w(x) = ex^x^ in Theorem 1.6, we obtain the following corollary. Corollary 1.7 Let p, a, c, WCyCt be as in Theorem 1.6, and let w{x) = ex^x^ for some A > 0 and 0 < 7 < 1. Then WCi J

1 and f G V , _„ -.(R). —

J

w\OL

-)K

'

Rewriting (1.1) as Wc,af(x)

= a2 f c(a(x - y))f(ay)dy,

(2.1)

./R

and repeated application of (2.1) n times gives W?J(x)

= [ Kn(x - y)anf(any)dy

for all n > 1,

(2.2)

where Kn(x) = (ac(o-)) * • • • * (« n c(a n -)), and / * g denotes the convolution of two integrable functions / and g.

(2.3)

57

Lemma 2.2 Let a and c be as in Theorem 2.1, K0 be chosen so that c is supported in [—(a - l)K0, (a - 1)K0], be the solution of (1.2) normalized so that fn(x)dx = 1, and Kn(x),n > 1, be as in (2.3). Then supp(Kn(-)) C [-Ko,Ko]

for all n > 1,

(2.4)

and 0.

lim \\Kn

(2.5)

n-4-oo k Proof. Note that akc(ak-) is supported in [-(a-l)a K0, (a-l)a~kK0] for any k > 0. Therefore Kn{x),n > 1, are supported in [ - ( a — a k a l ) ^ o E L i ~ , ( - l ) « o E L i 1. Lemma 3.4 Let w(x) be a weight function that satisfies (1.9) and (1.10). (i) If w satisfies (3.1), then for any 0 < 6 < 1/2 there exists a positive constant C\ independent of x such that w(x) < C i ( l + | z | ) - l n r i / l n Q + < 5

forallxGR.

(3.6)

60

(ii) Ifw satisfies (3.4), then for anyO < 5 < 1/2, there exist positive constants Ci and C3 independent of x and n such that w(anx) w(anx)

> C27vTn(l + 6)-nw(x), \x\ > 1, n n > C3 m i n ( l , r ^ ( l + Sy )w(x), \x\ < 1,

(3.7) (3.8)

for all n > 1.

Lemma 3.5 Let w(x) be a weight function that satisfies (1.9). Then ( mm w(x))\\f\\p < \\f\\p < ( maxw(x))\\f\\p xE\a,b\

(3.9)

x£[a,b\

for any function f G i^([a, 6]), 1 < p < oo. Proof of Theorem 3.1. To prove (3.2), we note that l?w([-K,K]) C L^,(R), for any 1 < p < oo and K > 0. Then every eigenvalue of the operator WCiCl restricted to L£,([—K, K]) is an eigenvalue of Wc,a restricted to L^(R). This together with Theorem 1.3 gives So = E(Wc,a,Ll([-K0,K0}))

C E(Wc,a,Ll(B.)).

(3.10)

Let A be any complex number that satisfies 0 < |A| < r\ and A 0 Eo- Then by (3.10), the proof of (3.2) reduces to proving that A G E(WCt is the solution of (1.2) normalized so that fR(x)dx = 1. Set OO

71=1

Then 4>A^0,

(3.14)

61

because ( • — an) are supported in the sets [—KQ, KQ] + an, n = 1,2,..., which are mutually disjoint for sufficiently large n. Let 5Q > 0 be chosen that a2So\X\ = n. Using (3.6), (3.12) and (3.13) with 50 chosen as 5 leads to

\\x\\P,v,x = -{Wc,a

- \I)il>x +4>(- - 1) = 0.

(3.16)

It follows from (3.14), (3.15) and (3.16) that A is an eigenvalue of Wc,a restricted to -L^,(R). This completes the proof of (3.2). To prove (3.3), recall that a(WCta,LPv{R)) is closed and contains {0}USo. Then, by (1.3) and (3.2), it suffices to prove that for any A ^ So with 0 < |A| < n a 1 - 1 / p , there exists / „ 1, such that / „ ^ 0 and

lim I K ^ - ^ I I ^ = 0. ™->°°

(3.17)

ll/n||p,tu

Let be t h e solution of (1.2) normalized so that JR(/>(x)dx = 1, a n d let TOO be t h e minimal positive integer so t h a t am°(l — a - 1 ) > 4KQ and sup

\\ 1 and f G LJ, (R), and

U^cVIU < C(l + b)nan-nlprZ||/|U

(3.28)

n

/or all n > 1 and / G £ „ ( R ) ura£/i support in R\[—ra ,aa™]. Proof. For any 0 < 5, a < 1, by (3.7) and (3.8), there exists a positive constant C independent of n such that ll/llp,»(a-«.) < C r J ( l +Jni/llp,™ for all / G U>w(R \ {-aan,aan}),

(3.29)

and l l / l l P , » ( a - - ) ^ C r m a x ( 1 . r 2 ( l + 5) n )||/|| P ,«, for all / G L*(R).

(3.30)

Thus (3.27) and (3.28) follow from (3.29), (3.30) and Theorem 2.1. * Lemma 3.7 Let p,a,w,c E(Wc,a,Ll(R))

and WC)Q be as in Theorem 3.2. Then c {A G C : |A| < rao 1 " 1 /"} U E 0 .

(3.31)

Proof. Suppose on the contrary that there exists a complex number A G £(W c , a ,L&(R)) with |A| > raa 1 " 1 /", A i S 0 . Then Wc,af = Xf

(3.32)

for some nonzero / G L^(R). Since A g" So, / does not have compact support by Theorem 1.3. Hence there exists an integer no such that |n 0 | > l/(a — 1) + 2K0/(a - l ) 2 and / ^ 0 on [n0,n0 + 1]. Define fi0 = [no, no + 1] and

64

flk = [akn0 - Ko^Ja^a^no + 1) + /f0 Ej=o ^'l f o r k ^ * > a n d s e t = /fc fxtik- Thenfi*.,fc> 1, are mutually disjoint. By Lemma 3.3, Wc,ag is supported in R\fifc_i for any function g with support in R\fifc, k > 1. Therefore from (3.32), = Afc/o + ( / - /o) - Wcka(f - fk) = Xkf0 + h,

(3.33)

where fk, k > 1, are supported in R\fio- This implies that \\WckJk\\p,w

> |A|fe||/o||PlU).

(3.34)

Since fk is supported in Q.k for any k > 1, by Lemma 3.6, there exists a positive constant C independent of k > 1 such that l|W c * a /*IU < C ( l + C ( l + 5o)*||/o|| P ,»,

Com(3-36)

for sufficiently large k, where C is a positive constant independent of k and / . Since ||/|| P)U , > ||/fc||p,«, for all k > 1, (3.36) implies that ||/|| p ,«, = oo, which is a contradiction. 4 Proof of Theorem 3.2. Let A be a complex number that satisfies |A| > ^a1-1^ and A £ S 0 . By Lemma 3.7, A 0 £(W C , Q , Z£, (R)). Therefore, (Wc>aXI) is injective on L^ (R). Then it remains to show that for any / £ Lg, (R), we can find g £ ££,(R) such that \\g\\P,w < C\\f\\p,w

(3.37)

(WClQ - XI)9 = / ,

(3.38)

and

where C is a positive constant independent of / . Write oo

/ = fX[-a,a] + ^2 (fX(*i,ai+1] + /X[- + i,-)) = 2_/ &' 3=1

JSZ

where / 0 = fx[-a,a], fj = fX(ai,&+i] and f-j = /X[-+i,-) for 3 > 1Then the support of / , , j € Z, are mutually disjoint and | | / I U = ll(ll/illP,u,)J-6z||/p

(3-39)

65

by the definition of the norm in L£,(R). Here and hereafter, for any countable index set A and 1 < p < oo, we let £p(A) := {D = (dj)i€A : dj G C } , and define the norm on £P(A) by

» o »--( Ei "K'r , ""! i -' ,0.

(3.46)

1=1

By Lemma 3.3, Wi.a{fi+j + f-i-j) is supported in [-al+1 - K0, -a1 + K0] U l+1 [a' — KQ, a +Ko], which have finite overlaps for any given j . This together with (3.39) and Lemma 3.6 leads to II&IIP,™

< CMWiMi+i

+

f-i-j)\\p,»)i>i\\t>

< C2r-ja^-l'^{l

+ Cs(l + \x\)a. This implies that i^,(R) C l £ ,, ,y(R) for any s > 1. Hence any eigenvalue of the operator WCtCl restricted to L£,(R) is an eigenvalue of the operator Wc>a restricted to ^fi+l-i).^)- Therefore by Corollary 1.5, E(Wc,a,L?w(R))

c E{Wc,a,L*1+H).(B.))

C

a ,LP (R)) since L£,(R) C L ^ R ) . This together with (4.7) and (4.8) lead to (1.16). * Acknowledgments. The research is supported by the Wavelets Strategic Research Programme, National University of Singapore, under a grant from the National Science and Technology Board and the Ministry of Education, Singapore. References 1. M. Baake and R. V. Moody, Self-similarities and invariant densities for model sets, in: Algebraic Methods and Theoretical Physics (Aubin Y St ed.), Springer, New York, In press. 2. M. Baake and R. V. Moody, Multi-component model sets and invariant densities, in: Aperiodic '97 (Verger-Gaugry J. L.), World Scientific, Singapore, In press.

70

3. A. Cohen and I. Daubechies, A new technique to estimate the regularity of refinable functions, Rev. Math. Iberoamericana, 12(1996), pp. 5 2 7 591. 4. A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision, Memoir Amer. Math. Soc, 93(1991), pp. 1 - 186. 5. C. K. Chui and X. Shi, Continuous two-scale equations and dyadic wavelets, Adv. Comp. Math., 2(1994), pp. 185 - 213. 6. W. Dahmen and C. A. Micchelli, Continuous refinement equations and subdivision, Adv. Comp. Math., 1(1993), pp. 1 - 37. 7. G. Derfel, N. Dyn and D. Levin, Generalized functional equations and subdivision processes, J. Approx. Theory, 80(1995), pp. 272 - 297. 8. T. N. T. Goodman, C. A. Micchelli and J. D. Ward, Spectral radius formulas for subdivision operators, in Recent Advances in Wavelets Analysis, L. L. Schumaker and G. Webb, ed., Academic Press, 1994, pp. 335-360. 9. T. N. T. Goodman, C. A. Micchelli and J. D. Ward, Spectral radius formulas for the dilation-convolution integral operator, SEA Bull. Math., 19(1995), pp. 95 - 106. 10. L. Hennion, Sur un theoreme spectral et ses applications aux noyaux lipschitziens, Proc. Amer. Math. Soc, 118(1993), pp. 627 - 634. 11. M. C. Ho, Spectra of slanted Toeplitz operators with continuous symbols, Michigan Math. J., 44(1997), pp. 157 - 166. 12. R. Q. Jia, Subdivision schemes in Lp spaces, Adv. Comp. Math., 3(1995), pp. 309 - 341. 13. R. Q. Jia, S. L. Lee and A. Sharma, Spectral properties of continuous refinement operators, Proc. Amer. Math. Soc, 126(1998), pp. 729 737. 14. K. Kabaya and M. Iri, On operators defining a family of nonanalytic C°°-functions, Japan J. Appl. Math., 5(1988), pp. 333 - 365. 15. W. Lawton, S. L. Lee and Zuowei Shen, Stability and orthonormality of multidimensional refinable functions, SIAM J. Math. Anal., 28(1997), pp. 999 - 1040. 16. D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9(1969), pp. 267 - 278. 17. D. Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes Etudes Sci. Publ. Math., 72 (1990), pp. 175 - 193. 18. D. Ruelle, Spectral properties of a class of operators associated with maps in one dimension, Ergod. Th. Dynam. Sys., 11(1991), pp. 7 5 7 - 767. 19. V. A. Rvachev, Compactly supported solutions of functional-differential equations and their applications, Russian Math. Survey, 45:1(1990), pp. 87 - 120.

71

20. G. Strang, Eigenvalues of (J, 2)H and convergence of cascade algorithm, IEEE Trans. Signal Processing, 44(1996), pp. 233 - 238. 21. Zhou Ding-Xuan, Spectra of subdivision operators, Proc. Amer. Math. Soc, 129 (2001), pp. 191-202.

DATA T U N I N G

P E T E R HALL Centre for Mathematics

and its Applications, Australian Canberra, ACT 0200, Australia E-mail: [email protected]

National

University,

Time was when altering one's data was sacrilegious. A major difficulty was that we didn't know how to do it objectively; altering the d a t a according to objective criteria turns out to be a surprisingly computer-intensive business, and in many instances wouldn't have been feasible a decade or two ago. Today, however, thanks to the ready availability of computing power, we can do all sorts of complex things to our data. Data-tuning methods alter the data so as to enhance performance of a relatively elementary technique. The idea is to retain the advantageous features of the simpler method, and at the same time improve its performance in specific ways. Different approaches to d a t a tuning include physically altering the d a t a (data sharpening), reweighting or tilting the data (the biased bootstrap), adding extra " pseudo data" derived from the original data, or a combination of all three. Tilting methods date back to the 1950's, although only recently have they become popular. Evidence is growing, however, that sharpening is more effective than tilting, since it doesn't reduce effective sample size.

1

Introduction

Data-tuning methods alter the empirical distribution so as to enhance performance of a relatively elementary technique. The idea is to retain the advantageous features of the simpler method, and at the same time improve its performance in specific ways. Different approaches to data tuning include physically altering the data (data sharpening), reweighting or tilting the data (e.g. the biased bootstrap), adding extra pseudo data derived from the original data, or a combination of all three. In isolation, and for particular applications, none of the above methods is particularly new. All three have found application to rendering statistical methods more robust. For example, trimming is an extreme form of data reweighting or tilting; winsorising is a more satisfactory analogy. If, when altering the position of a data value, one always moves it to an existing datum, then data sharpening and reweighting are often identical. Likewise, if pseudo data are always added on top of existing data then the result is generally indistinguishable from reweighting. More interesting approaches to the use of pseudo data include methods for reducing edge effects in nonparametric curve estimation; see for example Schuster (1985), Hall and Wehrly (1991) and Cowling and Hall (1996). 72

73

Perhaps the best-known recent application of tilting methods is empirical likelihood; see Efron (1981) and Owen (1988, 1990). However, Grenander's (1956) notion of "nonparametric maximum likelihood" is in the same spirit, as too are the ideas behind the Kaplan-Meier (1958) estimator. A more general biased-bootstrap approach, viewing tilting methods as a means of solving a very wide range of contemporary problems in nonparametric statistics under constraints (the biased bootstrap), has been suggested by Hall and Presnell (1999a). These ideas will be developed in section 2. An explicit method for perturbing data, for the purpose of bias reduction in nonparametric density estimation, was discussed by Samiuddin and elSayyad (1990); see also Jones and Signorini (1997). Choi and Hall (1999) suggested the name "data sharpening" for methods of this type, and Choi, Hall and Rousson (1999) developed variants of the idea for nonparametric regression. Implicit methods for data sharpening, where data are moved by the least amount subject to enforcing a constraint, have a very wide range of applications and may be viewed as competitors with the biased bootstrap; see Braun and Hall (1999). These methodologies will be surveyed in section 3.

