Volume 70, Number 10, November 2023 Notices of the American Mathematical Society 9781470474232

Table of contents :
November 2023 Front Cover
JMM 2024 Registrants
AMS Employment Center at JMM
Table of Contents
Masthead
Letters to the Editor
A Word From...Tyler Kloefkorn
Shape Optimization for Covering Problems by Ernesto G. Birgin and Antoine Laurain
Quantum Ergodicity in Theorems and Pictures by Seymon Dyatlov
The Next Generation
A Perspective on the Regularity Theory of Degenerate Elliptic Equations by Hector A. Chang-Lara
Threshold Phenomena for Random Discrete Structures by Jinyoung Park
Random Phenomena with Fractal-Like Features by Patricia Alonso Ruiz
AMS Career Fair at JMM 2024
Early Career
In Memory of Andrew J. Majda by Bjorn Engquist, Panagiotis Souganidis, Samuel N. Stechmann, and Vlad Vicol
In Memory of Steve Zelditch, Coordinated by Bernard Shiffman and Jared Wunsch
When Small Changes Lead to Big Impact: Hysteresis in Mathematics Teaching by Chris Rasmussen and Estrella Johnson
Concepts at the Heart of Mathematics--Through the Centuries, A Review by Katalin Bimbo
Bookshelf
AMS Bookshelf
AMS Congressional Fellowship 2024-2025
Walter Rudin Meets Elias M. Stein by Odysseas Bakas, Valentina Ciccone, and James Wright
SACNAS Turning 50 by Fabio Augusto Milner
MRCs
Some Colleges with Vulnerable Students Cull Math Programs by Susan D'Agostino
Calls for Nominations & Applications
Apply for AMS Fellowships
Biennial Overview of AMS Honors
MathSciNet for Developing Countries
AMS Reciprocity Agreements
AMS Updates
Mathematics People
Join SIAM
Classified Advertising
New Books Offered by the AMS
Meetings & Conferences of the AMS November Table of Contents
Meetings & Converences of the AMS
AMS Programs Booth at JMM 2024
Now Available from the AMS

Citation preview

ISSN 0002-9920 (print) ISSN 1088-9477 (online)

of the American Mathematical Society November 2023

Volume 70, Number 10

The cover design is based on imagery from “Quantum Ergodicity in Theorems and Pictures,” page 1592.

JMM 2024 Registrants:

Join us for member activities! JMM 2024 Grand Opening Reception Wednesday, January 3 | 6:00 p.m.– 8:30 p.m. Moscone Center South, Hall A Enjoy music, food stations, and a chance to socialize and connect with your colleagues on this first night in San Francisco. Find the AMS booth and these special freebies: 6:00 p.m.– 8:30 p.m. Photo booth—take a memory home!

6:00 p.m.–8:00 p.m. Wine tasting

6:00 p.m.– 8:30 p.m. Virtual Reality Tour—AMS members may enjoy a unique view of “The Golden City” provided to you by the AMS Membership Department!

Special JMM Offers on AMS Membership JMM 2024 registrant members of one of the JMM partner organizations* who have not been an AMS member since 2021, are eligible for 50% off a 2024 American Mathematical Society Membership! Note: to receive this discount please use the same email account you used to register for JMM. This special offer is available until February 1, 2024.** * Eligible JMM partner organizations: American Statistical Association (ASA), Association for Women in Mathematics (AWM), Mathematical Association of America (MAA), National Association of Mathematicians (NAM), and Society for Industrial and Applied Mathematics (SIAM). **This offer excludes Affiliate, Life, and Unemployed memberships.

Join or renew your membership on Thursday–Friday at the exhibit booth and receive a complimentary gift!

Professional Portraits for AMS members–back by popular demand! Thursday, January 4 –Friday, January 5 AMS members: Make your appointment to have your professional portrait taken for free. Make an appointment between 9:30 a.m.– 4:25 p.m. PST at the AMS membership booth. Your image will be emailed to you in just a few minutes! Schedule appointments at: amermathsoc.simplybook.me/v2/

www.ams.org/membership

INTERVIEW CANDIDATES AT THE

AMS EMPLOYMENT CENTER AT THE JOINT MATHEMATICS MEETINGS

Open Hours Wednesday, January 3

8:00 a.m.–3:00 p.m. Thursday, January 4

Moscone North/South Convention Center Exhibit Hall B San Francisco, CA January 3–5, 2024

8:00 a.m.–5:30 p.m. Friday, January 5

8:00 a.m.–5:30 p.m. EC

Look for the EC icon on MathJobs.org to find employers interviewing at the JMM!

2024

Only interviewing a few people? Try a one-day table.

Visit www.ams.org/emp-reg for registration instructions.

November 2023 Cover Credit: The image used in the cover design appears in “Quantum Ergodicity in Theorems and Pictures,” p. 1592, and is courtesy of Semyon Dyatlov.

FEATURES Shape Optimization for Covering Problems............................ 1584 Ernesto G. Birgin and Antoine Laurain

Quantum Ergodicity in Theorems and Pictures....................... 1592 Semyon Dyatlov

A Perspective on the Regularity Theory of Degenerate Elliptic Equations....................................................................... 1604 Héctor A. Chang-Lara

Threshold Phenomena for Random Discrete Structures...........1615 Jinyoung Park

Random Phenomena with Fractal-like Features....................... 1626 Patricia Alonso Ruiz

Letters to the Editor................................................... 1581

Book Review: Concepts at the Heart of

A Word from... Tyler Kloefkorn................................. 1582

Reviewed by Katalin Bimbó

Early Career: Math and the Real World.................... 1635

Bookshelf................................................................... 1694

BIG Article............................................................. 1635

Mathematics—Through the Centuries..................... 1690

Jesse Berwald

AMS Bookshelf.......................................................... 1695

John Holmes

History: Walter Rudin Meets Elias M. Stein.............. 1697 Odysseas Bakas, Valentina Ciccone, and James Wright

Barbara Giunti, Jānis Lazovskis, and Bastian Rieck

Communication: SACNAS Turning 50...................... 1704 Fabio Augusto Milner

Teaching Mathematical Finance.......................... 1637 DONUT: Creation, Development, and Opportunities of a Database........................ 1640 Thinking About Failure in Data Analysis and Beyond........................................................... 1644 Roger D. Peng

Dear Early Career.................................................. 1646 Memorial Tribute: In Memory of

Andrew J. Majda........................................................ 1648

Bjorn Engquist, Panagiotis Souganidis, Samuel N. Stechmann, and Vlad Vicol

Memorial Tribute: In Memory of Steve Zelditch....... 1667 Coordinated by Bernard Shiffman and Jared Wunsch Education: When Small Changes Lead to Big

Impact: Hysteresis in Mathematics Teaching........... 1684 Chris Rasmussen and Estrella Johnson

Math Tales: Some Colleges with Vulnerable

Students Cull Math Programs...................................1711

Susan D’Agostino

AMS Reciprocity Agreements.................................... 1727 News: AMS Updates.................................................. 1728 News: Mathematics People....................................... 1730

Classified Advertising................................................ 1733 New Books Offered by the AMS............................... 1736 Meetings & Conferences of the AMS.........................1742

FROM THE AMS SECRETARY Calls for Nominations & Applications....................... 1716

I. Martin Isaacs Prize for Excellence in Mathematical Writing........................................... 1716 2024 MOS–AMS Fulkerson Prize........................ 1716

Biennial Overview of AMS Honors...........................1719

INVITATIONS FROM THE AMS Member Activities at JMM 2024........inside front cover AMS Employment Center at JMM 2024.................. 1577 AMS Career Fair at JMM 2024.................................. 1634 AMS Congressional Fellowship 2024–2025............ 1696 Apply for 2024 Mathematics Research Communities..............................................1710 Apply for AMS Fellowships....................................... 1718 Math Variety Show: Seeking Performers.................. 1731 Visit the AMS Programs Booth at JMM 2024............................................inside back cover

Math is a Gift You Can Give Daily Your Daily Epsilon of Math Wall Calendar 2024

is now available! Rebecca Rapoport

Harvard University, Cambridge, MA Michigan State University, East Lansing, MI

Dean Chung

Harvard University, Cambridge, MA University of Michigan, Ann Arbor, MI

Keep your mind sharp all year long with Your Daily Epsilon of Math Wall Calendar 2024 featuring a new math problem every day and 13 beautiful math images! Let mathematicians Rebecca Rapoport and Dean Chung tickle the left side of your brain by providing you with a math challenge for every day of the year. The solution is always the date, but the fun lies in figuring out how to arrive at the answer. End the year with more brains than you had when it began with Your Daily Epsilon of Math Wall Calendar 2024. 2023; ISBN: 978-1-4704-7423-2; List US$20; AMS members US$16; MAA members US$18; Order code MBK/147

Learn more at bookstore.ams.org/mbk-147

Notices of the American Mathematical Society

EDITOR IN CHIEF

SUBSCRIPTION INFORMATION

Erica Flapan

Individual subscription prices for Volume 70 (2023) are as follows: nonmember, US$742, member, US$445.20. (The subscription price for members is included in the annual dues.) For information on institutional pricing, please visit https://www.ams.org/publications/journals/subscriberinfo. Subscription renewals are subject to late fees. Add US$6.50 for delivery within the United States; US$24 for surface delivery outside the United States. See www.ams.org/journal-faq for more journal subscription information.

ASSOCIATE EDITORS

Daniela De Silva Benjamin Jaye Reza Malek-Madani Chikako Mese Han-Bom Moon Emily Olson Scott Sheffield Laura Turner

Boris Hasselblatt, ex officio Richard A. Levine William McCallum Antonio Montalbán Asamoah Nkwanta Emilie Purvine Krystal Taylor

ASSISTANT TO THE EDITOR IN CHIEF

PERMISSIONS

Masahiro Yamada

All requests to reprint Notices articles should be sent to: [email protected].

CONSULTANTS

Jesús De Loera Ken Ono Bianca Viray

Bryna Kra Kenneth A. Ribet

ADVERTISING Notices publishes situations wanted and classified advertising, and display advertising for publishers and academic or scientific organizations. Advertising requests, materials, and/or questions should be sent to: [email protected] (classified ads) [email protected] (display ads)

Hee Oh Francis Su

SUBMISSIONS

MANAGING EDITOR

The editor-in-chief should be contacted about articles for consideration after potential authors have reviewed the “For Authors” page at www.ams.org/noticesauthors.

Meaghan Healy

The managing editor should be contacted for additions to our news sections and for any questions or corrections. Contact the managing editor at: [email protected].

CONTRIBUTING WRITER

Letters to the editor should be sent to: [email protected].

Elaine Beebe

To make suggestions for additions to other sections, and for full contact information, see www.ams.org/noticescontact.

COMPOSITION, DESIGN, and EDITING

Brian Bartling Craig Dujon Dan Normand Miriam Schaerf

John F. Brady Anna Hattoy John C. Paul Mike Southern

Nora Culik Lori Nero Courtney Rose-Price Peter Sykes

Supported by the AMS membership, most of this publication, including the opportunity to post comments, is freely available electronically through the AMS website, the Society’s resource for delivering electronic products and services. Use the URL www.ams.org/notices to access the Notices on the website. The online version of the Notices is the version of record, so it may occasionally differ slightly from the print version.

The print version is a privilege of Membership. Graduate students at member institutions can opt to receive the print magazine by updating their individual member profiles at www.ams.org/member-directory. For questions regarding updating your profile, please call 800-321-4267. For back issues see www.ams.org/backvols. Note: Single issues of the Notices are not available after one calendar year.

The American Mathematical Society is committed to promoting and facilitating equity, diversity and inclusion throughout the mathematical sciences. For its own long-term prosperity as well as that of the public at large, our discipline must connect with and appropriately incorporate all sectors of society. We reaffirm the pledge in the AMS Mission Statement to “advance the status of the profession of mathematics, encouraging and facilitating full participation of all individuals,” and urge all members to conduct their professional activities with this goal in mind. (as adopted by the April 2019 Council)

[Notices of the American Mathematical Society (ISSN 0002-9920) is published monthly except bimonthly in June/July by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2213 USA, GST No. 12189 2046 RT****. Periodicals postage paid at Providence, RI, and additional mailing offices. POSTMASTER: Send address change notices to Notices of the American Mathematical Society, PO Box 6248, Providence, RI 02904-6248 USA.] Publication here of the Society’s street address and the other bracketed information is a technical requirement of the US Postal Service.

© Copyright 2023 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability. Opinions expressed in signed Notices articles are those of the authors and do not necessarily reflect opinions of the editors or policies of the American Mathematical Society.

LETTERS TO THE EDITOR ICM 2026 in Philadelphia The next International Congress of Mathematicians will be held in Philadelphia on July 23–30, 2026. This coincides with the 250th anniversary of the Declaration of Independence and the 40th anniversary of the last ICM in the US which was held in Berkeley. The ICM offers a rare opportunity to meet some of the world’s leading mathematicians and get impressions of the vast diversity of today’s mathematics. Philadelphia provides the ICM with an excellent conference center conveniently located in the heart of the city. There is an abundance of choices of food within walking distance. There are top-quality hotels close to the conference center or for the budget-minded there will be a selection of dorm rooms at Philadelphia universities. The city offers fascinating history, world class art, and culture for your enjoyment. We hope to see many of you in Philadelphia! Sincerely, Jalal Shatah Chair of the ICM 2026 Local Organizing Committee

Letter to the Editor RE: Chandler Davis I am writing in response to Peter Rosenthal’s very nice tribute to Chandler Davis that was published in the August 2023 edition of the Notices. His tribute includes an error that is minor but I feel Professor Davis would appreciate having corrected. Rosenthal wrote, “[Horace B. Davis, Chandler’s father] was fired from his position at a university because he refused to answer questions asked by HUAC.” Horace B. Davis was not questioned by HUAC. Rather, he was questioned by the senate’s analogue, the Senate Internal Security Subcommittee, commonly called SISS or “Jenner’s Committee” (after its chairman, Senator William Jenner). Furthermore, Horace lost not one but two jobs over the SISS hearing: one at the University of Missouri–Kansas City and another at Benedict College. He

wrote about both experiences in his autobiography, Liberalism Is Not Enough, and additional details about the Kansas City incident are given in a report by the American Association of University Professors that was published in the April 1957 issue of the AAUP Bulletin. Best regards, Jesse Kass

Response To AMS Notices, Jesse Kass is correct in pointing out my error: Chan’s father was brought before SISS, not HUAC. In my defence, the Wikipedia entry for Horace Davis Senior makes the same mistake. I can’t blame Wikipedia though. I only learned the truth after the Notices article was published and Jesse Kass told me about it. Chan’s father was a very interesting man who had a lot in common with his son, in addition to their political views. I toyed with the idea of including more about Chan’s father in my Notices piece but decided it would not be appropriate in a mathematics journal. If any reader of this letter feels that they would like to consider writing such an article and wants some ideas that might be useful, I would be glad to suggest some (no kidding). Best, Peter Rosenthal

*We invite readers to submit letters to the editor at notices-letters @ams.org. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2808

NOVEMBER 2023

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A WORD FROM. . . Tyler Kloefkorn, AMS Associate Director for Government Relations

Photo courtesy of Michelle Schwalbe.