2 2.1

Tilting, and Biased Bootstrap Methods The Problem and its Solution

In standard settings, where the appropriate way of applying the bootstrap is relatively clear, the uniform bootstrap offers an unambiguous approach to inference. Therein lies part of its attraction; there are no tuning parameters to be selected, for example. However, the lack of ambiguity can also be a drawback. In particular, the rigidity of the conventional bootstrap algorithm makes it relatively difficult to modify uniform-bootstrap methods so as to include constraints on the parameter space. The tilted or weighted bootstrap offers a way around this difficulty, by providing an opportunity for biasing bootstrap estimators so as to fulfill constraints. Moreover, we may interpret the notion of a constraint in a very broad sense, much broader even than that of a parameter. For example, the constraint can be purely qualitative, asking that a nonparametric function estimate be monotone or have a given number of modes. Nevertheless, an unambiguous approach to choosing the weights is required. Biased-bootstrap methods provide a solution to that problem. The biased bootstrap requires two inputs from the experimenter: the distance measure, and the constraints. The first is generic to a large class of problems, and will be discussed from that viewpoint in section 2.2. The second is problem-specific, and will be briefly introduced through examples

74

in section 2.3. 2.2

Weights and Distance Measures

The generality of the standard, uniform bootstrap may be increased by allowing the resampling distribution to be a general multinomial. In this setting we shall use a dagger instead of the familiar asterisk notation, so that there will be no ambiguity about the procedure we are discussing. Thus, given a sample X — {Xi,..., X„} we draw a resample X^ = {X\,..., Xl} by sampling independently according to the following scheme: P{X\ =Xj\X)=Pj,

l 0, although for some distance measures the case where negative pi's are allowed has at least computational advantages, even though it cannot be readily interpreted in the context of (2.1). Given p ^ 0 or 1 we may measure the distance between p and pUnif by

Dp(p) =

{P(i-p)riL-J2(npir}. *•

4 = 1

>

This quantity is always nonnegative, and vanishes only when p = pUmf- For p = | , Dp(p) is proportional to Hellinger distance. Letting p —> 0 we obtain n D

o(p) = ~^2

log (npi),

which equals half Owen's (1988) empirical log-likelihood ratio. Similarly, D\ may be defined by a limiting argument; it is proportional to the KullbackLeibler divergence between p and pUnif, whereas Do(p) is proportional to the Kullback-Leibler divergence between punif and p.

75

In constructing a biased-bootstrap estimator we would select a measure of distance and then compute p = ( p i , . . . ,pn) from the sample X = {Xi,..., Xn} so as to minimise Dp(p), subject to the desired constraints being satisfied. If the parameter value that we wished to estimate was expressible as 6(F), then its biased-bootstrap estimator would equal 9(Fp), where Fp denotes the distribution function of the discrete distribution that has mass pi at data value Xi for 1 < i < n. Sometimes the value of 8(Fp) will not be computable directly, but it may always be calculated by Monte Carlo methods, resampling from X according to the scheme that places weight pi on Xi. In some applications, for example outlier reduction (Hall and Presnell, 1999b), there are advantages to taking p ^ 0, since Do(p) becomes infinite whenever some pi = 0. By way of comparison, Hellinger distance (i.e. the case p = | ) allows one or more values of pi to shrink to zero without imposing more than a finite penalty. However, in most other applications there is little to be gained, and sometimes, something to be lost, by using a value of p other than p = 0. 2.3

Examples

Empirical likelihood. The method of empirical likelihood, or EL (Owen, 1988, 1990), may be viewed as a special case of the biased bootstrap in which the constraint is 8(FP) = 0\, where Fp denotes the distribution function of the weighted bootstrap distribution with weights pi, and 9\ is a candidate value for 8. It is based on the value p = p(6\) of p that minimises Dp(p) subject to 8{FP) = 9L One EL approach to constructing an a-level confidence interval for the true value of 8 is to take ta to be the upper a-level quantile of the chi-squared distribution for which the number of degrees of freedom equals the rank of the limiting covariance matrix of the uniform-bootstrap estimator, 8(FPuni{); and to let the interval be the set of d^s such that Dp{8(Fp^l))} < ta. Under regularity conditions that represent only a minor modification of those of Hall and La Scala (1990), this interval may be shown to have asymptotic coverage equal to 1 — a, no matter what the value of p. Using methods of DiCiccio, Hall and Romano (1991) it may be shown that this generalised form of EL is Bartlett-correctable if and only if p = 0. (Strictly speaking, the term " likelihood" is appropriate for describing these generalised EL techniques only if p = 0.) See also Baggerley (1998) and Corcoran (1998). Variance stabilisation. Here we wish to choose, by empirical means, a transformation g which, when applied to a (scalar) parameter estimator 8, will

76

implicitly correct for scale. The method suggested here is a biased-bootstrap version of a conventional-bootstrap technique proposed by Tibshirani (1988). It has an advantage over the latter approach in that it does not require selection of any smoothing parameters, or any extrapolation. As in the previous example, choose p to minimise Dp(p) subject to 6(FP) = 0\. Let X^ = {X\,... ,X^} denote a resample drawn by sampling from X using the weighted bootstrap with weights pi, and let #t denote the version of 9 computed from X^ rather than X. Let v(9{) = var(0t| 1 is an integer and Xp = Xp(k) minimises Y^i Pi (-^» _ x)2k with respect to x. (Taking k = 1 we see that 7(p, X) is the variance of the biased-bootstrap distribution.) Put 7 = 7(pUnif,

Figure 5. Several trajectories of a simple random walk reflected between two barriers.

Figure 6. Several particles undergoing two-dimensional simple random walk reflected at the boundaries of a box.

run the simulation for until one can reasonably suppose that equilibrium has established itself? In fact the right kind of simulation can achieve equilibrium quite quickly (this is the case, for instance, for the point pattern example described earlier on), but on the other hand it can take quite a long time, and it can even be quite deceptive, in that one may feel the system has settled down when in fact it has still got quite a way to go. The big question is, How to be sure? Is there a way to figure this out, or

99 are we going to have to be content with the advice, it will take a long time so just guess what you think might be long enough? 4

The quest for perfect simulation

Till very recently the general view was, the only general way of solving this problem was to be good at guessing how long would be long enough. This isn't very satisfactory, because one could always end up committing a big mistake that way, but it is not an unfamiliar situation in mathematical science. Quite often the theory will take you a long way but not the whole way to the ultimate goal, and will then deposit you and say, it is up to you now, use your own skill and judgement and experience to do the best you can . . . So we all thought this would be as good as it could get. And then suddenly, just four years ago, a preprint started circulating, written by a couple of American mathematicians, Jim Propp and David Wilson. It showed that in potentially quite complicated situations one really could do a great deal better. Fundamentally they pointed out that one could often modify the random simulation using a bit of lateral thinking, so that it would no longer take a long time to get approximately to equilibrium, but instead would run for a random time, then stop, and deliver a result that was exactly in equilibrium! This seemed like a miracle, which is why it was christened "perfect simulation". I shall describe this idea for a very simple and picturesque example, the "dead leaves" model. (An animation in the talk also showed how to do this for an example based on the random walk.) The example is to do with a model used in mathematical geology, a model for the sort of image you see when you make a section through some rock. It is the "dead leaves" model, invented by workers at the Ecole des Mines in Fontainebleau near Paris. Fontainebleau is in the middle of a forest, and around September all the tree leaves fall down and produce beautiful random patterns on the ground. So it was very natural for people working there to propose the following model. Imagine an image produced by placing many overlapping objects (the "dead leaves") one after another at random on the ground. As the leaves continue to fall, the pattern keeps changing. However after a long time it will have settled into some kind of approximate equilibrium. The equilibrium itself is the dead leaves model. We never actually reach equilibrium this way; we just get closer and closer as time goes on. The model is very simple, and it is possible to compute exactly how close one gets to equilibrium after a finite time (and one does so fast). However with a minor change in perspective one can alter matters so

100

Figure 7. T h e "dead leaves" model (before equilibrium is achieved).

that one gets exactly to equilibrium in a finite amount of time. It is just a matter of freeing oneself from the prejudice of being a human being, rather than a rather smaller creature! Human beings walk among the dead leaves looking down on a constantly changing pattern. However imagine yourself to be a small furry animal in the forest. Falling leaves mean winter is close, so you hide in a hole for warmth and protection. As the leaves fall the hole is covered, and you look up at the pattern. At any fixed time the pattern is essentially the same as would be seen by a human; a pattern produced by leaves placed randomly and overlapping. However from the animal's point of view the pattern eventually stops changing (when the hole is completely covered by leaves) whereas from the human's point of view the pattern is constantly changing. There are three facts to bear in mind: 1. As the leaves fall, the pattern seen by the human approaches statistical equilibrium; 2. As the leaves fall, the pattern seen by the animal eventually stops changing; 3. Both patterns, whether seen from above or below, are the same from a statistical point of view. These three facts result in an intuitively reasonable conclusion; when the hole is completely covered and the pattern stops changing then the animal sees a perfect draw from statistical equilibrium.

101

+:+ +

+

4.+ +

+ ++ +

+ +

- .

;+ ^

+

+

+

^

+ •++ * , :+ +

++ +

+

•'++*-

•+ +

+ + •:"lt* + •-t-+ -1* +4 ' J. *' , " . '

+ ++"*"++ '+ -H+ + + + Figure 8. Perfect simulation of a point process with repulsive interaction between the points.

This is a very simple example, but the basic ideas can apply to a whole variety of situations. Much current work is about following through these ideas for more and more complicated (and useful!) situations. For example, I have worked with others on how to use this idea to produce perfect samples point patterns which are partly random, but are modified for example so as to produce fewer pairs of neighbouring points (so that in some sense the points repel each other). We modified the simulation described above in section 2 using ideas sparked off by reading Propp and Wilson's work. The diagram here was produced this way. This talk began with the question of how to calculate with uncertainty, moved on to the issue of updating such calculations when new information becomes available, sketched out how one can use simulation to help carry out the updating calculations, and finally described how some beautiful mathematics can allow one to carry out such simulations perfectly. My hope is that it has thus succeeded in communicating some of the excitement and challenge of this area of mathematical science.

A U T O M O R P H I C R E P R E S E N T A T I O N S A N D COHOMOLOGY OF A R I T H M E T I C G R O U P S J I A N - S H U LI Department

of Mathematics, HKUST and Zhejiang Clear Water Bay, Hong Kong E-mail: matom.Qust.hk

University,

JOACHIM SCHWERMER Institut

1

fur Mathematik, Universitat Wien, Strudlhofgasse A 1090 Wien, Austria E-mail: joachim.schwermer@univie. ac. at

4,

Introduction

Let G be a semisimple Lie group with associated (Riemannian) symmetric space X = G/K, where K is a maximal compact subgroup of G. Suppose T is a lattice in G (a discrete subgroup such that T\G has finite volume with respect to a non-zero G-invariant measure). Any finite dimensional representation of G on a complex vector space E defines a local system on T\X, and we are interested in the cohomology space H*(T\X, E) of T\X with coefficients in the local system associated to E. It is well-known that this agrees with the Eilenberg-MacLane cohomology space H*(T,E) of T with coefficients in E. On the other hand, there is an expression of the same space in terms of relative Lie algebra cohomology. If Q denotes the complexified Lie algebra of G one has the isomorphisms H*(T, E) ~ H*(T\X, E) ~ H*(g, K; C°°(T\G) ® E),

(1.1)

and J. Franke has shown recently that one may replace C°°(r\G) on the right hand side by the space of automorphic forms. When T\G is compact, the Matsushima formula (see Section 2 below) relates the above cohomology space to the spectral decomposition of L2(T\G) The first author was supported in part by NNSFC Grant no. 19928103, RGC-CERG G R A N T HKUST6169/99P, RGC-CERG grant HKUST6126/00P, and the Cheung Kong Scholars Programme. He would like to thank the organizers of the conference and the National University of Singapore for their wonderful hospitality. The second author would like to thank the Hong Kong University of Science and Technology and his host for their support and hospitality in February 2001 when this paper was finalized.

102

103

in a explicit fashion. In this case the study of cohomology is seen as a (far from solved) sub-problem of the spectral theory of square integrable automorphic forms. When T\G is not compact, Franke's result again shows that the study of cohomology is intimately related to automorphic forms. But in this case the precise relationship is much more subtle. This paper is intended as a survey of some recent results concerning the cohomology of arithmetic groups and its relationship with the theory of automorphic representations. Thus in Section 2 we review the Vogan-Zuckerman theory of unitary representations with non-zero cohomology, and deduce from it some vanishing results which are local in the sense that they can be deduced from abstract representation theory and the Matsushima formula. We will also discuss a few examples of vanishing results which are global in nature, in that they are arithmetic and depend on some non-trivial understanding of irreducible subspaces of L2(T\G). Section 3 contains a discussion of the result of Franke and some of its consequences. First, it gives an isomorphism between the cohomology of congruence subgroups of a connected reductive algebraic group G over Q and appropriate cohomology spaces attached to the space of automorphic forms on G(Q)\G(A). Second, based on a decomposition theorem for functions of uniform moderate growth (established by Langlands in 1972), one obtains a decomposition of the cohomology as a direct sum parameterized by the set C of classes of associate parabolic Q-subgroups. Next, we discuss as a refinement the decomposition along the cuspidal support. This gives a structural description of the cohomology in terms of summands parameterized by C and, for a given {P} £ C, the set of classes of associate irreducible cuspidal automorphic representations of the Levi components of elements of {P}. This is related to the fact that each automorphic form can be written as a linear combination of appropriate residues or derivatives of Eisenstein series attached to cuspidal automorphic forms on the Levi components. The internal structure of these cohomological subspaces in terms of the actual construction of Eisenstein cohomology classes is discussed in Section 4. This leads to an analysis of the analytic properties of certain Euler products attached to cuspidal automorphic representations of groups of smaller rank. In Section 5 we review some of the methods that have been used to prove the existence of automorphic forms of various types and the non-vanishing of corresponding cohomology classes. In the non-compact case we mainly focus on the construction of cuspidal cohomology classes. The subject of this survey is related in various ways with number theory, geometry and arithmetic algebraic geometry. Thus, this expository account is incomplete in several aspects. Some other lines of approach to the study of the cohomology of arithmetic groups are indicated in the final Section 6.