The opinions expressed here are not necessarily those of the Notices or the AMS. During his invited address at JMM 2023, Jordan Ellenberg asserted that “all mathematics is mathematical communication.” This resonated with me. I’ll go a step further and claim that mathematical communication is mathematical education. For many of us, a passion for mathematics and a passion for education go hand in hand. Our passion is often translated into action. I am overwhelmed with the number of books, op-eds, research papers, podcasts, and conferences for disseminating work in education. These efforts come from teachers, researchers, students—truly from the entire community of mathematical scientists, at all career stages. Sometimes this passion causes great tension within our community (hello math wars!). We collectively have strong opinions about active learning, teacher training, curriculum, standardized testing—you name it. But, on the whole, this passion is an asset; it incentivizes us to constantly grow and improve. I think we have the most impact on mathematical sciences education when we 1) have respectful and productive public forums within our community and, when possible, 2) speak with one voice to other communities. Seeing an opportunity to contribute to both, I joined the AMS Office of Government Relations (OGR) in 2021. The role of Associate Director was designed to focus on education policy and advocacy with the executive and Tyler Kloefkorn is the associate director for government relations at the AMS. His email address is [email protected]. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2791

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legislative branches of the federal government. In this position, I have an opportunity to supplement the work of the AMS community and amplify it at the national level.

The AMS Office of Government Relations’ Education Advocacy Foci As per the mission of the OGR, “Our activities strengthen the involvement of mathematics in federal science policy considerations, and simultaneously support mathematicians at all stages of their careers.” Thus, we support students and educators in the mathematical sciences, and we must be mindful of the priorities of each administration and of each Congress. Here, I’ll share a few examples of this work. The Biden administration—via the Department of Education (DoEd) and the Office of Science and Technology Policy (OSTP)—places a strong emphasis on STEM education. For example, in December 2022, DoEd hosted a day-long, in-person event at its headquarters called “You BELONG in STEM.” This was a call-to-action for organizations to improve inclusiveness in STEM from Pre-K, to workforce training, and everything in between. Meanwhile, and for many years, DoEd has facilitated the Graduate Assistance in Areas of National Need (GAANN) Program. Its annual budget for the past several years has been approximately $23 million, having fallen from $31 million in 2012. Awardees are academic departments and they provide funding for eligible graduate students in—as the name suggests—academic areas of national need. In 2012, 20 different departments in the mathematical sciences were awardees. But since 2016, the mathematical sciences were not designated to be an area of national need and most of our departments could not apply. Seems a little inconsistent, right? This policy inconsistency provides an opportunity for OGR—we know there’s a clear consensus in our

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 70, NUMBER 10

A Word From. . . community for robust federal support for graduate students. We work with past GAANN awardees and with DoEd about including the mathematical sciences in future solicitations. This is an ongoing effort and we hope our advocacy on this front will be successful in 2023. You have likely read stories about mathematics curriculum reform and teacher shortages in national media outlets. It’s difficult to find consensus—at the national but any level—on curriculum, and recent trends in teacher recruitment and retention are tremendously challenging. Our office can’t speak on behalf of the community here. However, we can be a connector to those in the community who are doing important work and we can partner with other societies and associations (e.g., NCTM, MAA, CBMS). In May 2022, and with guidance from the AMS Committee on Education and partner societies, OGR hosted the Forum on Evolving Curriculum in High School and Early Undergraduate Mathematical Sciences Education. This event was designed as a respectful and productive conversation on difficult topics (e.g., data science in high school mathematics pathways). We invited panelists who had recently led curriculum reforms and/or worked with their states. On high school curricula, we won’t come to consensus, but through such conversations we can better understand the work of others and gain empathy for others’ opinions. In the exhibit hall at JMM 2023 in Boston, OGR organized an interactive board for advocacy. We informally polled JMM attendees “to help inform AMS advocacy in Washington DC” by asking “which issues matter most to you: federal research funding; visas and immigration; artificial intelligence; diversity, equity, and inclusion in math; math modeling for policy; K–12 education and support for math teachers; and other?” Hundreds of attendees participated by placing stickers—colored based on their career stage and sector—on preferred issues. With each participant having up to three stickers to use, JMM ended and we counted 1,095 stickers on the board. The top two issues were diversity, equity, and inclusion in math and K–12 education and support for math teachers. Not far behind was federal research funding and math modeling for policy. This was the first time OGR facilitated something like this. But based on the participation that far exceeded our expectations, we will absolutely do this at future JMMs. By no means is it a definitive survey, but we seldom get the opportunity to hear from a large portion of the community. Please look for us in San Francisco at JMM 2024!

NOVEMBER 2023

Things to Watch in 2023 It’s critical for our community to watch the implementation of CHIPS and Science Act, a bill that was signed into law by President Biden in the summer of 2022, and related appropriations (i.e., annual funding) in 2023 and beyond. For example, in the 2023–2024 award cycle and pending appropriations, the National Science Foundation will be able to award significantly more (as many as 3,000) Graduate Research Fellowships (GRFs) than in years past (∼2,000). Typically, 100 awards are made to students in the mathematical sciences. In theory, there should be 150 awards for the mathematical sciences in 2023. However, the number of awards in a given discipline is determined, at least in large part, by the relative number of applications among all disciplines. Alert, alert, alert: in order for us to see more NSF GRFs in mathematics, we need more of our students to apply. Second, keep an eye on NSF’s soon-to-be-piloted STEM Teacher Corps. This 10-year program aims to recruit and retain high-quality STEM teachers. Having some similarities with Math for America, this program will give teachers stipends and provide professional development. Teachers in the program must be devoted to “teaching and coaching underserved students to increase STEM student achievement and STEM participation rates for students from rural and high-need schools.” Awards will be made to collaborators from educational institutions and nonprofit organizations. If you want to learn more about CHIPS and Science and its education-related provisions, visit: http://www.stemedcoalition.org/2022/08/08 /stem-education-in-the-chips-science-and -competitiveness-bill/.

How You Can Get Involved It is incredibly important that mathematical scientists talk to government agencies and legislators about the value and impact of mathematics education. In other words, I encourage you to share out your passion for mathematics education to elected representatives and appointees. We, at OGR, are here to help you do that. Visit our webpage at https://www.ams.org/government. We’d love for you to participate in our take-actions (minimal time required). We’d love to help you plan a visit with staff of your congressional representatives (more time required). Email us, say hello at meetings, and stop by if you’re ever in DC.