104

Notation 1.2 Following a fairly common practice, the real Lie algebra of a Lie group is denoted by the corresponding gothic letter with a subscript 0, and its complexification by the same gothic letter without subscript. In Section 2 we use G to denote a Lie group. However starting from Section 3 the same letter will denote an algebraic group defined over the field of rational numbers, and G(R) is the corresponding group of real points. The real and complexified Lie algebras of G(R) will now be denoted g0 and $j respectively. Analogous notations apply to groups denoted by other letters. All algebraic groups considered are linear. If G is an algebraic group defined over some field k and A is an commutative algebra over k, we denote by G{A) the group of A-valued points of G. Given an algebraic group defined over some number field, we (almost always) view it as a group over Q by restriction of scalars. Here are some specific notations concerning parabolic subgroups. Let G be a connected reductive group defined over Q. We assume that a minimal parabolic Q-subgroup PQ of G and a Langlands decomposition PQ = MQAQNQ have been fixed. Unless otherwise indicated, parabolic subgroups of G will be assumed to be rational and standard with respect to PQ. For any such parabolic subgroup P there is a unique Langlands decomposition P — MpApNp with AP C A0, Mp 2 M0. The Levi component is denoted by Lp = MpAp. 2

The Matsushima Formulas and Vanishing Results

2.A. Matsushima Formulas. Suppose G is a reductive Lie group with finitely many connected components. Fix a maximal compact subgroup K, and set X = G/K. Let T C G be a lattice in G. If E is a finite dimensional representation of G, the corresponding local system on T\X will again be denoted E. The L2-cohomology space HT2JT\X,E) is defined to be the cohomology of the complex of E-valued smooth differential forms LJ such that u) and du are both square integrable; see for example [11]. This space is related to the decomposition of the regular representation of G on L 2 ( r \ G ) via a Matsushima type formula as follows. Write L^isc(T\G) for the discrete part of L 2 ( r \ G ) : the sum of all irreducible subspaces of L 2 ( r \ G ) . We have ^Lc(r\G) = 0

m(7r,r).H f f

(2.1)

7T6G

where G is the unitary dual of G and the multiplicity m(7r, T) is a non-negative integer for each n. When certain conditions on T are satisfied, one has the formula

105

tf(*2)(r\X,£)~0

m(Tr,r)-H*(9,K;H«®E)

(2.2)

TT€G

where (by a slight abuse of notation) we have identified Hn with its underlying (Q, If )-module. In general the right hand side is a subspace of H?2JT\X, E). We know from Borel-Casselman [11] that 2.2 is always valid when G has a compact Cartan subgroup. It is also valid when T is cocompact; in that case HT2s (T\X, E) agrees with the ordinary cohomology space H* (T\X, E) and 2.2 reads H*(T\X,E)~

0

m(7r,r)-H*(fl,K;H n ®E)

(T\G compact)

(2.3)

weG

This is the usual Matsushima formula (see [13] p.235). In general we have H*(T,E)

~ H*(T\X,E)

~ H*(a,K;C°°(T\G)

® E)

Let Llusp(T\G) C L^isc(T\G) be the G-invariant subspace of cusp forms [34]. Using the superscript oo to denote subspaces of smooth vectors, one has the obvious inclusions L2cusp(T\G)°°

C L L c ( r \ G ) ° ° C C°°(r\G)

which lead to homomorphisms of cohomology groups H*(g, K; L2cusp(r\G)°°

® E) —)• H*(g, K; L2disc(T\Gr —>ff*(r\X,E)

® ^) (2.4)

The second homomorphism is not injective in general. But the composition of the two homomorphisms gives rise to an inclusion map H*(g, K- L2cusp(T\G)°°

®E) H*(T\X, E)

The image of this last map will be denoted H*usp(T\X,E) and is called the cuspidal cohomology of T\X (or of T) with coefficients in E. Let mc(n, T) be the multiplicity of 7r in L^ u s p (r\G), i.e. ic«sp(r\G) = 0

m c (7r,r)-H»

TTGG

Then H*cusp(T\X, E) = 0 7T6G

mc(7r, r ) • H*(Q, K; Hn ® £ )

(2.5)

106

2.B. Cohomological Representations. The representations that could (possibly) contribute to the right hand sides of 2.2, 2.3 and 2.5 are those IT with

H*(g,K;H„®E)?o for some E. They are usually referred to as unitary representations with nonzero cohomology. Through the work of Parthasarathy [74] [75], Kumaresan [51], Enright [20], Speh [93], and finally Vogan and Zuckerman [100], one has a fairly complete understanding of these representations. The paper by Vogan and Zuckerman is now the standard reference for information on unitary representations with non-zero cohomology. We recall some basic facts from [100]. Let K° be the connected component of K. Then K acts on H*(Q,K°; Hn E), and it is known [13] that H*(Q, K• Hv ® E) = i T ( 0 , K°; Hn ®

E)K'K°

For simplicity we shall assume for the rest of this section that K = K°, i.e. G is connected. Let Q = t + p be the Cartan decomposition determined by K, and let 6 be the corresponding Cartan involution. The unitary representations with non-zero cohomology are associated to various ^-stable parabolic subalgebras of Q. Let q C Q be one of them, with a (0-stable) Levi decomposition q = l + u. Then [ is the complexification of a real subalgebra to C go- The normalizer of q in G is connected since G is ([47], Lemma 5.10), and coincides with the connected Lie subgroup of G with Lie algebra to- It will be denoted L. The relevant representations arising from q are obtained via cohomological induction from one dimensional unitary representations of L. If A € L we denote its differential also by A. It must satisfy a positivity condition ([100], p. 73). The resulting representation of G is denoted ^4q(A) (we shall not distinguish an irreducible unitary representation from its underlying (g,lf)-module). Then every irreducible unitary representation with non-zero cohomology is equivalent to some Aq(A). Let t)o C [0 be a ^-stable fundamental Cartan subalgebra (of both fo and 0o). Then to = f)0 n to is a Cartan subalgebra of both 6o and fo H 6o- Let A + (g, h) be a positive root system containing all roots of f) in u. Let p be the half sum of all roots in A + (g, h). Assume that E is irreducible. We summarize some properties of Aq (A). Theorem 2.6 (Vogan and Zuckerman [100]) In the above setting one has (a) The representation ^4q(A) has infinitesimal character defined by A|(, +p (via the Harish-Chandra homomorphism). If the contragredient representation

107

E* has the same infinitesimal character (this is the same as saying E* has highest weight A|f,, or equivalently, E has lowest weight — X\t)), then H\Q,

K- Aq(X) ® E) ~ HomLnK(Ki-R{l

n p),C),

(2.7)

where R = R(q) = dim (u n p) Otherwise, we have H1(Q, K; Aq(X) (g> E) = 0 for all i. (b) Suppose G is of hermitian type. Then H*(g,K;An(X) E) admits a Hodge decomposition. Let p = p + ®p~ be the usual decomposition determined by the complex structure of X = G/K, and set R± = i ? ± ( q ) = d i m ( u O p ± ) Assume that E has lowest weight — A|f,. If p — R+ = q — R~ = j > 0 one has H™(S,K;Ati(\)®E)~HomLnK(A2i(lnv),C),

(2.8)

Otherwise, H™(Q,K;A,(\)®E)=0

(c) The representation Aq(X) is in the discrete series if and only if L is compact. It is tempered if and only if L contains no non-compact simple factors. These two conditions are equivalent when G has a compact Cartan. Thus if G has a compact Cartan then the only tempered Aq (A) 's are actually in the discrete series. Let ra be the smallest integer of the form dim(u n p), where u is the unipotent radical of a proper ^-stable parabolic subalgebra of Q. It suffices to calculate re when G is simple. This is done in [51] and [20] for complex groups, and in [100] for real groups. The number ra is a useful invariant attached to G. It is not clear if one could derive a general formula in terms of the structure of the Lie algebra go (rather than listing it via a table). In general one knows ro > the split rank of G. More precisely one has re = split rank of G for G = SL(n), SU(p,q),SO(p,q),Sp(n) (split symplectic groups of rank n), ra > split rank of G in all other cases. We take this opportunity to point out a misprint in the table on p. 88 of [100]. For the simple Lie group G = iJ 6 (_ 14 ) of Cartan label EIII (type Eg, rank 2 and of hermitian type), it is stated that rg = 8 with dim(u Pi p) = 8 achieved by a maximal parabolic q = I + u, where [lo, lo] — so (2, 8). But in fact one has ra = 6 with dim(u (1 p) = 6 when [[0, to] — so*(10). We thank David Vogan for confirming this point.

108

2.C. Vanishing Results. In order to avoid trivialities, we assume for the rest of this section that G is semi-simple with no compact simple factors. Then the only finite dimensional unitary representation of G is the trivial one. The results of Vogan and Zuckerman discussed above immediately imply a "vanishing theorem" for the cohomology of arithmetic groups. In order to state it in a uniform fashion, we shall take the convention that when Y\G is compact, the space of cusp forms Llusp(T\G) consists of all functions in L^isc(T\G) which are orthogonal to the constant functions. We take the corresponding interpretation for the cuspidal cohomology H*usp(T\X, E). The Matsushima formulas 2.3, 2.5 now give Proposition 2.9 For any finite dimensional representation E of G, one has Wcusp(F\X,E)

=0

V i 2n — 2.

•

112

Remark 2.21 (a) For n = 2 the results of [92] imply that one could have non-trivial cuspidal cohomology at the degree TG = n = 2n — 2. (b) 2.20 is still not sharp for general n. With a little more work, one could prove a much better version (perhaps sharp) for Sp(n) as well as other classical groups. We include 2.18 here just to illustrate that the general bound 2.10 is not sharp when T\G is not compact. Example 2.22 Here is an example that is much deeper than 2.19. Let G = SU(p,q). Let T be an arithmetic subgroup of G defined by certain division algebra over a CM field with an involution of the second kind. (In the notation of [95], T arises from an absolutely simple group of type 2 ^4^Q o v e r some number field k, where d = n + 1). Then in particular T is cocompact. It has been shown by Rapoport and Zink [77] for the case p = 1, q = 2 and Clozel [18] in general that with some further restriction (cf. [18] for the precise conditions), H*(T\X,E) consists entirely of contributions from the trivial representation and discrete series representations of G. This means that away from the middle dimension, one has only "trivial" cohomology classes coming from the trivial representation. 3

Cohomology and Automorphic Forms

3.A. The Adelic Framework. Let G be a connected reductive algebraic group defined over Q and let ZQ/Q be the identity component of the center of G/Q. Then G is the almost direct product of ZQ and the connected derived group Gder of G defined over Q. The maximal Q-split torus in ZG is denoted by AG- Let K^, be a maximal compact subgroup of G(R). Let A be the ring of adeles of Q, and let A/ be the subring of finite adeles. For a given open compact subgroup Kf of G(A) we consider the space (where K = K^ • Kf) XK = G{Q)AG(R)0\G(A)/Koo

• Kf

This space has only finitely many connected components, each of which has the form

for an appropriate arithmetic subgroup of G. Let (v, E) be a finite dimensional algebraic representation of G(C). For the sake of simplicity we assume that Ac acts on E by a central character XE- This representation {v, E) provides a sheaf E on XK in a natural way. Its sections on an open subset V are given by E{v) = {s : 7r _ 1 (y) —> E | s locally constant, s^v)

= -ys(v) V 7 e G(Q)}

113

where 7T: AG(R)0\G(A)/KooKf

G(Q)AG(R)0\G(A)/KooKf

—+

denotes the natural projection. The objects we are interested in are the cohomology groups H* (XK , E) for a given choice of an open compact subgroup Kf. Given another open compact subgroup Kf C Kf we have a finite covering XK. —+ XK which induces an inclusion H*(XK,E) —* H*{XK,,E). This is a directed system of cohomology groups, and the inductive limit H*(G,E)

:=\im

H*(XK,E)

carries a natural G(A/)-module structure. For any g £ 7r0(G(K)) X G(A/), there is a natural map XK —> Xg-iKg given by right translation. This map extends to a map between the sheaves on both sides. This induces an action of 7To(G(K)) x G(A/) on the directed system of cohomology groups and defines a 7r0(G(]R)) x G(A/)-module structure on H*(G, E). For a given open compact subgroup Kf C G(Af) one may recover the cohomology of XK by taking ^/-invariants. Note that this limit H*(G, E) of cohomology groups is in fact equal to the cohomology of the limit YWIXK (cf. [78]). For a fixed level given by an open compact subgroup Kf C G(Af) there is a natural action of the Hecke algebra HK = Cc(G(Af)//Kf) of if/-biinvariant, Q-valued functions with compact support on H*(XK,E) = H*(G, E)Kf given by convolution. There is also a natural action by this Hecke algebra HK on the cohomology with compact supports, and the natural map

j":H:(XK,E)^H*(XK,E) is compatible with the action of the Hecke algebras. These cohomology groups can be computed via the de Rham complex which in turn can be interpreted in terms of relative Lie algebra cohomology. Let G o o (G(Q)^ G (R) 0 \G(A)) be the space of all complex valued G°°-functions on G(Q)A G (R)°\G(A) (i.e. G°° in the usual sense in the variable g^, in G(R) and right invariant under the transformations of a suitable open compact subgroup of G(A/)). The intersection of ker(dx) for all rational characters x of G is denoted by mg C Q. Then there is an isomorphism of G(A/)-modules between the (IHG, i^oo)-cohomology of the (0,^00), G(A / )-module attached to G°°(G(Q)A G (1R) 0 \G(A)) and the

114

de Rham complex of ^-valued currents on G(Q)AG(R)0\G(A)/Koo computes the inductive limit H*(G,E), i.e. H*(G, E) = H*(mG, Koc; Cco(G(Q)AG(R)°\G(A))

® E){XE)

which (3.1)

Here the symbol (XE) represents a twist of the G(A/)-action to be defined as follows. Let n be an integer such that XE c o m e s from a character ip G X*(G) of G. Then for any G(A/)-module V, V(XE) is the G(A/)-module whose underlying vector space coincides with the one for V, and such that the action of g G G(Af) on V(\E) is equal to its action on V multiplied by the character YlP\i)(9p)\p ( P running over all primes). With this definition, 3.1 is an isomorphism of G(Ay)-modules. 3.B. Cohomology and spaces of automorphic forms. Let VG :=

C~mg(G(Q)AG(R)°\G(A))

be the space of G°°-functions on G(Q)AG(R)°\G(A) of uniform moderate growth. We refer to [12] for the definition in the adelic setting. Let C be the set of classes of associate parabolic Q-subgroups of G. For {P} 6 C denote by VG({P}) the space of elements of VG which are negligible along Q for any parabolic Q-subgroup Q g' {P}, i-e., the constant term / Q with respect to Q = MQAQNQ is orthogonal to the space of cusp forms on MQ. We recall the following result ([54], Lemma 3.7 and its corollary; [71], Proposition 1.3.4): If a function / G VG is negligible along all parabolic Q-subgroups it is zero, and if it is negligible along all proper parabolic Q-subgroups it is cuspidal. As a consequence, one has that ^ VG({P}), {P} G C, forms a direct sum. It was first proved by Langlands [55] that one has a direct sum decomposition

VG = 0

VG({P})

(3.2)

{P}ec

(A variant of the original proof using the truncation operator and a result of Dixmier-Malliavin as new ingredients is given in [12], 2.4). Let Z(Q) be the center of the universal enveloping algebra of the Lie algebra Q of G, and let J C Z(g) be the annihilator in Z(g) of the dual representation E*. Let AE C CZig(G(Q)AG(R)°\G(A))

(3.3)

be the space of if-fmite G°°-functions of uniform moderate growth which are annihilated by a power of J, and set AE,{P} •= AE n VG({P})

(3.4)

115

for {P} G C. As a consequence of 3.2 one has a decomposition as a direct sum

AE=

0

AEAP}

(3.5)

{P}ec

For {P} = {G}, AE,{G} i s the space of cuspidal automorphic forms. This space is semi-simple as a module under the natural action of G(A). By using the theory of Eisenstein series as developed by Langlands [54], Franke has shown in [21] that each automorphic form can be written as a linear combination of appropriate residues or derivatives of Eisenstein series attached to cuspidal automorphic forms on the Levi components of parabolic Q-subgroups of G. In other words, AE has a description in terms of the spaces of cuspidal automorphic forms on the Levi components of parabolic Q-subgroups of G. As a cohomological consequence it is shown in [21], Theorem 18, that the inclusions of the space AE of automorphic forms into the space VQ of functions of uniform moderate growth (resp. of the latter into the space of C°°-functions on G ( Q ) A G ( R ) ° \ G ( A ) ) induce isomorphisms in cohomology, i.e., one gets (in view of 3.5) T h e o r e m 3.6 (Franke) Let G be a connected reductive algebraic group defined over Q. We have H*(G, E) = i T ( m G ) # « , ; AE ®