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Shape Optimization for Covering Problems

Ernesto G. Birgin and Antoine Laurain In his 1915 pioneering paper [Nev15], Neville describes a game played at fairs where a large disk is painted on a cloth and five smaller, identical circular disks of thin metal are available. An award is offered to the person who is able to completely cover the large disk with the small disks. Neville then proceeds to show that the problem can be modeled by a system of nonlinear equations, and discusses a numerical method for an approximation of the solution. He also provides a figure for the covering of a large disk with five small disks that looks identical to the solution obtained with our algorithm shown in Figure 1. Many other works followed that dealt with the problem of covering a disk with smaller disks of minimum radius or a convex body with smaller homothetic copies. In general, planar geometry techniques are used in these works Ernesto G. Birgin is a professor of computer science at the University of S˜ao Paulo, Brazil. His email address is [email protected]. Antoine Laurain is a professor of mathematics at the University of DuisburgEssen, Germany. His email address is [email protected]. Communicated by Notices Associate Editor Reza Malek-Madani. For permission to reprint this article, please contact: [email protected].

to, given a fixed and small number of disks, find the optimal radius or a bound for the optimal radius. See, for example, [Kro93] and the references therein.

Figure 1. Covering of a disk of unit diameter with five identical disks of minimum radius approximately 0.3023, obtained with the method introduced in [BLMS21].

Nowadays, this type of problem is called a covering problem. The covering of the whole 𝑛-dimensional space with minimally overlapping identical balls has often been investigated in parallel to the problem of packing nonoverlapping spheres with the highest possible density [CS99].

DOI: https://doi.org/10.1090/noti2799

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VOLUME 70, NUMBER 10

It is well-known that the optimal covering of the plane is achieved by the hexagonal lattice, which realizes the thinnest covering, i.e., the plane covering with the least possible density of disk intersection; see Figure 2.

Figure 2. In the left subfigure, the centers belong to a square lattice, while in the right subfigure they belong to a hexagonal lattice. The hexagonal lattice is visibly more efficient as it leads to less overlapping between the disks.

The covering of a bounded set with overlapping identical balls minimizing the number of balls with a fixed radius, or minimizing their radius with a fixed number of balls, as in Figure 3 in two dimensions, also represents a challenging question with a wide variety of practical applications, ranging from the configuration of a gamma ray machine radiotherapy equipment unit [LMZ09] to the placement of base stations [DDNS06]. Compared to the problem of covering the whole space, in which case the solution is a lattice, covering a bounded set naturally yields a less regular solution, as the covering depends on the shape of the covered set. Numerical investigations hence play an important role in studying such problems. The covering of specific shapes such as squares, rectangles, disks, triangles, and polygons with a fixed number 𝑚 of small disks has naturally been the focus of various papers on this topic; see for instance [HM97].

Shape optimization is the study of optimization problems where the variable is a geometric object, such as a subset of R𝑛 or a manifold; see [SZ92]. The shape sensitivity analysis is usually performed using strong regularity conditions on the geometry, in order to parameterize the perturbation of the geometry to compute derivatives. For instance, one often works with sets of class 𝐶 𝑘 with 𝑘 ≥ 2, i.e., sets whose boundary can be locally represented by a function of class 𝐶 𝑘 . Still, many relevant shape optimization problems depend on mildly nonsmooth shapes such as curvilinear polygons, which means that their boundary is a union of smooth curves and it can have vertices. The covering of a set 𝐴 may be naturally formulated as a nonsmooth shape optimization problem, since 𝐴 may be nonsmooth, and the union of balls covering 𝐴 can be seen, except for degenerate cases, as a curvilinear polygon, as shown in Figure 4. The shape optimization viewpoint on the covering problems opens up new perspectives as the tools of shape calculus become available, which allows us to numerically handle the case of a large number 𝑚 of balls. We describe now the approach that we have developed in [BLMS21] and [BLMS22]. We focus here on a description in two dimensions for the sake of simplicity, but we emphasize that the theoretical part of the shape optimization approach is relatively independent of the dimension. Let 𝐴 be an open 𝑚 bounded subset of R2 and Ω(𝒙, 𝑟) = ⋃𝑖=1 𝐵(𝑥𝑖 , 𝑟), where 2 𝒙 ≔ {𝑥𝑖 }𝑚 𝑖=1 and 𝐵(𝑥𝑖 , 𝑟) is an open disk with center 𝑥𝑖 ∈ R and radius 𝑟. We consider the problem of covering 𝐴 using a fixed number 𝑚 of closed disks 𝐵(𝑥𝑖 , 𝑟) with minimal radius 𝑟, where 𝐴 denotes the closure of 𝐴. In other words, we are looking for a vector (𝒙, 𝑟) ∈ R2𝑚+1 such that 𝐴 ⊂ Ω(𝒙, 𝑟) with minimal 𝑟. The optimization problem can be formulated as Minimize 𝑟 subject to 𝐺(𝒙, 𝑟) = 0,

(1)

𝐺(𝒙, 𝑟) ≔ Area(𝐴 ⧵ Ω(𝒙, 𝑟))

(2)

(𝒙,𝑟)∈R2𝑚+1

where

Figure 3. Region 𝐴, in green, to be covered by a union of disks Ω(𝒙, 𝑟) (left) and Ω(𝒙, 𝑟) ∩ 𝐴, in gray (right).

Shape optimization approach. Even though various numerical methods have been introduced to solve the covering problem, the natural approach of considering the shape of the union of balls as the optimization variable has been generally overlooked, except in a few specific cases, see for instance [HLE03]. In this framework, the tools of shape optimization and shape calculus are employed to investigate the sensitivity with respect to variations of the union of balls, as the balls’ centers or radii are perturbed. NOVEMBER 2023

and Area(𝐴 ⧵ Ω(𝒙, 𝑟)) denotes the two-dimensional measure of 𝐴⧵Ω(𝒙, 𝑟). It can be shown that 𝐺(𝒙, 𝑟) = 0 is equivalent to 𝐴 ⊂ Ω(𝒙, 𝑟), thus a solution to problem (1) yields a solution to the covering problem. Covering problems are usually formulated with closed sets, but for the shape optimization framework it is convenient to work with the open sets 𝐴 and Ω(𝒙, 𝑟). The function 𝐺 can be interpreted as the composition of a so-called shape functional 𝐴 ⧵ Ω ↦ Area(𝐴 ⧵ Ω) with a function (𝒙, 𝑟) ↦ 𝐴 ⧵ Ω(𝒙, 𝑟). Under some geometric conditions detailed in [BLMS21] and [BLMS22], the derivative of such a function can be computed using techniques of shape calculus and in particular via the concept of shape

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𝑇𝑡

Ω(𝒙 + 𝑡𝛿𝒙, 𝑟) = 𝑇𝑡 (Ω(𝒙, 𝑟))

Ω(𝒙, 𝑟)

𝑇𝑡−1 Figure 4. A small perturbation of the disks’ centers modifies the shape of the union Ω(𝒙, 𝑟). The corresponding shape perturbation Ω(𝒙 + 𝑡𝛿𝒙, 𝑟) of Ω(𝒙, 𝑟) can be parameterized with the help of a bi-Lipschitz transformation 𝑇𝑡 .

derivative [SZ92]. Using these techniques, we proved in [BLMS21] that the partial derivatives of 𝐺 with respect to the centers 𝑥𝑖 and the radius 𝑟 are given by 𝜕𝑥𝑖 𝐺(𝒙, 𝑟) = − ∫

𝜈(𝑧) 𝑑𝑧,

(3)

𝜕𝐵(𝑥𝑖 ,𝑟)∩𝜕Ω(𝒙,𝑟)∩𝐴

𝜕𝑟 𝐺(𝒙, 𝑟) = ∫

𝑑𝑧,

(4)