= 0

E)(XE)

ff*(mG,JCx,;ABi{P}®£;)(xB)

(3.7)

{P}gC

The method is to construct a descending filtration of finite length on the space AE and to use the description of its successive quotients in terms of Eisenstein series. The filtration is defined by imposing certain vanishing conditions on the constant terms of functions in AE3.C. The decomposition along the cuspidal support. Let {P} € C be a class of associate parabolic Q-subgroups of G represented by a standard parabolic Q-subgroup P with Langlands decomposition P = MpApNp. Its Levi component in this decomposition is denoted by Lp = MpAp. We consider a class

@H*(Q,E) denote the morphism induced by Q C G, where in the right hand side the sum ranges over the elements in the associate class of P up to conjugacy (for a definition of H*(P, E) see below). Then H*(mo, K; ^ E , { P } ® E) restricts isomorphically under rp onto I m r p (still under the assumption of 4.3) and

dimH*{mG,K00;AEt{P}®E)

= \ 0 dimtf'uap(Q, E) {P}

(4.4)

122

Interlude: In general, the cohomology H*{P,E) of an arbitrary parabolic Q-subgroup P of G is defined as follows: Given P one considers (as in 3.A.) the spaces XK =

P(Q)\G(A)/KocKf

with an open compact subgroup Kf c G(A). Passing over to open compact subgroups K'j C Kf one forms the inductive limit H*{P,E)

:=limH*(Xg,E)

It carries a natural G(Ay)-module structure. Set is on P(R), r>/m\ Z{P — K^ := Koo

TSP Kf

:=KfnP

The natural map P ( Q ) \ P ( A ) / t f £ • Kf

—• P(Q)\G(A)/ J ff 0O • # ,

is an embedding and induces an isomorphism of G(A/)-modules, and we have in the limit H*(P, E) = I n d ^ ; j t f * ( l i m P ( Q ) \ P ( A ) / i C • Kf,

E)

Furthermore, if we let N be the unipotent radical of P then the canonical projection P —> P/N ~ L induces a fibration of the corresponding spaces P ( Q ) \ P ( A ) / < • Kf —> L(Q)\L(A)/K^

• Kf =: XLK

with fibre N(Q)\N(A)/Kf in obvious notations. This fibration gives rise to a spectral sequence in cohomology which degenerates at E2 ([86], §2). The cohomology of the fibre can be interpreted as the Lie algebra cohomology H*(n, E) of the Lie algebra n of N(R) endowed with the natural L(R)-module structure. Thus, we have as a G(A/)-module H*(XP, E) = IndGp{AA

A(GL2,E)

126

from the set of equivalence classes of irreducible automorphic representations of GL2(AF) to that of GL2(AB) (see [57] and [43]). If TT e A(GL2,F) is a non-dihedral cuspidal representation, then $>(TT) is cuspidal as well. The map is compatible (in a sense to be made precise) with cohomology at the archimedean places. Based on this method some results concerning the existence of cuspidal cohomology classes for SL2/E, E an imaginary quadratic extension of F, inner forms thereof or SL3/F were obtained in [53]. By continuing this approach with a quite preliminary form of what should be the stabilization of the twisted trace formula one could obtain nonvanishing results for the cuspidal cohomology in the case of an absolutely almost simple algebraic group G over some number field k (of strictly positive fc-rank) that admits a Cartan type automorphism. This last assumption is satisfied in the following case ([12], 10.6, 10.7). Proposition 5.2 Suppose that G is an almost simple split connected group over some totally real algebraic number field k (or G is obtained by restriction of scalars from a group G' defined over a quadratic totally imaginary extension of k), then H*usp(G,C) ^ 0 via a base change construction. This result could be extended to the case of GLn over more general number fields. Some further progress in the stabilization of the trace formula and the existence of cyclic base change (at least for specific cuspidal automorphic representations) has lead to the following result generalizing the one for GLn in [12]. Theorem 5.3 [52] Let E/F be an extension of a totally real algebraic number field F such that there is a tower E = FmDFm-1D---DF0

=F

of intermediate fields so that Fj+i/Fj is cyclic of prime degree, and let G be a semisimple connected simply connected split algebraic group over E. Then the cuspidal cohomology H*usp(G,C) does not vanish. 5.C. Lefschetz Numbers. The assertion in 5.B. and the methods used bear some resemblance to some of the results proved by Rohlfs and Speh in specific cases in [81], where they deal with the Lefschetz number of a Cartan type automorphism on the total cohomology of arithmetic groups and prove a non-vanishing result. In combining this with a re-writing of the geometric side of the simple form of the twisted trace formula they also obtain results in the cuspidal cohomology ([82], 3.4).

127

To be more precise, let G be a connected simply connected semisimple group defined over Q, such that G(R) is non-compact. Let r be a finite order rational automorphism of G. The Lefschetz number L(T,E) is denned as the alternating sum over the traces of the homomorphisms induced on the cohomology groups Hj(G,E). In [82] an explicit formula for L(T,E) is given. It involves (among other things) the Lefschetz number of certain twisted r-actions on the compact dual space attached to G(K), volume terms corresponding to the compact real forms of the group of fixed points of these actions in G, and stabilized r-twisted orbital integrals. If r comes from a cyclic base change this formula takes a simple form. From the topological point of view this formula may be interpreted as a r-twisted variant of Hirzebruch's proportional principle. On the other hand, this result fits into the framework of the conjectural form of the stabilization of the twisted trace formula. This is discussed in [83] and in the case of cyclic base change for purely imaginary extensions in [84]. The general explicit formula for the Lefschetz number given in [82] is used in [80] to obtain in the case of Siegel modular varieties (i.e. G = Spn/F with F a totally real number field) an arithmetic formula for the Lefschetz number attached to the Cartan involution. It is expressed as a product of special values of the zeta function CF(S) and the L-function Lp(x-, S) attached to the non-trivial character \ arising from the extension F(y/—1)/F by class field theory. Another issue is to obtain via the Arthur-Selberg trace formula or a topological trace formula an explicit expression of the Lefschetz number of Hecke operators. We refer to [1] [30] [21] [8] 5.D. Theta Series. The theory of theta series and reductive dual pairs as propounded by Weil [102] and Howe [40], [41] has been a powerful method for constructing automorphic forms in general and cohomology classes in particular. Among the numerous results related to the construction of automorphic forms via theta series, we will state just one (implicit in [58]) that is particularly relevant to the present survey. Many special cases of this result were known before [58] was written, and we refer to the references in that paper for various earlier works by other authors. Before stating the result, we recall some notations from [95]. Let k be a number field and H an absolutely simple algebraic group over k. The classification of such groups is given in [95]. The first two infinite families have Dynkin diagrams of type An and denoted 1An,l and 2Ant'r respectively. Groups of type 1A\l?r are given (up to isogeny) by SLT+\{D) where D is a

128

division algebra of dimension d2 over k, and n -\-1 = d(r -\-1). The groups of the second type 2An,r, on the other hand, are given by special unitary groups defined by non-degenerate hermitian forms of dimension (n + l)/d over some division algebra D (of degree d over some quadratic extension of k) with an involution of the second kind. When d = n+ l,r = 0, such groups (with appropriate additional restrictions) define the Shimura varieties studied in [49] and [18]. Theorem 5.4 [58] Let G be an algebraic group defined over Q. Suppose that (a) G is almost Q-simple and of classical type: this means G = ReS),/Q(H), where H is absolutely simple over the number field k, and is of classical type. (b) H is neither of type 1Ani'r, nor of type 2An,r with d > 1. (c) G(R) is a real simple Lie group up to compact factors. Let Aq(X) be a unitary representation o/G(R) with non-zero cohomology at degree re- Assume that A is trivial on the (intersection of L with the) compact simple factors of G(R). Then there is an irreducible automorphic representation 7r = TT^ itf which occurs discretely in L 2 (G(Q)\G(A)), such that 71-QO ~ j4q(A). Consequently, for any arithmetic group T C G(Q) there is a subgroup Ti C F of finite index, such that A^(\) C L 2 (r\G(R)). The same statement is in fact true for a whole range of ^4q(A)'s of a certain type, having cohomologies at degrees different from re- We refer to [58], §5. for more details. In the recent years, there have been attempts to extend the "classical" theory of theta series to a more general setting that (among other things) allows the inclusion of exceptional groups. In this direction there have been some definitive progresses in both the local and global theory (see for example [14] [23] [25] [46] [42] [62] [97]). For the purpose of constructing new cohomology classes, it seems most effective to combine this theory with the Burger-Sarnak method, to be discussed next. 5.E. The Burger-Sarnak Method. This was developed via a striking new technique based on the study of matrix coefficients. Let G be a connected reductive group denned over Q. The unitary dual of G(R) is endowed with the usual Fell topology. Any unitary representation of a reductive Lie group has a unique spectral decomposition into a direct integral of irreducible representations. Suppose 7r and IT' are unitary representations of the same group, with 7r irreducible. We say that 7r is weakly contained in n' if it belongs to the support of the spectrum of n'.

129 Definition 5.5 (For this subsection only) An irreducible unitary representation of G(R) is said to be automorphic for G, if it is weakly contained in L2(T\G(R)) for some congruence subgroup T C G(Q). Note that this differs somewhat from what we usually call an automorphic representation. A trivial but important remark is that if TT is automorphic in the present sense, and is isolated in the unitary dual of G(R), then in fact it occurs in the discrete spectrum of L 2 (r\G(R)) for some T. Let now H be a reductive Q-subgroup of G. We write Ind for unitary induction from H(R) to G(R), and Res for restriction from G(R) to H(R). Then the main facts that we wish to make use of here can be stated as Theorem 5.6 ([16], [17]) Let TT and a be irreducible unitary representations of G(R) and H(R) respectively. (a) Suppose a is automorphic for H, and IT is weakly contained in Ind( 2) or else is a multiple of some prime number p > 2. Since integers of the first kind are divisible by 4 — an exponent for which Fermat himself proved Fermat's Last Theorem — it suffices to consider exponents that are odd prime numbers when one seeks to prove Fermat's Last Theorem. In other words, after verifying Fermat's assertion for n = 4 and n = 3, mathematicians were left with the problem of proving the assertion for the exponents 5, 7, 11, 13, 17, and so on. Progress was slow at first. The case n = 5 was settled by Dirchlet and Legendre around 1825, while the case n = 7 was treated by Lame in 1832. In the middle of the nineteenth century, E. Kummer made a tremendous advance by proving Fermat's Last Theorem for an apparently large class of prime numbers, the regular primes. The definition of this class may be given quickly, thanks to a numerical criterion that was established by Kummer. Namely, one considers the expression 691a;12 1307674368000 xi and defines the ith Bernoulli number B; to be the coefficient of — in this i\ 691 expansion. Thus B\2, for example is — ; the denominator is 2 • 3 • 5 • 7 • 13, the product of those primes p for which p—1 divides 12. A prime number p > 7 is regular if p divides the numerator of none of the even-indexed Bernoulli numbers B2, B4,..., Bp-z- The primes p < 37 turn out to be regular. On the other hand, 37 is irregular (i.e., not regular) because it divides the numerator of B32: the numerator is 7709321041217 = 37 • 683 • 305065927. (We may conclude that 683 and 305065927 are irregular as well.) A proof of Fermat's Last Theorem for regular primes, along the lines given by Kummer, may be found in [11, Ch. 5]. See also [19] and [2] for alternative discussions. In these books, the reader will find a proof that there are infinitely many irregular prime numbers; see, for example, [19, Ch. VI, §4] or [2, Ch. 5, §7.2]. Although heuristic probabilistic arguments suggest strongly that regular primes should predominate, the set of regular primes is currently not known to be infinite. Over the years, Kummer's work was refined repeatedly. Aided by machine calculation, mathematicians employed criteria such as those presented in [19] to verify Fermat's Last Theorem for all prime exponents that did not exceed ever increasing bounds. Most notably, four mathematicians proved Fermat's Last Theorem for all prime exponents below four million in an article that was published in 1993 [4]. It is striking that the calculations in that article T x e - 1

x2 2 12

x

x4 720

x6 30240

x8 1209600

x10 47900160

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were motivated by questions involving Bernoulli numbers and the arithmetic of cyclotomic fields; the proof of Fermat's Last Theorem for a large set of prime numbers came almost as an afterthought. Readers used to dealing with experimental sciences might well now ask why a mathematician would insist on a rigorous proof of a statement, depending on a parameter n, that can be verified by calculation for all n < 4,000,000. A statement that is true in this range seems very likely to be true for all n. To answer this question, it suffices to point out the logical possibility that an assertion that is true experimentally may have one or more counterexamples that happen to be very large. In fact, assertions that realize this possibility are not hard to find in number theory. As Fermat himself knew, the first solution to x2 — 109y2 = 1 in positive integers x and y is given by x = 158070671986249, y = 15140424455100. (See [30, Ch. II, §XII] for an illuminating discussion of Fermat's study of x2 — Ny2 = ±1.) If we set out to examine x2 — 109y2 = 1 with a computer, we might look for solutions with x and y non-zero, find no such solutions, and conclude incorrectly that this equation has only the trivial solutions (—1,0) and (1,0). Here's another example: Euler conjectured in the eighteenth century that a perfect fourth power cannot be the sum of three perfect fourth powers. Noam Elkies [12] found the first counterexample to Euler's conjecture in 1988: 26824404 + 153656394 + 187967604 = 206156734. These examples illustrate the fact that numerical evidence in number theory can be misleading. 4

Modern History

The proof of Fermat's Last Theorem at the end of the last century hinges on a connection between putative solutions of Fermat's equation and cubic equations with integer coefficients (elliptic curves). To have a solution to Fermat's equation is to have positive integers a and b for which an + bn is a perfect nth power. (We shall suppose that n is at least 5 and that n is a prime number. The results of Fermat and Euler imply that these assumptions are harmless.) Given a and b, we consider the equation E:y2=x(x-an){x

+ bn),

in which x and y are new variables. This equation defines an elliptic curve. The connection between Fermat and elliptic curves was noticed by several mathematicians, including Yves Hellegouarch and Gerhard Frey. In a recent

166

book [13], Hellegouarch recounts the history of this connection. It was Frey who had the decisive idea that E could not possibly satisfy the ShimuraTaniyama conjecture, which states that elliptic curves are modular. (We shall discuss this crucial property in §5 below.) Frey's suggestion became known to the mathematical community in the mid 1980s. In 1986, I proved that elliptic curves associated solutions to Fermat's equation are non-modular, thereby showing that Fermat's Last Theorem is a consequence of the Shimura-Taniyama conjecture [21], [22]. Said differently: each solution to Fermat's Last Theorem gives a counterexample to the Shimura-Taniyama conjecture. Thus if that conjecture is true, so is Fermat's Last Theorem. As the reader is no doubt aware, Andrew Wiles worked in his Princeton attic from 1986 to 1993 with the goal of establishing the Shimura-Taniyama conjecture. Although the conjecture per se was a central problem of number theory, Wiles has stated that he was drawn to this problem because of the link with Fermat's Last Theorem. In June, 1993, Wiles announced that he could prove the Shimura-Taniyama conjecture for a wide class of elliptic curves, including those coming from Fermat solutions. This announcement implied that the proof of Fermat's Last Theorem was complete. After a short period of celebration among mathematicians, Wiles's colleague Nicholas Katz at Princeton found a "gap" in Wiles's proof. Because the gap's severity was not appreciated at first, it was months before the existence of the gap was known widely in the mathematical community. By the end of 1993, however, the fact that Wiles's proof was incomplete was reported in the popular press. The proof announced by Wiles remained in doubt until October, 1994, when Richard Taylor and Andrew Wiles released a modified version of the proof that circumvented the gap. The new proof was divided into two articles, one by Wiles alone and one a collaboration by Taylor and Wiles [32], [29]. The two articles were published together in 1995. The proof presented in those articles was accepted quickly by the mathematical community. As a result of his work, Wiles has been honored repeatedly. For example, in December, 1999, he was knighted by the Queen: he received the "KBE/DBE" along with Julie Andrews, Elizabeth Taylor and Duncan Robin Carmichael Christopher, Her Majesty's ambassador to Jakarta 0 . After the manuscripts by Wiles and Taylor-Wiles were written in 1994, the technology for establishing modularity became increasingly more sophisticated and more general. The class of curves to which the technology can be °http://files.fco.gov. uk/hons/honsdec99.shtml