𝜕Ω(𝒙,𝑟)∩𝐴

where 𝜈 is the outward unit normal vector to Ω(𝒙, 𝑟). The partial derivative 𝜕𝑥𝑖 𝐺(𝒙, 𝑟) is a vector with two components since 𝜈 is vector-valued. Note that the formulas for the partial derivatives of 𝐺 are boundary integrals involving the unit normal vector 𝜈 to the boundary of Ω(𝒙, 𝑟). This illustrates a fundamental property in shape optimization known as the structure theorem, which essentially states that the first derivative with respect to the shape of a shape functional only depends on the boundary perturbations in the normal direction. This is a well-known property of the shape derivative which is known since the pioneering work of Hadamard [Had68]. In order to prove these results, the main task is to build a transformation 𝑇𝑡 between the union of disks Ω(𝒙, 𝑟) and its perturbation Ω(𝒙 + 𝑡𝛿𝒙, 𝑟), where 𝛿𝒙 is a perturbation of the disks’ centers. The transformation needs to be biLipschitz, which means that 𝑇𝑡 and its inverse are both Lipschitz functions, in order to perform a change of variables in the integral on Ω(𝒙+𝑡𝛿𝒙, 𝑟) that appears in 𝐺(𝒙+𝑡𝛿𝒙, 𝑟). The small perturbation Ω(𝒙 + 𝑡𝛿𝒙, 𝑟) and the corresponding mapping 𝑇𝑡 are illustrated in Figure 4. In the case of a perturbation of the radii, one considers the set Ω(𝒙, 𝑟+𝑡𝛿𝑟) and the procedure is similar. In [BLMS22] we have also computed the second-order 2 partial derivatives 𝜕𝑥2𝑖 ,𝑥𝑗 𝐺(𝒙, 𝑟), 𝜕𝑥2𝑖 ,𝑟 𝐺(𝒙, 𝑟), 𝜕𝑟,𝑟 𝐺(𝒙, 𝑟). Their expressions are more involved compared to (3),(4) and we refer to [BLMS22] for their detailed description, but their computation is based on similar shape calculus techniques as for the calculation of (3),(4). 1586

Numerical methods and illustrations. In [BLMS21], nonpolygonal sets 𝐴 were considered. We even supposed that the set 𝐴 was defined by an oracle that, given a point in the plane, answers whether the point is inside 𝐴 or not. This level of generality forced us to compute 𝐺 and its gradient approximately. (At that time we were not yet considering second derivatives.) For the calculation of 𝐺, which can be written as an area integral, we simply partitioned the plane into small squares and counted the area of the squares whose center was at the same time in 𝐴 and in the union of the disks; see [BLMS21, Alg. 4.1]. For the calculation of the partial derivatives of 𝐺 in (3),(4), we discretized 𝜕𝐵(𝑥𝑖 , 𝑟) ∩ 𝜕Ω(𝒙, 𝑟) ∩ 𝐴 for 𝑖 = 1, . . . , 𝑚 and 𝜕Ω(𝒙, 𝑟) ∩ 𝐴, which basically consists of considering a finite number of points on the boundary of each disk and of verifying, for each point, whether it lies within 𝐴 and is not in the interior of any disk. With the points satisfying these conditions, we approximated the line integrals using a quadrature rule; see [BLMS21, Alg. 4.2]. This approach allowed us to consider a large variety of sets 𝐴, but resulted in time-consuming and relatively imprecise calculations that enabled us to find approximate solutions for problems with up to 25 disks. Figure 5 illustrates some solutions; see [BLMS21] for details. The highlight of the pictures is that the sets 𝐴 are not polygons. It should also be noted that the discrete calculation of 𝐺 together with the minimization of the radius of the disks produces solutions in which boundary spikes may remain uncovered if the discretization is not sufficiently fine, as can be seen in Figure 5. This is due to the fact that spikes have a large perimeter and a small area, therefore their contribution to 𝐺 is much smaller than the contribution of the smoother regions of 𝐴, and a very fine discretization is necessary to obtain a visually satisfying covering. Numerically, a domain 𝐷 containing 𝐴 and Ω(𝒙, 𝑟) is partitioned into small squares, and the area of 𝐴 ∩ Ω(𝒙, 𝑟) is approximated by the sum of the areas of the squares whose centers are both in 𝐴 and in Ω(𝒙, 𝑟). If 𝐴 has a thin spike, as in Figure 5, the squares have to be very small for any of them to have their center inside the spike. A numerical study showing how the covering improves when the size of the small squares used in the discretization of 𝐺 decreases is presented in [BLMS21]. In [BLMS22], using shape calculus techniques, we obtained the formulas of the second-order partial derivatives of 𝐺. Moreover, by limiting the numerical experiments to sets 𝐴 that are the union of disjoint convex polygons, we were able to calculate 𝐺, its gradient, and its Hessian exactly, disregarding the machine precision. The main tool for the calculation of 𝐺 and its derivatives was the calcu𝑚 lation of Voronoi diagrams 𝒱(𝒙) ≔ {𝑉𝑖 (𝒙)}𝑖=1 , with the

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VOLUME 70, NUMBER 10

Figure 5. Heart covered by 15 radius-minimizing identical disks with 𝑟 ≈ 0.3476 (left) and curved star covered by 9 radius-minimizing identical disks with 𝑟 ≈ 0.1300 (right).

Voronoi cells 𝑉𝑖 (𝒙) ≔ {𝑥 ∈ R2 ∶ ‖𝑥 − 𝑥𝑖 ‖2 ≤ min ‖𝑥 − 𝑥𝑗 ‖2 } 1≤𝑗≤𝑚

for 𝑖 = 1, . . . , 𝑚, where the generators 𝑥𝑖 are the disks’ centers. Voronoi cells 𝑉𝑖 are disjoint (possibly unbounded) polyhedra which, intersected with the convex polygons composing 𝐴, become disjoint convex polygons. In turn, the intersection of each convex polygon with its corresponding disk results in a convex plane figure, whose sides are determined by segments and arcs of a circle, as in Figure 6. The area of each of these plane figures is easy to calculate with basic concepts of plane geometry. This partition of Ω(𝒙, 𝑟) ∩ 𝐴 as well as access to all vertices, edges, and arcs that compose the partition is all we need to calculate exactly not only 𝐺 but also its first- and second-order derivatives; see [BLMS22, Algs. 4.1–4.3]. These tools, by enabling the exact evaluation of 𝐺 and its first and second derivatives, have allowed us to calculate very precise solutions to problems with up to 100 disks in a fraction of the time required by the method introduced in [BLMS21], which uses costly approximations of 𝐺 and its gradient, and does not use the second derivative of 𝐺. Figure 7 illustrates some solutions; see [BLMS22] for details. The highlight of the pictures is that the corners of the regions are “well covered” by the disks. The description of the tools needed for the exact computation of 𝐺 and its derivatives makes more or less clear that the choice to consider sets 𝐴 given by the union of disjoint convex polygons was an arbitrary choice that simplifies the implementation of the calculations. Other types of sets 𝐴 could be considered, as long as one is willing to implement the required tools of plane geometry. So far we have not mentioned how the optimization problems were solved. Problem (1) is a nonlinear programming problem with a single hard-to-compute constraint. The familiar tool for solving problems of this type is the augmented Lagrangian method; see [BM14]. In particular, we used Algencan [ABMS07] which is the implementation of an augmented Lagrangian method with safeguards. Very roughly speaking, an augmented Lagrangian NOVEMBER 2023

Figure 6. Voronoi diagram generated using the centers of the disks as generating points that allows, by partitioning Ω(𝒙, 𝑟) ∩ 𝐴, to calculate its area.

Figure 7. Minkowski island fractal (left) and eight-pointed star (right) covered by 100 minimizing radius identical disks with 𝑟 ≈ 0.2753 and 𝑟 ≈ 0.2791, respectively.

method solves a sequence of subproblems in which the violation of shifted constraints is penalized. In the specific case of the augmented Lagrangian method implemented by Algencan, bound constraints are not penalized and remain in the subproblems. However, since problem (1) has no bound constraints, the subproblems that Algencan solves are unconstrained. In Algencan, leaving aside other issues such as availability of a linear system solver and subproblem size, when the subproblems are unconstrained and second derivatives of the functions defining the problem are available, the subproblems are solved with a globally convergent line search Newton’s method. For this reason we can say that the work developed in [BLMS22] is an application of a shape-Newton method in a genuinely nonsmooth setting, a notable fact since Newton’s method is rarely used in shape optimization, even in smooth settings. It should also be emphasized that we are actually seeking a global solution of problem (1). There are no practical deterministic global optimization methods that are

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capable of dealing with problem (1). Thus, the stochastic options remain and, among them, the most natural is to use a multistart approach. This is precisely what we did in [BLMS21] and [BLMS22], using randomly generated starting points. In [BGL], based on hexagonal lattices, we developed a way to generate better than random starting points. With that, we managed to improve all the solutions reported in [BLMS22] using fewer initial points and, consequently, with lower computational cost. Asymptotic analysis of the optimal radius. Following the numerical approximation of solutions of covering problems of bounded set, a theoretical question that naturally arises is the asymptotic behavior and bounds on the optimal radius, solution of problem (1), as the number of disks grows to infinity. The possibility of running numerical experiments with large 𝑚 allows us to verify the sharpness of asymptotic estimates for the optimal radius and to formulate new hypotheses for the bounds. This asymptotic analysis creates a bridge between the problems of covering the whole plane and covering a bounded set, as the optimal arrangement for a given 𝑚 converges in some appropriate sense toward the solution for the covering of the whole plane, which is the hexagonal lattice ℒ𝑟 ≔ {𝑘𝑣 𝑟 + ℓ𝑤 𝑟 | (𝑘, ℓ) ∈ ℤ2 } 𝑟

𝑟

with 𝑣 𝑟 ≔ (3, √3), 𝑤 𝑟 ≔ (3, −√3). This convergence 2 2 toward a lattice can be observed numerically using a large number of discs, as in the results in Figure 7. One observes that the disks located farthest from the boundary tend to align in a hexagonal lattice pattern, whereas the disks intersecting the boundary display a more intricate and less predictable behavior. Kershner [Ker39] pioneered the topic in 1939, providing an asymptotic result on the smallest number of disks of fixed radius that are necessary to cover an arbitrary region of the plane, a result that was improved ten years later by Verblunksy [Ver49]. We have investigated a similar question recently in [BGL] for the covering of a general class of sets. Using honeycombs, defined as unions of 𝑚 regular hexagons whose centers belong to a regular hexagonal lattice, we have shown that the solution 𝑟∗ (𝑚) to the minimization problem (1) satisfies ∗

𝑟 (𝑚) = [

2 Area(𝐴) 3√3𝑚

1/2

]

+ 𝑅(𝑚)

(5)

where 𝑅(𝑚) ≤ 𝑅(𝑚) ≤ 𝑅(𝑚),

(6)

with the following asymptotic expansions, as 𝑚 goes to 1588

infinity, 𝑅(𝑚) = − 𝑅(𝑚) =

2 Per(𝜕𝐴)

3√3𝑚 2 Per(𝜕𝐴) 3√3𝑚

+𝑂(

+𝑂(

1 ), 𝑚3/2

1 ), 𝑚3/2

where Per(𝜕𝐴) denotes the perimeter of the boundary of 𝐴. We have then computed numerical solutions of covering problems with large 𝑚 in order to verify the sharpness of these asymptotic expansions. Numerically, we have observed that the asymptotic order 𝑚−1 of 𝑅(𝑚) is sharp, but that the constant 2 Per(𝜕𝐴)/(3√3) could probably be improved by a factor roughly equal to 0.2. We also observed in all our numerical experiments that the lower bound 𝑅(𝑚) is always positive, which suggests that our asymptotic estimate for the lower bound is not sharp. To improve the lower bound estimate, a finer study of the behavior of the disks in the neighborhood of the boundary of the optimal covering set Ω(𝒙∗ (𝑚), 𝑟∗ (𝑚)) will have to be performed, as this behavior is the primary driver of the asymptotic expansions of the bounds 𝑅(𝑚) and 𝑅(𝑚). Minimizing eigenvalues with respect to a union of disks. So far we have discussed several features of the covering of a bounded set with a union of 𝑚 disks. This problem does not involve the solution of a partial differential equation. Many shape optimization problems actually involve the solution of a partial differential equation, which usually models a physical phenomenon. For instance, the optimization of eigenvalues of differential operators with respect to geometrical features is a topic of high interest in pure and applied mathematics but also in engineering and natural sciences, such as in structural mechanics for the control of vibration frequency, in mathematical biology, acoustics or electromagnetism. In particular, the optimization of Laplacian eigenvalues is a popular topic in mathematics as these problems are often simple and elegant to formulate, but are also challenging and require deep mathematical tools from a large spectrum of disciplines such as partial differential equations, spectral theory, and differential geometry. The celebrated Rayleigh–Faber–Krahn inequality, conjectured by Lord Rayleigh in the 19th century and proved several decades later by Faber and Krahn, states that the ball minimizes the first Dirichlet eigenvalue under a volume constraint. Since then, many shape optimization problems of this nature have been considered, such as the minimization of the 𝑘th eigenvalue of the Dirichlet Laplacian for 𝑘 > 2, or the minimization of eigenvalues with other types of partial differential equations and boundary conditions; see [Hen06].

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Let Ω ⊂ R2 be a bounded open set and introduce the function spaces 𝐻 1 (Ω) ≔ {𝑣 ∈ 𝐿2 (Ω) | ∇𝑣 ∈ 𝐿2 (Ω)}, 𝐻01 (Ω) ≔ {𝑣 ∈ 𝐻 1 (Ω) | 𝑣 = 0 on 𝜕Ω}, where 𝐿2 (Ω) denotes the space of functions whose square is integrable on Ω. The first Dirichlet Laplacian eigenvalue is defined as ∫ |∇𝑢| 𝜆(Ω) ≔

min

ᵆ∈𝐻01 (Ω)⧵{0}

Ω

𝜕𝑖 𝜆(𝒙) = − ∫ |∇𝑢|2 𝜈,

2

(10)

𝒜𝑖

.