167

applied was enlarged in three stages [10], [5], [3]. In the last stage, four mathematicians— Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor — announced in June, 1999 that they had proved the full ShimuraTaniyama conjecture, i.e., the modularity of all elliptic curves (that are defined by equations with integer coefficients). Although their proof is not yet published, it is available from h t t p : / / w w w . m a t h . h a r v a r d . e d u / ~ r t a y l o r / , Richard Taylor's Web site at Harvard. In addition, the proof has been the subject of a substantial number of oral presentations. In particular, the proof was explained by the four authors in a series of lectures at a conference held at the Mathematical Sciences Research Institute in Berkeley, California in December, 1999. 5

The Shimura-Taniyama Conjecture

The conjecture hinges on the notion of "arithmetic mod p," p being a prime number. When working mod p, we ignore all integers that are multiples of p. In other words, when we interact with an integer m, we care only about the remainder when m is divided by p. This remainder is one of the numbers 0, 1, 2 , . . . , p — 1. For example, the integers mod 5 are 0, 1, 2, 3 and 4. Suppose that we are given an equation with integer coefficients. Then for each prime number p, we can use the equation to define a relation mod p. As an illustration, the simple equation x2 + y2 = 1 gives rise to a relation mod 2, mod 3, mod 5, and so on. This type of relation is best illustrated by a concrete example. Suppose that we take p = 5, so that the numbers mod 5 are the five numbers that we listed above. There are thus 25 pairs of numbers (x, y) mod 5. For each pair (x,y), we can ask whether x2 + y2 is the same as 1 mod 5. For (0,4), the answer is "yes" because 16 and 1 are the same mod 5. For (2,2), the answer is "no" because 8 and 1 are not the same mod 5. After some calculation, one finds that there are four pairs of numbers mod 5 for which the answer is in the affirmative. These pairs are (0,1), (0,4), (1,0) and (4,0). After we recognize that 4 is that same as —1 mod 5, we might notice that the four solutions that we have listed have analogues for every prime number p > 2. There are always the four systematic solutions (0,1), (0, - 1 ) , (1,0) and (-1,0) for each such prime. We can make a similar calculation mod 7. It is fruitful to begin by listing the squares of the seven numbers mod 7: a 0 123456 a2 0 1 4 2 2 4 1.

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In which ways can we write 1 as the sum of two numbers in the bottom row (possibly the sum of a number and itself)? We can write 1 as 0 + 1 = 1 + 0, and we can also write 1 as 4 + 4 (since 8 is the same as 1 mod 7). We end up with the four "new" solutions (±2, ±2) in addition to the four systematic solutions that we listed in connection with the case p = 5. As a consequence, there are eight solutions to x2 + y2 = 1 mod 7. After experimenting with other primes (p = 11, p = 13, etc.), you will have little trouble guessing the general formula for the number of solutions to x2 + y2 = 1 mod p. When p = 2, there are the two solutions (0,1) and (1,0). When p is bigger than 2, there are either p+1 or p— 1 solutions to x2 +y2 = 1 mod p, depending on whether p is 1 less than or 1 more than a multiple of 4. This simple recipe was known centuries ago. It can be established in various ways; perhaps I should leave its proof as an exercise for the interested reader. The equation x2 + y2 = 1 was intended as a warm-up; we shall now consider the superficially analogous equation x3 + y3 = 1. Here again we study the number of solutions to x3 + y3 = 1 mod p and seek to understand how this number varies with p. It turns out that the quantity p mod 3 plays an important role here—just as the behavior of p mod 4 was significant for x2 +y2 = 1. When p = 3, the quantity x3 mod p coincides with x mod 3; this is a special case of what is called "Fermat's Little Theorem" in textbooks. Hence the solutions to a;3 + y3 = 1 mod 3 are the same as the solutions to x + y = 1; there are three solutions, because x can be taken arbitrarily, and then y is 1 — x mod 3. If p is 2 mod 3, i.e., if p is 1 less than a multiple of 3, one shows by an elementary argument that there are again p solutions. (If p is 2 mod 3, then every number mod p has a unique cube root.) The interesting case for this equation is the remaining case where p is 1 more than a multiple of 3. This case was resolved by Gauss in the nineteenth century. To see what is going on, we should look at a few examples: First off, we take p = 7. The cubes mod 7 are 0, 1 and 6 = - 1 . If two cubes sum to 0, one is 1 and the other is 0. Also, 1 has three cube roots: 1, 2 and 4. Thus there are six solutions to x3 + y3 = 1, namely (0,1), (0,2), (0,4) and the analogous pairs with x and y reversed. When p - 13, the cubes are 0, 1, - 1 , 5 and 8. There are again only six solutions because the only way to write 1 as a sum of two cubes is to take 0 + 1 as before. Now try p = 19. It turns out that there are 24 solutions here—the six that we knew about already, together with 18 unexpected ones arising from the equation 1 = 8 + (-7) and the fact that 8 and - 7 are both cubes mod 19. (Since 4 3 = 64 = 7 mod 19, - 7 is the cube of - 4 . ) We get 18 solutions by taking x to be one of the 3 cube roots of 8 and y to be one of the 3 cube roots

169

of —7, or vice versa. When p = 31, there are 33 solutions. (Note that 33 = 6 + 18 + 9.) There are 6 solutions coming from 0 + 1 = 1, 18 coming from 2 + (-1) = 1 and 9 from 16 + 16 = 1. Summary: p 7 13 19 31 # solns. 6 6 24 33 How does this table continue? What is the number of solutions that we get when p is, say, 103? It is hard to imagine the rule that expresses the number of solutions in terms of p. Gauss found an expression for the number of solutions that we can view as a "generalized formula" [14, p. 97]. Namely, when p = 1 mod 3, Gauss showed that one has 4p = A2 + 27B2 for some integers A and B. These integers are uniquely determined except for their signs. We can and do choose A so that A = 1 mod 3. Then Gauss's formula states: # solns. =p-2

+ A.

For example, if p = 13, then 4p = 52 = 5 2 + 27 • l 2 . Thus A = - 5 . We have p - 2 + A = 6. When p = 31, 4p = 124 = 4 2 + 27 • 2 2 . Thus A = 4 and p - 2 + A = 33. When p = 103, 4p = 13 2 + 27 • 3 2 , so A = 13 and the number of solutions is 114. The equation x3 + y3 = 1 defines one of the simplest possible elliptic curves. Gauss's explicit recipe shows in particular that x3 + y3 = 1 defines a modular elliptic curve. The Shimura-Taniyama conjecture states that there's an analogous "formula" for every elliptic curve. Because this formula involves modular forms, the Shimura-Taniyama conjecture is usually paraphrased as the statement that elliptic curves are modular. For a random elliptic curve, the formula provided by the associated modular form is not as explicit as Gauss's formula for x3 +y3 = 1. Here is a famous example that begins to give the flavor of the general case: We consider first the formal power series with integral coefficients J^ anXn that is obtained by expanding out the product oo

X

Y[(l~Xm)2(l-XUm)2.

170

For all n > 1, an is an integer. In fact, the numbers an are the coefficients of the Fourier expansion of a well known modular form. At the same time, we consider the elliptic curve defined by the equation y2 +y = x3 — x2. Then a theorem of M. Eichler and G. Shimura states that, for each prime p (different from 11), the number of solutions to this equation mod p is p — ap. The connection between the number of solutions and the pth coefficient of a modular form shows that the elliptic curve defined by y2 + y = x3 — x2 is a modular elliptic curve. (A coffee mug that celebrates this relation is currently available from the Mathematical Sciences Research Institute. Go to http://www.msri.org/search.html and search for "coffee cup.") 6

Another Formula of Gauss

For a third example, we look at the elliptic curve defined by the equation y2 = x3 — x. Although its equation recalls the equation y2 + y = x3 — x2 of the second example, this third example is much more analogous to the first example. To explain the analogy, it is important to recall a theorem of Fermat about sums of squares. Namely, suppose that p is a prime number and that we seek to write p is the form r2 + s2, where r and s are integers. If p is 2, we can write p = l 2 + l 2 . If p is congruent to 3 mod 4, then it is impossible to write p &s r2 + s2. Indeed, squares are congruent to either 0 or 1 mod 4; it is therefore impossible that a sum of two squares be congruent to 3 mod 4. The interesting case is that where p is congruent to 1 mod 4, i.e., where p is 1 plus a multiple of 4. Fermat proved in that case that p may be written as a sum of two squares: we have p = r2 + s2 with r and s whole numbers. The pair (r, s) is clearly not unique because we can exchange r and s and we can change the signs of either or both of these integers. However, there is no more ambiguity than that: the integers r and s become unique up to sign after we require that r be odd and that s be even. Accordingly, r and s are determined completely if we require that r be odd, that s be even and that both integers be positive. This theorem of Fermat is proved in most elementary number theory books; see, e.g., [14, Ch. 8] for one proof. (The uniqueness is left as an exercise at the end of the chapter.) A beautiful proof of the existence of r and s, due to D. Zagier, is presented in "Proofs from the Book" [1], a volume celebrating Paul Erdos's idea that there is frequently an optimally beautiful proof of a given proposition in mathematics. Following Gauss, we will now adjust the sign of r (if necessary) to ensure that the sum r + s is congruent to 1 mod 4. For example, suppose that p = 5,

171

so that (r, s) = (1,2) under the initial choice that has both r and s positive. With this choice, r + s = 3 is not 1 mod 4. Accordingly, we change the sign of r and put r = — 1. The sum r + s is then 1, which of course is 1 mod 4. For another example, we take p = 13, so that (r, s) = (3,2) with the initial choice. Here r + s = 3 + 2 = 5, which is already 1 mod 4. We therefore leave r positive in this case. It is perhaps enlightening to tabulate the values of r and s for the first primes that are 1 mod 4. In doing so, we write "r p " instead of "r" and "s p " instead of "s" to stress that r and s depend on p: p 5 13 17 29 37 41 53 61 73 89 rp -1 3 1 -5 -1 5 7 -5 -3 5 Sp

2 2 4 2 6 4 2 6 8 8

We return now to y2 = x3 — x with the idea of calculating the number of solutions to the mod p congruence defined by this equation. If p = 2, there are two solutions: (0,0) and (1,0). If p is congruent to 3 mod 4, it turns out that there are exactly p solutions. More precisely, if a: is 0, 1 or —1 (i.e., p — 1) mod p, then y = 0 is the one value of y for which (x, y) is a solution. For each value of x different from 0 and ± 1 , there are either two values of y or no values of y for which (x, y) is a solution mod p. (If a non-zero number mod p has a square root, it has exactly two square roots, which are negatives of each other.) An elementary argument shows that if there are two y for a given x, then there are no y for —x, and vice versa. The point here is that a non-zero number mod p has a square root mod p if and only if its negative does not; this observation is valid when p is 3 mod 4 but fails to be true when p is 1 mod 4. The end result is that, on average, there is one value of y that works for each x. Thus the number of solutions is p, as was stated. The interesting case for y2 = x3 - x is that where p is congruent to 1 mod 4. We assume now that this is the case. To get a feel for the situation, we can calculate the number of solutions mod 5 and mod 13; these are the first two primes that are 1 mod 4. Suppose that p = 5. The values x = 0, x = 1 and x = 4 make x3 - x congruent to 0, so that they give rise to exactly one solution each; y must be 0. If x = 2, then a;3 - x is congruent to 1, a number that has two square roots mod 5, namely ± 1 . Thus x = 2 gives rise to two solutions. Similarly, if x = 3, then a;3 - x is congruent to 4 mod 5, and 4 has two square roots. Thus x = 2 also gives rise to two solutions. As a result, there are seven solutions to y2 = x3 — x mod 5. Suppose now that p = 13. The three values x = 0,1, - 1 give rise to a single solution each as before; in each case, y is again 0. The ten remaining

172

values of x (namely, x = 2, 3 , . . . , 11) each give rise either to two or to no solutions: the quantity x3 — x is non-zero mod p and we have to decide in each case whether or not it is a square (i.e., a number with square roots mod p). The quantities are respectively 6, 11, 8, 3, 2, 11, 10, 5, 2 and 7 mod 13. On the other hand, the non-zero squares mod 13 are 1, 3, 4, 9, 10 and 12. It happens, then, that only two of the numbers x between 2 and 11 are such that x3 — x is a square. Thus we find — one again — that there are seven solutions to y2 = x3 — x mod p. One could easily guess from these two examples that there are always seven solutions to y2 — x3 — x mod p when p is 1 mod 4, but these two examples are misleading. Theorem 1 (Gauss) Suppose that p is a prime that is 1 mod 4- Then the number of solutions to y2 — x3 — x mod p is p — 2rp, where rp is chosen as above. The theorem is compatible with the two examples that we presented. When p — 5, we have rp = —1, so that p — 2rp = 7. When p = 13, rp is 3, and 13 — 2 - 3 = 7. Since rj3 = —3, the number of solutions to y2 — x3 — x mod 73 is 73 + 6 = 79. Here is an example with a p that is considerably larger than the primes that have appeared thus far: Suppose that p is the prime number 144169. We can write p as the sum 3152 + 2122. It follows that rp = ±315. Since 315 + 212 = 527 is 3 mod 4, we must take rp = —315. Gauss's formula then asserts that the number of solutions to y2 — x3 — x mod p is 144169 + 2 • 315 = 144799. A variant of Gauss's formula is proved in [14, Ch. 11, §8]. The connection between the variant given there and the formula of Theorem 1 is made in Exercise 13 at the end of (14, Ch. 11]. 7

Binomial Coefficients

Because I have written extensively about Fermat's Last Theorem, I have received a number of letters about number theory from amateur mathematicians. Several years ago, I received a letter about binomial coefficients. These are the numbers that appear in the expansion of (x + y)n when n is a positive integer. Recall, for example, that (x + y)2 = x2 + 2xy + y2, {x + y)3 = x3 + 3x2y + 3xy2 + y3, {x + yf = xi+ Ax3y + 6x2y2 + 4xy3 + y4, (x + yf = x5 + 5x 4 y + 10x3y2 + 10x2y3 + 5xy4 + y5,

173

The coefficient of xn~lyl

in the expansion of (x + y)n is usually denoted ("). n! It may be expressed as the fraction -7777> where A;! is used to denote the ( n — i)\i\

product of the first k positive integers. (By convention, 0! = 1.) Looking at the expansion for (a; + y)A, say, we see that (2) = 6 and that (3) = 4. The letter that I received concerned the central coefficient in the expansion of (x + y)n when n = {p — l ) / 2 and p is a prime congruent to 1 mod 4. If p is 5, for example, this coefficient is 2. In general, it is the binomial coefficient

(r -1I/4)' a

nuim:)er t n a t w e c a n c a u

^p f° r short:

37 p 5 13 17 29 bp 2 20 70 3432 48620 These numbers grow large very quickly, but my correspondent was considering them modulo p in order to keep their size manageable. For reasons that I no longer recall, he hit upon the scheme of representing them modulo p as even numbers between — (p — 1) and +(p— 1). In the table that follows, I've written cp for the unique even number in this range that is congruent to bp modulo p: p 5 13 17 29 37 41 Cp

53 2 -6 2 10 2 10 -14

He noticed that the residues cp were related to the integers rp that we introduced above in connection with Gauss's formula. Because he did not have Gauss's formula in mind, he tabulated the rp as positive numbers: p 5 13 17 29 37 41 Cp Tp

53 61 73 89 2 -6 2 10 2 10 -14 10 -6 10 1 3 1 5 1 5 7 5 3 5

It was clear to him empirically from his calculations that cp = ±2r p , but the sign in this equation seemed completely opaque. He asked me to determine the sign and to explain to him why the identity is true. Although the identity cp = ±2rp has been known at least since the nineteenth century, it was new to me. However, I realized that the sign that appears in this identity becomes significantly less mysterious once we again

174

endow rp with the sign that we introduced in connection with Gauss's formula: p Gp

rp

5 13 17 29 37 41 53 61 73 89 2 -6 2 10 2 10 -14 10 -6 10 -1 3 1 -5 -1 5 7 -5 -3 5

The rule relating cp and rp in this latter table is as follows: J cp = +2rp [ cp = —2rp

if p is 1 plus a multiple of 8 if p is 5 plus a multiple of 8.

Notice here that p is assumed going in to be of the form 1 + At. If t is even, p is congruent to 1 mod8; if t is odd, p is congruent to 5 mod 8. It is not too hard to establish this rule if one takes Gauss's formula as a starting point. Indeed, suppose that we seek to calculate the number of mod p solutions to y2 = x3 — x. We can let x run over the set { 0 , 1 , 2 , . . . ,p — 1} of numbers mod p. For each x, the number of y satisfying y2 = x3 — x is: 1 if x3 — x is 0 mod p 2 if a;3 — x is a non-zero square mod p 0 if x3 — x is not a square mod p. This number may be written 1 + [~^), where (-) is the traditional Legendre symbol whose values are 0, + 1 , —1 according as the argument is 0, a non-zero square, or a non-square mod p. The number of solutions to y2 = x3 — x mod p is then

go+(^))Thus

by Theorem 1. A standard congruence for (-) states that (|) is congruent mod p to a ( p - 1 ) / 2 for each integer a. Using this congruence, we get

2rp = - ] [ > 3 - z ) ( p - 1 ) / 2 , where "=" denotes congruence mod p. The expression (a;3 - x)^1^2 be expanded out as a sum that involves the binomial coefficients (

,

can )

175

(i — 0 , . . . , (p - l)/2). After changing the order of summation, we get from this expansion 2rp

= - iPJ2\-iy

| £ ({p -. 1 )/ 2 V 3 «p- 1 )/ 2 )- 2i |.

Now an elementary fact about sums of powers states that we have p—X

E

j _ / 0 if p — 1 does not divide j ~ \ - 1 if p - 1 does divide j .

It follows that only one of the inner sums is non-zero mod p: this is the sum corresponding to the choice i — (p — l)/4. We thus get 22rr

P~

=_(_I\(P-I)/4/'(P-1)/2V_:M 1J

I !)

V(p-l)/4/

2rp =

(-1)W%.

'

so that

Since the sign in this expression is 1 if and only if p is congruent to 1 mod 8, we find that c p is either 2rv or — 2r p , with the choice of sign as stated above. 8

Sums of Squares mod p

During my "lecture to schools" at Victoria Junior College, I discussed several issues that were brought up by undergraduate students and amateur mathematicians. One was the binomial coefficient identity that is treated in the previous section — it turned out to be a corollary of a formula of Gauss that a number of mathematicians had been featuring in their lectures on Fermat's Last Theorem. A second question involves squares mod p; it was posed by a freshman (i.e., first-year student) at Yale University. Recall from our discussion before that the non-zero squares mod 13 are 1, 3, 4, 9, 10 and 12. The sum of these six numbers, considered as positive integers, is 1 + 3 + 4 + 9 + 10 + 12 = 39, which is 13 • 3. Suppose, more generally, that p is a prime different from 2. It is a standard fact from elementary number theory that there are precisely (p — l ) / 2 different non-zero squares mod p. (It's easy to see that there are at most this number because (— x)2 is the same as x2 for each x mod p. The point is that if x2 = y2, then x = ±y mod p.) Regarding the squares as integers between 1 and p— 1, we form their sum and call the resulting positive integer S(p). Then 5(3) = 1, S(5) = 1 + 4 = 5, S(7) = 1 + 2 + 4 = 7,

176

5(11) = 1 + 3 + 4 + 5 + 9 = 22, etc. We can guess from these examples that S(p) is divisible by p for p > 5, and indeed this divisibility is relatively easy to establish. Let's assume then that p is at least 5 and set L(p) = S(p)/p. Thus L(5) = L(7) = 1, L ( l l ) = 2, and so on. The question concerns L(p): can we find a formula for it?

L{P)

11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 7 10 10 13 13 15 16 14

After looking at this table (or perhaps an extension of it), one comes to the realization that p - \ L(P) when p is congruent to 1 mod 4. This formula is given frequently as an exercise in number theory classes, and the proof is not difficult. The key fact in this case is that if a is a square mod p, then so is —a. (The analogous statement for p congruent to 3 mod 4 is that —a is never a square mod p if a is a non-zero square!) Eqivalently, if a is a square between 1 and p — 1, then p — a is again such a number. Notice also that a and p — a are distinct numbers since p is odd. Thus the squares mod p between 1 and p—1 can be partitioned into pairs {a,p — a}. The sum of the numbers in each pair is p. Since there are (p — l ) / 2 squares mod p, there are (p — l ) / 4 pairs. It follows that the sum 1 of all the squares is p Pas was claimed. For example, if p = 17, then the squares are 1, 2, 4, 8, 9, 13, 15 and 16. We re-write the sum of these eight numbers as (1 + 16) + (2 + 15) + (4 + 13) + (8 + 9) = 4 • 17. Knowing the formula for L(p) when p = 1 mod 4, one might anticipate a similar formula in the complementary case p = 3 mod 4. The values of L(p) for p = 7, 11 and 19 suggest that one has

I was asked whether this formula was true in general. In fact, I saw quickly that the formula is false by continuing the computation. Indeed, we have L(23) = 4 when the formula predicts L(23) = 5. On the other hand, the formula seemed perhaps to be not so far from the truth: in the table, the formula is correct for p = 43 and p = 67, as well as for the small values 7, 11 and 19. This behavior is somewhat striking, since false formulas tend to fail in a much more spectacular way. Although I was initially puzzled by what was going on, I realized after orienting myself that I already knew how to write down a correct version of

177

the formula. In fact, the information needed is available in a classic text on algebraic number theory, Hermann Weyl's "Algebraic Theory of Numbers" [31]. The corrected formula reads

where h{p) is a certain odd positive integer that depends on p. The fraction (p - 3)/4 coincides with (p - 1 — 2/i(p))/4 if and only h{p) = 1. The term h(p) is a class number that has been studied extensively at least since the time of Gauss. One way to introduce the class number is to return to the proof of Fermat's Last Theorem for exponents 3 and 4. In the seventeenth century, Fermat proved that the equation a4 + b4 = c4 has no solution in positive integers by proving a stronger statement involving squares and fourth powers. As recapitulated by Ireland and Rosen [14, p. 272], the idea is to show that there are no solutions to a4 + b4 = c2 by a method of descent: one assumes that there is a solution to this equation in positive integers, chooses a ("minimal") solution for which the number c is as small as possible, and then parlays the chosen solution into a new solution whose c-value is even smaller than that for the minimal solution. The method hinges on properties of unique factorization of positive integers; unique factorization is the statement that an integer bigger than 1 can be written as a product of prime numbers in essentially only one way. To exploit unique factorizaton, one re-writes the equation a4 + b4 = c2 as the statement that c2 — a4 = (c — a2)(c + a2) is a perfect fourth power (namely, b4). One then recalls the principle that if a product of two positive integers A • B is a fourth power, then A and B must each be perfect fourth powers — provided that A and B have no common factor. The (c — a?) and (c + a 2 ) may share a common factor, so that the principle cannot be used without modification. However, one can exchange a and b in the minimal solution (if necessary) so as to make 2 the only divisor > 1 that is common to the two factors (c — a2) and (c + a2). After using unique factorization in a judicious way, one emerges with a new solution whose c-value is smaller than the c-value of the minimal solution. Euler treated the equation a3 + b3 = c3 by a method that is similar to Fermat's. However, he needed to work with quantities that involve a complex cube root of 1 [14, Ch. 17, §8]. The key idea is to introduce quantities involving y/—3 that act as generalized integers and to establish for these quantities an analogue of the unique factorization theorem for positive integers. These

178

quantities are expressions of the form n+m

-1 + ^33

,

where n and m are usual integers. These expressions can be added, subtracted and multipled to form an arithmetical system like the system of ordinary integers. Suppose now more generally that p is a prime number that is congruent to 3 mod 4. Then one considers in an analogous way the system of quantities n+m

-1+V =p , ^

where n and m are again integers. The class number h{p) measures the extent to which unique factorization fails for this system; h(p) = 1 if and only if unique factorization can be established. Gauss proved that h(p) is always an odd number and examined the behavior of h{p) numerically. He conjectured that h(p) = 1 if and only if p is one of the prime numbers 3, 7, 11, 19, 43, 67, 163. This conjecture was established only in the twentieth century! See [28] for a discussion of work by A. Baker, K. Heegner and H. Stark that resolved this question. Thus the proposed formula is correct precisely for the values of p that we listed above — namely, 7, 11, 19, 43 and 67 — plus the larger prime p = 163. 9

Fermat-like Equations

At Victoria Junior College, I discussed a third question that was sent to me by an amateur mathematician: he asked whether the methods that proved Fermat's Last Theorem would shed any light on the Fermat-like equation an + bn = 2c™, where n is a positive integer. In this equation, we can suppose first that a, b and c are positive integers. We observe immediately that this equation, in contrast to Fermat's equation, does have solutions. Indeed, we can take a to be an arbitrary positive integer and set b and c equal to a! We can call these solutions trivial and ask whether there are non-trivial solutions to the equation. If n = 1, then the equation states simply that c = a + b, so of course there is no obstacle to having solutions. As with Fermat's equation, there are non-trivial solutions if n = 2; for example, we can take (a, 6, c) = (1,7, 5). It is an interesting exercise to find a description of the nontrivial solutions that is analogous to the familiar description of all Pythagorean triples — the solutions to a2 + b2 = c2. The special appeal of an + bn = 2cn stems from its reformulation as the statement that cn is the average of an and bn. In other words, if an+bn — 2c n ,

179

then an, cn and bn form an arithmetic progression. In the example (1,7,5) with n = 2, 25 is the average of 1 and 49; equivalently, the differences 2 5 - 1 and 49 — 25 are equal. As I gradually learned while researching this equation, its history is parallel to that of Fermat's equation. As with Fermat's equation, the discussion for exponents n > 2 may be reduced to the two cases n = 4 and n = p, where p is a prime greater than 2. Fermat proved that four distinct perfect squares cannot form an arithmetic progression, and he showed that there are no non-trivial solutions to a 4 + bA = 2c 4 . Accordingly, it sufficed to consider the case n — p with p > 3. Euler and Lengendre treated the case p = 3. In 1952, P. Denes showed that there are no non-trivial solutions for p < 29 and conjectured that there are no non-trivial solutions for all prime exponents bigger than 2. In an article that was published in 1997 [24], I adapted the technology that was used in proving Fermat's Last Theorem to establish Denes's conjecture for primes p that are congruent to 1 mod 4. Subequently, H. Darmon and L. Merel settled Denes's conjecture completely (in the affirmative) by introducing new techniques to deal with the case p = 3 mod 4 [9]. There has been a substantial literature about Fermat-like equations ever since the connection between Fermat solutions and elliptic curves was uncovered in the 1980s. See [8] for information in this direction.

10

Further Reading

During the course of this article, I have mentioned some of my favorite articles and books about number theory, especially those that touch on Fermat's Last Theorem. Here are a few more references that I have not yet had occasion to cite. First, a summary of "elementary" approaches to Fermat's Last Theorem is provided by P. Ribenboim in his book [20]. Secondly, an interesting discussion of elliptic curves and modular forms is contained in A. van der Poorten's book [16]. Next, the recent "diary" by C.J. Mozzochi [15] contains photos of the mathematicians who participated in the proof of Fermat's Last Theorem, along with detailed descriptions of lectures and other events that are associated strongly with the proof. Finally, several accounts of the details of the proof of Fermat's Last Theorem have been written for professional mathematicians [7], [18], [6]. What is missing from the literature, at least so far, is an extended account of the proof that is accessible to a scientifically literate lay reader and does justice to the mathematics behind the proof.

180

Acknowledgments The author thanks the University of Singapore and the Isaac Newton Institute for Mathematical Sciences at Cambridge for organizing this conference and for their kind hospitality in Singapore. He was partially supported by the US National Science Foundation during the preparation of this article. References 1. M. Aigner and G.M. Ziegler, Proofs from the Book. New York-BerlinHeidelberg: Springer-Verlag, 1998. 2. Z.I. Borevich and I.R. Shafarevich, Number Theory. New York: Academic Press, 1966. 3. C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q. To appear. 4. J. Buhler, R Crandall, R. Ernvall, and T. Metsankyla, Irregular primes and cyclotomic invariants to four million, Math. Comp. 61 (1993), 151— 153. 5. B. Conrad, F. Diamond, and R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), 521-567. 6. G. Cornell, J.H. Silverman and G. Stevens, eds. Modular forms and Fermat 's last theorem, Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9-18, 1995. Berlin-Heidelberg-New York: Springer-Verlag, 1997. 7. H. Darmon, F. Diamond and R. L. Taylor, Fermat's last theorem. In "Elliptic curves, modular forms & Fermat's last theorem," Proceedings of the Conference on Elliptic Curves and Modular Forms held at the Chinese University of Hong Kong, Hong Kong, December 18-21, 1993, J. Coates and S.T. Yau, eds., second edition. Cambridge, MA: International Press, 1997. 8. H. Darmon and A. Granville, On the equations zm = F(x, y) and Ax9 + By* = Czr, Bull. London Math. Soc. 27 (1995), 513-543. 9. H. Darmon and L. Merel, Winding quotients and some variants of Fermat's last theorem, J. Reine Angew. Math. 490 (1997), 81-100. 10. F. Diamond, On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), 137-166. 11. H. M. Edwards, Fermat's Last Theorem: a genetic introduction to modern number theory. Graduate Texts in Mathematics, volume 50. New YorkBerlin-Heidelberg: Springer-Verlag, 1977.