∫ 𝑢2 Ω

The corresponding eigenfunction 𝑢 satisfies −Δ𝑢 = 𝜆(Ω)𝑢 in Ω,

(7)

𝑢 = 0 on 𝜕Ω,

(8)

and we impose the normalization condition ‖𝑢‖𝐿2 (Ω) = 1. The name “Dirichlet” refers to the boundary condition (8). We consider the following eigenvalue minimization problem: Minimize 𝜆(𝒙) 2𝑚 with respect to 𝒙 ≔ {𝑥𝑖 }𝑚 , 𝑖=1 ∈ R

(9)

where 𝜆(𝒙) ≔ 𝜆(Ω(𝒙)) denotes the solution to (7),(8) with 2 Ω = Ω(𝒙) ≔ ∪𝑚 𝑖=1 𝐵(𝑥𝑖 , 𝑟), with 𝑚 and 𝑟 fixed, 𝑥𝑖 ∈ R for all 𝑖 ∈ {1, … , 𝑚}. The minimizers of problem (9) produce an interesting geometrical configuration of the centers {𝑥𝑖 }𝑚 𝑖=1 . First, ⋆ 𝑚 ⋆ when 𝒙 ≔ {𝑥𝑖 }𝑖=1 is a solution of problem (9), Ω(𝒙⋆ ) must be connected due to a monotonicity property of Dirichlet eigenvalues. Second, Ω(𝒙⋆ ) achieves an equilibrium between two competing tendencies. On the one hand, Ω(𝒙⋆ ) strives to maximize its area, as the first Dirichlet eigenvalue tends to decrease as the area of the domain increases. On the other hand, Ω(𝒙⋆ ) seeks to minimize the angles (measured from the interior of Ω(𝒙⋆ )) at circle intersections, and to prevent small gaps from appearing, as these create strong singularities in the eigenfunction 𝑢 in the neighborhood of the circle intersections, which increase the eigenvalue. Here, “singularities” roughly means that ∇𝑢 is unbounded in these neighborhoods. Furthermore, considering that the minimizer of 𝜆 with respect to a free-form shape under a volume constraint is a disk due to the Rayleigh–Faber–Krahn inequality, one expects the small disks 𝐵(𝑥𝑖⋆ , 𝑟) to agglomerate and approximate a large disk as 𝑚 → ∞, and to minimize their overlapping while avoiding gaps in Ω(𝒙⋆ ), i.e., Ω(𝒙⋆ ) should be simply connected. Note that the minimizing of overlapping is also characteristic of solutions of covering problems [BLMS21, BLMS22, CS99], as discussed in the previous sections. NOVEMBER 2023

In [BFHL23] we have developed an algorithm to find approximate solutions of problem (9). The approach is similar to the method described in the previous sections for the covering problem. Here we can also compute the derivative of the eigenvalue 𝜆(𝒙) with respect to 𝒙 using techniques of shape calculus. We have shown that the partial derivative of the eigenvalue is given by

where 𝒜𝑖 ≔ 𝜕𝐵(𝑥𝑖 , 𝑟) ∩ 𝜕Ω(𝒙) for 𝑖 ∈ {1, … , 𝑚} and 𝜈 is the outward unit normal vector to Ω(𝒙, 𝑟). Notice the similarity between (10) and formula (3) which was obtained in the case of the covering problem. This similarity is due to the structure theorem of shape optimization that we have discussed above. There is nevertheless an important difference between (10) and (3), as 𝜕𝑖 𝜆(𝒙) depends on the gradient of the eigenfunction ∇𝑢. Indeed, ∇𝑢 is unbounded in the neighborhood of the circle intersections due to the nonsmoothness and the concavity of the shape at these points, and this makes the accurate numerical approximation of 𝜕𝑖 𝜆(𝒙) challenging. Figure 8 shows the results obtained by our algorithm for 𝑟 = 1 and 𝑚 ≤ 10. We can clearly see how the shape of Ω(𝒙⋆ ) converges to a disk as the number 𝑚 of disks grows. We also observe symmetries and regular patterns formed by the disks’ centers at the solutions. However, the solutions sometimes have less regularity and symmetries than expected. For instance, in the case 𝑚 = 4, the disks’ centers have two symmetries, but they are the vertices of a rhombus and not of a square. In the cases 𝑚 = 6, 7, 8, the results suggest that the exterior layer of disks should form a regular pentagon, hexagon, and heptagon, respectively. The case 𝑚 = 5 is interesting as it only possesses one axis of symmetry, and the pentagon formed by the disks’ centers is not regular. The structure of the solution is also remarkably similar to the solution of the covering problem with five disks shown in Figure 1. Starting from 𝑚 ≥ 9, regular patterns are more difficult to observe as the shape of Ω(𝒙⋆ ) becomes more complex due to the appearance of two disks in the inner layer. As in the covering problem, the asymptotic behavior of Ω(𝒙∗ (𝑚)) is an interesting theoretical question. As discussed in the previous sections, for the covering problem, the disks’ centers converge in some sense toward the hexagonal lattice. For the eigenvalue minimization problem, numerical results also suggest that the optimal placement of the disks’ centers probably converges, in some appropriate sense, toward a subset of the hexagonal lattice. This conjecture is also supported by a simple argument: asymptotically, the hexagonal lattice configuration is a trade-off between maximizing the total area of the union of balls,

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and keeping Ω(𝒙∗ (𝑚)) simply connected, i.e., avoiding any gaps in Ω(𝒙∗ (𝑚)).

this notice represents a first step in this direction, and other nonsmooth shape optimization problems involving partial differential equations will be investigated in the near future. ACKNOWLEDGMENT. This work has been partially supported by FAPESP (grants 2013/07375-0, 2018/24293-0, and 2022/05803-3) and CNPq (grants 302073/2022-1, 303243/2021-0, 304258/2018-0, and 408175/2018-4). Antoine Laurain also acknowledges the support, since May 2023, of the Collaborating Researcher Program of the Institute of Mathematics and Statistics at the University of S˜ao Paulo. References

Figure 8. Numerical approximations of minimizers of the first Dirichlet eigenvalue for 𝑚 = 2, 3, 4, 5, 6, 7, 8, 9, 10.

Conclusions and future research. Shape calculus and optimization are a powerful set of techniques for the sensitivity analysis of functions depending on the geometry. There exists an extensive literature in the smooth setting, but shape calculus still requires an active development in the nonsmooth case. Nonsmooth shape optimization has a variety of relevant applications such as the modeling of evolving nonsmooth sets and the optimization of complex geometries such as the union and intersection of moving components, curvilinear polygons, tessellations, generalized Voronoi diagrams and minimization diagrams [BLM23]. Applied to covering problems, it provided a new perspective on the problem and allowed us to design efficient numerical methods. Here we have presented results with a union of balls of identical radius, but the shape optimization approach is versatile and union/intersection of sets of various shapes can be treated in a similar way, also in dimension greater than two. Of particular interest is nonsmooth shape optimization involving partial differential equations, as irregular geometries appear naturally in applications. Both theoretical and numerical challenges arise in this context, one of the main issues being the singularities that appear in the corners of the domain, which need to be carefully studied and handled numerically. The eigenvalue problem presented in