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12. N. Elkies, On A4 + B4 + C4 = D4, Math. Comp. 51 (1988), 825-835. 13. Y. Hellegouarch, Invitation aux mathematiques de Fermat- Wiles. Paris: Masson, 1997. 14. K. F. Ireland and M. I. Rosen, A Classical Introduction to Modern Number Theory, section edition. Graduate Texts in Mathematics, volume 84. New York-Berlin-Heidelberg: Springer-Verlag, 1990. 15. C.J. Mozzochi, The Fermat Diary. Providence: American Math. Soc, 2000. 16. A. van der Poorten, Notes on Fermat's Last Theorem. New York: John Wiley & Sons., 1996. 17. J. Lynch and S. Singh, The Proof. A television documentary written and produced by John Lynch and directed by Simon Singh. See http://www.pbs.org/wgbh/nova/proof/ for more information, and for a transcript. 18. V.K. Murty, ed., Seminar on Fermat's Last Theorem, Papers from the seminar held at the Fields Institute for Research in Mathematical Sciences, Toronto, Ontario, 1993-1994. Providence, American Math. Soc, 1995. 19. P. Ribenboim, 13 Lectures on Fermat's Last Theorem. New York-BerlinHeidelberg: Springer-Verlag, 1979. 20. P. Ribenboim, Fermat's Last Theorem for Amateurs. New York-BerlinHeidelberg: Springer-Verlag, 1999. 21. K.A. Ribet, On modular representations o/Gal(Q/Q) arising from modular forms, Invent. Math. 100 (1990), 431-476. 22. K.A. Ribet, From the Taniyama-Shimura conjecture to Fermat's last theorem, Ann. Fac. Sci. Toulouse Math. (5) 11 (1990), 116-139. 23. K. A. Ribet, Galois representations and modular forms, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 375-402. 24. K.A. Ribet On the equation ap + 2aW + cp = 0, Acta Arithmetica 79 (1997) 7-16. 25. K. A. Ribet and B. Hayes, Fermat's Last Theorem and modern arithmetic, American Scientist (March-April, 1994), 144-156. 26. S. Singh, Fermat's Enigma: The epic quest to solve the world's greatest mathematical problem, with a foreword by John Lynch. New York: Walker and Co., 1997. 27. S. Singh and K.A. Ribet, Fermat's Last Stand, Scientific American 227 (1997), 68-73. 28. H. Stark, On the "gap" in a theorem of Heegner, J. Number Theory 1 (1969), 16-27. 29. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke alge-

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bras, Annals of Math. 141 (1995), 553-572. 30. A. Weil, Number Theory: an approach through history, From Hammurapi to Legendre. Boston, Mass: Birkhauser Boston 1984. 31. H. Weyl, Algebraic Theory of Numbers. Annals of Mathematics Studies, volume 1. Princeton: Princeton University Press 1940. 32. A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Annals of Math. 141 (1995), 443-551.

CHROMATIC G R A P H THEORY CARSTEN THOMASSEN Department of Mathematics Technical University of Denmark DK-2800 Lyngby Denmark E-mail: [email protected] We survey some results and problems on the chromatic number of a graph: The 4-color problem and its extension to other surfaces, some specific graphs for which the chomatic number is hard, the computational complexity, and some recent results and problems on the chromatic polynomial and list colorings.

1

Introduction: The 4-Color-Problem

One of the best known problems in mathematics is the A-Color-Problem: Can every map in the plane or, equivalently, on the sphere, be colored in only 4 colors. Every country (region) should receive one color, and neighboring countries should receive different colors. The problem may be reformulated in terms of graphs as follows. First, a graph G consists of a set V(G) of elements called vertices and a set E(G) of unordered pairs xy of vertices called edges. The complete graph Kn is the graph with n vertices and all possible edges. The chromatic number x(G) is the smallest number k such that the vertices of G can be colored in colors 1,2,... ,k in such a way that neighbors always receive distinct colors. Clearly, x{Kn) = n. A planar graph is a graph that can be drawn in the plane such that the vertices are points and every edge xy is a simple arcs joining x and y, and no edges intersect except at a common end. An affirmative solution to the 4-Color-Problem follows from the following: Theorem 1.1 x(G) < 4 for every planar graph G. The 4-Color-Problem goes back to the first half of the 19th century and remained open for more that a century. An affirmative answer was provided by Appel and Haken in 1976 ([3],[4]). The proof is very complicated, and its completeness has been questioned. A simpler (but still complicated) proof has been found by Robertson, Seymour, Sanders and Thomas [21]. Both proofs involve a computer search, and it still remains a challenge to find a proof that is not computer based. 183

184

2

The Heawood Map Color Problem

The Heawood map Color Problem is the counterpart of the 4-Color Problem for higher surfaces. For example, there is a 7-color theorem for the torus: Every graph on the torus has chromatic number at most 7. And this is best possible because Kj can be drawn on the torus, see e.g. [18]. Let Sg denote the surface obtained from the sphere So by adding g handles, and let Nk be obtained from So by adding k crosscaps. We say that Sg and Nk have Euler genus 2g and k, respectively. Already in 1890 Heawood [13] proved the following Theorem 2.1 Let G be a graph drawn on a surface of positive Euler genus k. Then x(G)< 1(7+ VT+~24k)/2\. Heawood then claimed that this is best possible since Euler's formula allows the complete graph on [(7 + \/l + 2Ak) /2\ vertices to be drawn on any surface of Euler genus k. Although this claim is essentially correct, it was not verified until about 80 years later by Ringel and Youngs, see [20], [18]. Theorem 2.2 Let S be a surface of Euler genus k. Then the complete graph on [(7 + y/1 + 24k)/2\ vertices can be drawn on S, except that K-j cannot be drawn on the Klein bottle Ar2It is known that every graph on the Klein bottle has chromatic number at most 6. Thus the maximum chromatic number of a graph on a surface S equals the maximum number of vertices in a complete graph that can be drawn on S. It is perhaps remarkable that the sphere is the only surface for which the upper bound of the maximum chromatic number is extremely difficult, while for surfaces of large Euler genus, the lower bound is the difficult one. 3

The Chromatic Number as a Scheduling Problem

Many real life scheduling problems may be formulated in terms of the chromatic number. Consider for example a school with teachers t\, t-i,..., tm and classes Ci,C2,... ,cp. Assume that teacher ti teaches class Cj rriij times during a week. Assume that the school has plenty of classrooms, but no teacher can teach two different classes at the same time and no class can be taught by two teachers at the same time. What is the smallest number of periods needed during a week? We form a new graph G whose vertices are the teachers and the classes and we add mitj edges between tl and Cj for % = 1,2,... ,m, j = 1,2,... ,p. We think of every edge of G as a country in a map. Two countries are said to be neighbors if and only if the two corresponding edges are incident with the same vertex. The above question is then precesely the

185

smallest number of colors needed to color the map. That map cannot in general be realized on the sphere. But it can be realized on some surface. Many similar examples occur. Unlike coloring problems in general, the present optimization problem is very well understood. The answer is simply the maximum degree (valency) of the graph G. 4

The Chromatic Number and Gallery Problems

The chromatic number shows up in many other contexts. Consider for example a closed polygonal (not necessarily convex) curve in the plane with n corners. We think of each line segment as a wall in a gallery viewed from above. In that gallery we select a number of points (in each of which we place a guard) so that every wall can be seen from one of the points. What is the number of guards needed in the worst case? It is easy to give examples where n / 3 guards are needed. Chvatal proved that n / 3 are always sufficient, see [1]. The following beautiful argument is due to Fisk, see also [1]: We think of the galleri as a cycle with n vertices drawn in the plane. We add successively noncrossing straight edges in the interior of the cycle until the interior is triangulated. Then we color the vertices of the resulting graph G. Clearly, we need at least three colors for that . On the other hand, it is easy to color G with only three colors: We color first a triangle with colors 1,2,3. In the general step we choose a triangle in which precisely two vertioces are colored, and then we give the third vertex the third color. It is easy to see that this results in a 3-coloring of G. The smallest color class has at most n / 3 vertices, and we may place a guard in each vertex of that color class. There are other intriguing gallery problems, some of which are unsolved, see e.g. [1]. 5

The Chromatic Number of Abstract Graphs

There are examples of very specific graphs for which the chromatic number is hard to find. To illustrate this, let m,n be natural numbers, m < n, and let K(m,n) denote the graph whose vertices are all m-element subsets of { 1 , 2 , . . . , n} such that two vertices are neighbors if and only if they are disjoint. These graphs are called Kneser graphs. It is easy to see that K(m, 2m + k) has a (k + 2)-coloring: If a vertex of the graph contains one of the elements 1, 2 , . . . , k + 1, then we give the vertex that color. All the remaining vertices are pairwise intersecting and may therefore receive the same color k + 2. In 1955 Kneser [16] conjectured that this is best possible, and this was verified over 20 years later by Lovasz [17]:

186

Theorem 5.1 x(K(m> 2m + fc)) = fc + 2. Lovasz' proof is based on Borsuk's theorem saying that, if the fc-dimensional unit sphere is covered by a collection of k + 1 open (respectively closed) sets, then at least one of them contains a pair of antipodal points. The problem of finding the chromatic number of the so-called unit distance graph U has been open for over 50 years. It is sometimes called the Nelson problem or the the Hadwiger-Nelson problem, see e.g. [9]. The graph U is easy to describe: Its vertices are the points of the Euclidean plane, and two vertices are neighbors if and only the Euclidean distance between them is 1. Clearly, x(^0 > 3, as £/ contains triangles. It is easy to find seven points pi,P2,P3,P4,P5,P6,P7 such that the following pairs have distance 1: PlP2,PlP3,PlP4,P3P4,P2P5,P2P6,P5P6,P3P7,P4P7,P5P7,P6P7-

As this Subgraph

of U has chromatic number 4, x(U) ^ 4. The upper bound x{U) £ 7 is obtained from a regular hexagonal tiling of the plane. The tiling is chosen such that every hexagon has diameter slightly less than 1, and the tiling is colored like a map with the additional constraint that two hexagons are allowed the same color if and only if the distance between them is greater than 1. It is easy to see that 7 colors are necessary and sufficient for this particular type of coloring. Let us call a coloring of U nice if it is a map coloring such that every country has diameter less than 1 and two regions (countries) are allowed the same color if and only if the distance between them is greater than 1. This generalizes the above hexagonal map coloring. In [27] I proved that every nice coloring of U needs at least 7 colors, even if the Euclidean metric is replaced by any other (unbounded) metric. It even generalizes to higher surfaces provided the surface has no contractible curves with small diameter (like a thin 2-way infinite cylinder) and no small contractible curve whose interior contains many disjoint discs of radius 1 (like a thin 1-way infinite cylinder with a disc pasted on the bottom). Although this is far from solving the Nelson problem, it suggests that the optimum coloring may be hard to find (except of course, if the the above hexagonal map coloring is optimum.)

6

Computational Complexity of the Chromatic Number

The difficulties in connection with the 4-color problem and the chromatic numbers of the Kneser graphs and the unit distance graph indicate that the chromatic number is hard to find in general. This has been made more precise by the theory of TVP-completeness of combinatorial problems, see [12]. Consider for example the following question. Given a graph G with n vertices and a natural number k, k < n. Is x{G) ^ &? This can be answered by considering all partitions of V(G) into k parts. However, we seek an algorithm that

187

is polynomially bounded, that is, there exists a fixed number q such that the algorithm needs only nq steps in the worst case. If such an algorithm exists, then the problem "Is x(G) < kV is said to belong to the class of problems denoted P. The broader class of problems denoted by NP are those problems for which it is possible to present a solution that can be checked by a polynomially bounded algorithm. Thus the problem "Is x(G) < fc?" clearly belongs to NP since it is easy to verify if a suggested fc-coloring really is a fc-coloring. For three decades the problem of deciding whether NP — P has been regarded as one of the most important unsolved problems in discrete matehmatics. In recent years it has become recognized as one of the main challenges in mathematics. Thus Smale [29] included it in his list of important problems for the next century along with the Riemann Hypothesis and the Poincare Conjecture, and it is also included in the list of the Clay Institute's seven Millenium Prize Problems. This complexity problem has a natural formulation in chromatic graph theory. Let us first consider the question whether x(G) < 2 for a given graph G. It is sufficient to consider the question for each connected component H of G. We shall try to color the graph H in colors 1,2. We first give a vertex the color 1. In the general step we choose a noncolored vertex x joined a colored vertex y, and we give x the color distinct from that of y. Either this simple procedure results in a 2-coloring of H or else we consider the first vertex x that cannot be colored. This means that x is joined to two vertices y, z of distinct colors. The colored subgraph is connected and has therefore a path between y and z. Together with x this path forms a cycle of odd length, and since such a cycle has no 2-coloring, H and G have no 2-coloring. While the 2-color problem, that is the question of deciding whether x{G) < 2, is trivial, the 3-color problem is hard, even when restricted to planar graph. The 3-color problem for planar graphs belongs to P if and only if NP = P. The 3-color problem is a so-called NP-complete problem, for details, see [12]. A graph G is k-critical if x{G) = k, and x{H) < k for every proper subgraph H of G. Thus the 3-critical graphs are precisely the cycles of odd length. In order to solve the 3-color problem it would suffice to find a precise description of the 4-critical graphs. However that approach to the 3-color problem does not seem feasible. Erdos [8] proved the following counter-intuitive result: Theorem 6.1 For every natural numbers k,q, there exists a k-chromatic graph with no cycle of length less than q. In particular, there exist 4-critical graphs with no cycle of length less than 1000. However, such graphs are not trivial to describe, and to characterize them seems hopeless. Even to characterize the 4-critical subgraphs of highly specialized graphs, like the unit distance graph, may be very hard. However,

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if we consider a family of graphs which is closed under taking subgraphs, and which contains only finitely many (fe + l)-critical graphs, then there is a polynomial time algorithm for deciding if a graph in the family can be kcolored. 7

The Chromatic Polynomial

The chromatic polynomial P{G,t) was introduced by Birkhof [5] in 1912. If G is a graph and t is a nonnegative integer, then P(G, t) denotes the number of colorings of V{G) such that all colors are one of the integers 1, 2,..., t and neighbors receive different colors. Clearly, P(Kn,t) = t(t - 1 ) . . . (t - n + 1). If x, y are nonadjacent vertices in a graph G, and G' is obtained from G by identifying x and y, then clearly, P{G,t) = P{G U {xy} ,t) + P(G',t). So it follows by induction on the number of missing edges in G that P(G, t) is a polynomial of degree n, where n is the number of vertices of G. By the same argument, a number of other properties of the chromatic polynomial can be derived. For example, the leading coefficient is 1, and the coefficients alternate in sign. However, one encounters quickly open questions. For example, an old conjecture of Read says that the sequence of absolute values of the coefficients is unimodal, that is, it first increases, then decreases, see [19]. The chromatic polynomial was introduced in order to understand the chromatic number better, specifically to solve to 4-color problem. The 4-color theorem is equivalent to the statement that P(G, 4) is positive for every planar triangulation. Tutte came close, see [19]: Theorem 7.1 P(G, (5 + \/5)/2) = P(G, 3.618...) is positive for every planar triangulation. The reason that the chromatic polynomial has fascinated discrete mathematicians for years is not its potential to find the chromatic number. Ironically, there seems to be no family of graphs, perhaps even not one single graph, for which the easiest or most natural way to determine the chromatic number goes via the chromatic polynomial. It would be highly interesting to construct such a graph. The main reason for studying it is probably that it shows up in surprising contexts. One striking example is the following result of Stanley, see [30]: If G is a graph, then an acyclic orientation of G is obtained by numbering it vertices 1, 2 , . . . , n and then orient every edge from the small end to the large end. Clearly, Kn has n! acyclic orientations, and every other graph G on n vertices has fewer. To calculate the exact number is known to be hard in general. Stanley proved that it equals \P(G, - 1 ) | , that is the absolute value of the chromatic polynomial evaluated at —1.

189 In recent years the chromatic polynomial has also been studied by theoretical physicists, see e.g Shrock and S-H Tsai [30]. In a letter of April 26, 1999, Shrock has informed me that P(G, q) is equal to the zero-temperature limit of the partition function of a certain spin model called the q-state Potts antiferromagnet on the graph G (which is often a subgraph of a regular lattice). Also, the roots of the chromatic polynomials are studied by physicists. Thus Alan Sokal has recently informed me that he has proved that the collection of complex roots of all chromatic polynomials is a dense set in the complex plane. We close this section with some comments on real roots. We say that t is a chromatic root of G if P(G,t) = 0. Clearly, 0 is a chromatic root, and so is 1 unless G has no edges. It is well known and easy to see that there are no other roots less than 1. An old (still unsettled) conjecture by Birkhoff and Lewis [6] says that a planar graph has no chromatic root less than or equal to 4. More recently, Jackson [14] proved the following fascinating result: T h e o r e m 7.2 All chromatic roots distinct from 0,1 are greater than 32/27. Jackson also proved that 32/27 cannot be replaced by any larger number. This was strengthened in [25] where it is proved that the set of chromatic roots consists of 0 and 1 and a dense subset of the interval from 32/27 to infinity. Inspired by these results, I found in [26] a new connection between colorings and the problem of finding a Hamiltonian cycle that is, a cycle containing all vertices of a graph. This problem is a special case of the Travelling Salesman Problem, that is, the problem of minimizing the length of a closed walk in a graph visiting every vertex at least once. The first link between colorings and hamiltonian cycles was perhaps the observation made by Tait in 1880 (see [7], page 160) that, if a planar graph has a hamiltonian cycle, then the corresponding map is 4-colorable. The proof is easy: The Hamiltonian cycle divides the map into two parts, the interior and the exterior. The countries in each part can be colored in two colors, so that a total of only four colors are used. A stronger link is provided by the fact that both the problem of deciding if a graph is 3-colorable and that of deciding if a graph has a hamiltonian cycle are ./VP-complete, and thus these two problems are, in some sense, equivalent from a computational point of view. In [26], I proved: T h e o r e m 7.3 / / the chromatic polynomial of a graph has a noninteger root less than or equal to t0 = §+A\/26 + 6\/33+- 1 ^ 2 6 - 6 ^ = 1.29559..., then the graph has no hamiltonian path.

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This result is best possible in the sense that it becomes false if to is replaced by any larger number. Since 1.295... is only slightly larger than 32/27 = 1.185..., this result is not easy to apply. However, it shows that, if we know how many colorings (with any number of colors) that a graph has, then in some instances we may conclude that the graph has no Hamiltonian path and hence also no Hamiltonian cycle. For Hamiltonian cycles a slightly better result may be possible: I conjecture that no graph with a Hamiltonian cycle has a chromatic root distinct from 0,1 and less that 2. 8

List Color Techniques

If G is a graph and, for every vertex v, L{v) denotes a set of so-called available colors at v, then an L-coloring of G is a coloring of the vertices such that neighbors get different colors and every vertex gets an available color. A graph is k-list-colorable if it has an L-coloring whenever each vertex has k available colors. The list-chromatic number Xi(G) is the smallest number k such that G is fc-list-colorable. If all lists of available colors are equal to the set { 1 , 2 , . . . , k}, then list coloring reduces to fc-coloring. Hence x(G) < Xi{G). While the problem "Is "x(G) < k" clearly belongs to NP it is not known if the problem "Is Xi{G) < k" belongs to NP. So it is perhaps not surprising that list coloring gives rise to many difficult questions. During the past decade some list color methods have been developed. And, unlike the chromatic polynomial, list coloring has turned out to be very useful to attack more traditional coloring questions. One striking example is the solution of the following conjecture of Erdos which turned out to be equivalent to an old conjecture of Schur, see [10]: Let C be a cycle with 3m vertices. Add m pairwise disjoint triangles so that we obtain a graph H in which each vertex has degree (valency) 4. As H has triangles, x{H) > 3. Erdos conjectured that x{H) = 3 for each such graph H which is called a cycle plus triangles graph. The affirmative answer by Fleischner and Stiebitz [10] is a striking application of a list color theorem by Alon and Tarsi [2]. If D is a directed graph, that is, every edge has an orientation, then D is called Eulerian if, for every vertex, the indegree equals the outdegree. Every directed graph with no edge is considered to be Eulerian. Alon and Tarsi proved the following remarkable result: Theorem 8.1 If D is a directed graph such that the number of Eulerian subgraphs with vertex set V(D) and an even number of edges is distinct from the number of Eulerian subgraphs with vertex set V(D) and an odd number of

191

edges, then the list chromatic number of D is at most the maximum outdegree plus 1. The two numbers of Eulerian subgraphs in Theorem 8.1 are not easy to deal with in general. However, Fleischner and Stiebitz [10] showed that the cycle plus triangles graph H satisfies the assumption of the theorem of Alon and Tarsi when the initial cycle and each of the triangles is made into a directed cycle. Hence Theorem 8.2 If H is a cycle plus triangles graph, then x{H) = Xi{H) = 3. Consider again the graph class-teacher graph G in Section 3. The minimum number of periods equals the smallest number of colors needed to color the edges of G such that two edges incident with the same vertex always receive different colors. This number is called the chromatic index of G and is denoted x'{G). (It is the usual chromatic number of the so-called line graph of G.) As we pointed out, x'(G) equals the maximum degree of G. The proof is easy, see e.g. [7]. This depends heavily on the fact that G is bipartite (that is, 2-colorable). For graphs G that are not necessarily bipartite, x'(G) equals either the maximum degree of G or the maximum degree of G plus 1, by Vizing's theorem, see again [7]. The list chromatic index x'i(G) is defined in the obvious way, and it reflects to a higher degree than x'(G) the real life schedulling problems. The complete regular bipartite graph Km}m is the bipartite graph with m vertices in each part and all possible edges between the parts (representing m classes and m teachers, each of which teaches each class precisely once.) The conjecture that x'i{Km,m) = m became known as The Dinitz problem and was open for 15 years, see [1]. Galvin [10] then proved much more: Theorem 8.3 x[{G) = x'{G) for every bipartite graph G. One of the most prominent open problems on list coloring is to generalize this result to non-bipartite graphs. Vizing conjectured in 1975 that every planar graph has list chromatic number at most 5. This conjecture was rediscovered by Erdos, Rubin and Taylor in 1980, who also conjectured that there exist planar graphs that are not 4-list-colorable. The former conjecture was verified by the present author, and the latter by Margit Voigt, see [1]. We shall here present the proof from [23]. Theorem 8.4 x\{G) < 5 for every planar graph G.

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Proof It is sufficient to prove the theorem for maximal planar graphs, that is, graphs that triangulate the plane. We consider a slightly larger class of planar graphs: A near-triangulation is a planar graph containing a cycle C : v\V2 • • • vmv\ called the outer cycle such that all other vertices and edges are in the interior of C and triangulate the interior of C. Assume that every vertex in the interior of C has at least 5 available colors, every vertex on C has at least 3 available colors, and v\,v% are already colored. We now prove, by induction on the number of vertices, that the coloring of i>i, i>2 can be extended to a list coloring of the whole near-triangulation. Consider the vertex V3. This vertex has three available colors. Choose two of them, say i,j distinct from the color of i>2 . Now delete V3, and delete also i,j from the list of available colors from all neighbors of V3 in the interior of C. These vertices used to have five available colors, and now they may have only three. But, they are now on the outer cycle so we may apply the induction hypothesis to the resulting graph. Finally, we may also color V3 by i or j since we only have to avoid the color of v\. This results in a list coloring unless V3 has a neighbor vp on C which is distinct from V2:v$. In that case we argue differently. We first apply the induction hypothesis to the cycle VzV2V\... vpV3 and its interior. In particular, vp,V3 become colored. Then we apply the induction hypothesis to the cycle V3U4U5 . . . vpV3 and its interior where now vp, V3 play the role of i>i, i>2- This completes the proof. The proof of Theorem 8.4 does not use Euler's formula (as do most coloring results on planar graphs). It is perhaps the simplest known proof of the 5color theorem. More important, the strengthening in the formulation, which was introduced for technical purposes (to make the induction work) turns out to be useful for higher surfaces. In Theorem 2.1 the colors needed tends to infinity as the Euler genus tends to infinity. Yet, there is a 5-color theorem for each surface, as the following theorem in [24] shows: Theorem 8.5 For each surface S, there are only finitely many 6-critical graphs on S. In other words, if we wish to color a graph on S in 5 colors only, then it may not be possible. (As mentioned earlier, on the torus we may need 7 colors.) However, there is only a finite number of (minimal) obstructions that prevent us from color a graph on S in 5 colors. The presence of one of those (as a subgraph in the graph in consideration) can be checked in polynomial time. A classical result of Grotzsch says that every planar triangle-free graph can be colored in 3 colors only. A simple list color proof of that is the first step in the proof of the following theorem in [28]:

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Theorem 8.6 For each surface S, there are only finitely many ^.-critical graphs without triangles and qudrilaterals on S. Theorem 8.5 implies that the chromatic number of graphs without triangles and qudrilaterals on S can be found in polynimial time. Theorem 8.5, 8.6 are, in a sense, best possible. Theorem 8.5 becomes false if 5 is replaced by 4. And Theorem 8.6 becomes false if qudrilaterals are allowed. Nevertheless it would be interesting to decide if there is a general 4-color theorem in the sense that 4-colororability can be checked in polynomial time on each surface. Jensen and Toft [15] gave a comprehensive collection of beautiful open coloring problems. Many of these have natural list-color counterparts. The simple list color proofs of Grotzsch' theorem and Theorem 8.5 above raise the question if the 4-color theorem has a relatively short list color proof. As pointed out above, not every planar graph is 4-list-colorable. Not even if all lists are subsets of {1,2,3,4,5} as Horst Sachs has informed me. But, there might be a version where all the lists in the interior are {1, 2, 3, 4}, and the lists on the outer cycle are subsets of {1,2,3,4}. In order to make the induction work, some of them should be allowed to be proper subsets. However, even mild extensions of the 4-color theorem lead to false statements. References 1. M. Aigner k. G. Ziegler, Proofs from THE BOOK, Springer-Verlag, Berlin (1998). 2. N. Alon & M. Tarsi, Colorings and orientations of graphs, Combinatorica, 12 (1992), pp. 125-134. 3. K. Appel & W. Haken, Every planar graph is four colorable. Part I. Discharging, Illinois J. Math., 21 (1977), pp. 429-490. 4. K. Appel & W. Haken & J. Koch, Every planar graph is four colorable. Part II. Reducibility, Illinois J. Math., 21 (1977), pp. 491-567. 5. G.D. Birkhoff, A determinant formula for the number of ways of coloring a map, Ann. of Math., 12 (1912), pp. 42-46. 6. G.D. Birkhoff & D.C. Lewis, Chromatic polynomials, Trans. Amer. Math. Soc, 60 (1946), pp. 355-451. 7. J. A. Bondy & U. S. R. Murty, Graph Theory with Applications, Macmillan, London (1976). 8. P. Erdos, Graph theory and probability, Can. J. Math., 11 (1959), pp. 3438. 9. P. Erdos & G. Purdy, Extremal problems in combinatorial geometry,

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in Handbook of Combinatorics (eds. R. L. Graham, M. Grotschel k. L. Lovasz), North-Holland, Amsterdam (1985), pp. 809-874. 10. H. Fleischner & M. Stiebitz, A solution to a coloring problem of P. Erdos, Discrete Math., 101 (1992), pp. 39-48. 11. F. Galvin, The list chromatic index of a bipartite multigraph, Journal of Combinatorial Theory, Series B, 63 (1995), pp. 153-158. 12. M. R. Garey & D. S. Johnson, Computers and Intractability, A guide to the Theory of NP-completeness, Freeman, San Francisco (1979). 13. P.J. Heawood, Map-color theorem, J. Math. Oxford, 24 (1890), pp. 332338. 14. B. Jackson, A zero-free interval for chromatic polynomials of graphs, Combinatorics, Probability and Computing, 2 (1993), pp. 325-336. 15. T. Jensen & B. Toft, Graph Coloring Problems, John Wiley, New York (1995). 16. M. Kneser, Aufgabe 360, Jahresbericht. Deutchen Math. Ver., 58 (195556), pp. 27. 17. L. Lovasz, Kneser's conjecture, chromatic number, and homotopy, J. Combinatorial Theory, Series A, 25 (1978), pp. 319-324. 18. G. Mohar & C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press, Baltimore (2000). 19. R. C. Read & W. T. Tutte, Chromatic polynomials, in Selected Topics in Graph Theory (eds. L. W. Beineke & R. M. Wilson), Academic Press, New York (1988), pp. 15-42. 20. B. Ringel, Map Color Theorem, Springer-Verlag, Berlin (1974). 21. N. Robertson & D. Sanders & P. Seymour & R. Thomas, The four-color theorem, Journal of Combinatorial Theory, Series B, 70 (1997), pp. 2-44. 22. R. P. Stanley, Enumerative Combinatorics, Wadsworth and Brooks/Cole, Monterey, CA (1986). 23. C. Thomassen, Every planar graph is 5-choosable, Journal of Combinatorial Theory, Series B, 62 (1994), pp. 180-181. 24. C. Thomassen, Color-critical graphs on a fixed surface, Journal of Combinatorial Theory, Series B, 70 (1997), pp. 67-100. 25. C. Thomassen, The zero-free intervals for chromatic polynomials of graphs, Combinatorics, Probability and Computing, 6 (1997), pp. 497506. 26. C. Thomassen, Chromatic roots and hamiltonian paths, Journal of Combinatorial Theory, Series B, 80 (2000), pp. 218-224. 27. C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly, 106 (1999), pp. 850-853.. 28. C. Thomassen, The chromatic number of a graph of girth 5 on a fixed

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surface, MAT-report, Technical University of Denmark, 2000-34 (2000), pp. 1-34. 29. S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), pp. 7-15. 30. R. Shrock & S-H. Tsai, Ground state entropy of Potts antiferromagnets on homeomorphic families of strip graphs, Physica A, 259 (1998), pp. 315348.

GLIMPSES OF N O N L I N E A R PARTIAL D I F F E R E N T I A L EQUATIONS I N T H E T W E N T I E T H C E N T U R Y . A P R I O R I ESTIMATES A N D T H E B E R N S T E I N P R O B L E M NEIL S. TRUDINGER School of Mathematical Sciences, The Australian National University, CANBERRA, ACT 0200, AUSTRALIA E-mail: Neil. TrudingerQmaths. anu. edu. au

1

Introduction

The basic models of linear partial differential equations were formulated in the eighteenth and early nineteenth centuries. In order of appearance these were: the wave equation, (D'Alembert 1752), d2u

d2u

the Laplace equation, (Laplace 1780), d2u AU:

=

M

d2u +

„

=0

W '

,„ „. (L2)

and the heat equation, (Fourier 1810),

5 "S •