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[ABMS07] R. Andreani, E. G. Birgin, J. M. Mart´ınez, and M. L. Schuverdt, On augmented Lagrangian methods with general lower-level constraints, SIAM J. Optim. 18 (2007), no. 4, 1286–1309, DOI 10.1137/060654797. MR2373302 [BFHL23] E. G. Birgin, L. Fernandez, G. Haeser, and A. Laurain, Optimization of the first Dirichlet Laplacian eigenvalue with respect to a union of balls, J. Geom. Anal. 33 (2023), no. 6, Paper No. 184, 28, DOI 10.1007/s12220-023-01241w. MR4572197 [BGL] E. G. Birgin, J. L. Gardenghi, and A. Laurain, Asymptotic bounds on the optimal radius when covering a set with minimum radius identical balls, Mathematics of Operations Research, to appear. [BLM23] E. Birgin, A. Laurain, and T. Menezes, Sensitivity analysis and tailored design of minimization diagrams, Math. Comp. 92 (2023), no. 344, 2715–2768, DOI 10.1090/mcom/3839. MR4628764 [BLMS21] E. G. Birgin, A. Laurain, R. Massambone, and A. G. Santana, A shape optimization approach to the problem of covering a two-dimensional region with minimum-radius identical balls, SIAM J. Sci. Comput. 43 (2021), no. 3, A2047–A2078, DOI 10.1137/20M135950X. MR4267494 [BLMS22] E. G. Birgin, A. Laurain, R. Massambone, and A. G. Santana, A shape-Newton approach to the problem of covering with identical balls, SIAM J. Sci. Comput. 44 (2022), no. 2, A798–A824. MR4404468 [BM14] E. G. Birgin and J. M. Mart´ınez, Practical augmented Lagrangian methods for constrained optimization, Fundamentals of Algorithms, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014, DOI 10.1137/1.9781611973365. MR3186234 [CS99] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov, DOI 10.1007/978-1-4757-65687. MR1662447

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[DDNS06] G. K. Das, S. Das, S. C. Nandy, and B. P. Sinha, Efficient algorithm for placing a given number of base stations to cover a convex region, Journal of Parallel and Distributed Computing 66 (2006), no. 11, 1353–1358. [Had68] J. Hadamard, M´emoire sur le probleme d’analyse relatif a l’´equilibre des plaques e´ lastiques, M´emoire des savants e´ trangers, 33, 1907, Œuvres de Jacques Hadamard, Editions du C.N.R.S., Paris, 1968, pp. 515–641. [Hen06] Antoine Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. MR2251558 [HLE03] C. Ho-Lun and H. Edelsbrunner, Area and perimeter derivatives of a union of disks, Computer Science in Perspective: Essays Dedicated to Thomas Ottmann (Rolf Klein, Hans-Werner Six, and Lutz Wegner, eds.), Springer Berlin Heidelberg, Berlin, Heidelberg, 2003, pp. 88–97. [HM97] Alad´ar Heppes and Hans Melissen, Covering a rectangle with equal circles, Period. Math. Hungar. 34 (1997), no. 1-2, 65–81, DOI 10.1023/A:1004224507766. 3rd Geometry Festival: an International Conference on Packings, Coverings and Tilings (Budapest, 1996). MR1608319 [Ker39] Richard Kershner, The number of circles covering a set, Amer. J. Math. 61 (1939), 665–671, DOI 10.2307/2371320. MR0000043 [Kro93] S. Krotoszynski, ´ Covering a disk with smaller disks, Studia Sci. Math. Hungar. 28 (1993), no. 3-4, 277–283. MR1266811 [LMZ09] Leo Liberti, Nelson Maculan, and Yue Zhang, Optimal configuration of gamma ray machine radiosurgery units: the sphere covering subproblem, Optim. Lett. 3 (2009), no. 1, 109–121, DOI 10.1007/s11590-008-0095-4. MR2453509 [Nev15] E. H. Neville, On the solution of numerical functional equations: Illustrated by an account of a popular puzzle and of its solution, Proceedings of the London Mathematical Society s2-14 (1915), no. 1, 308–326. [SZ92] Jan Sokołowski and Jean-Paul Zol´esio, Introduction to shape optimization: Shape sensitivity analysis, Springer Series in Computational Mathematics, vol. 16, SpringerVerlag, Berlin, 1992, DOI 10.1007/978-3-642-58106-9. MR1215733 [Ver49] S. Verblunsky, On the least number of unit circles which can cover a square, J. London Math. Soc. 24 (1949), 164– 170, DOI 10.1112/jlms/s1-24.3.164. MR33552

NOVEMBER 2023

Ernesto G. Birgin

Antoine Laurain

Credits

All figures, including the opening image, are courtesy of the authors. Photo of Ernesto G. Birgin is courtesy of Bruno Datan. Photo of Antoine Laurain is courtesy of Rosana Miliorini Souza Laurain.

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Quantum Ergodicity in Theorems and Pictures

Semyon Dyatlov A popular culture notion of chaos was summed up by Edward Lorenz: it occurs “when the present determines the future, but the approximate present does not approximately determine the future” (or more dramatically “a butterfly flapping its wings in Brazil could set off a tornado in Texas”). In quantum mechanics there is no clear definition of quantum chaos but its manifestations include properties of eigenvalues and eigenfunctions. Here eigenfunctions are interpreted as pure quantum states, yielding the simplest, time-harmonic, solutions to the Schrödinger equation. It is natural then to look for distinguishing properties between quantum systems with underlying completely integrable (that is, organized and nonchaotic) and chaotic classical dynamics. At high energies or small wavelengths, Semyon Dyatlov is an associate professor of mathematics at MIT. His email address is [email protected]. In memory of Steve Zelditch. Communicated by Notices Associate Editor Scott Sheffield. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2801

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such classical effects would manifest themselves most clearly. We should stress though that the validity of such asymptotics almost always becomes accurate right away. One classical notion, present in many chaotic systems, is that of ergodicity. A classical system is ergodic if almost all classical trajectories equidistribute—see Definition 1. This article focuses on the corresponding topic in quantum chaos: macroscopic behavior of high energy eigenfunctions for systems with ergodic or more strongly chaotic classical dynamics. We cannot do justice here to the extensive literature on quantum ergodicity but we refer to the reviews by Sarnak [Sar11] and Zelditch [Zel19], as well as the author’s ICM proceedings [Dya21], for more references, and for yet another perspective to the article of Rudnick [Rud08]. To see animated versions of the figures illustrating both classical and quantum phenomena, the reader is encouraged to visit https://math.mit.edu/~dyatlov/chaos -movies.html.

1. Eigenfunctions on Planar Domains Eigenfunctions and eigenvalues of the Laplacian on bounded planar domains, with either Dirichlet or

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Figure 1. Numerically computed high energy Dirichlet eigenfunctions for two domains: a disk and a stadium. Here darker shading corresponds to larger values of |𝑢𝑗 (𝑥)|. The eigenfunctions for the stadium here and in Figure 4 are computed using the method developed by Barnett, see [BH14].

Neumann boundary conditions, are familiar across mathematics and science. Their investigation goes back to the experiments by Chladni over two hundred years ago and includes such popular questions as “Can one hear the shape of a drum?” formulated by Kac over sixty years ago. In the Dirichlet case, these eigenfunctions are solutions to the eigenvalue problem −Δ𝑢𝑗 (𝑥) = 𝜆𝑗2 𝑢𝑗 (𝑥), 𝑥 ∈ Ω,

𝑢𝑗 |𝜕Ω = 0.

(1)

Here Ω ⊂ ℝ2 is a bounded open set with smooth enough boundary 𝜕Ω, Δ = 𝜕𝑥21 + 𝜕𝑥22 is Laplace’s operator, and we choose, as we may, 𝑢𝑗 ’s to form an orthonormal basis of the space of square integrable functions, 𝐿2 (Ω). Moreover, 𝜆𝑗 ↑ ∞. The eigenfunction 𝑢𝑗 can be thought of as a pure state of a quantum particle confined to the domain Ω, with energy 𝜆𝑗2 . Since ‖𝑢𝑗 ‖𝐿2 (Ω) = 1, the expression |𝑢𝑗 (𝑥)|2 𝑑𝑥 defines a probability measure on Ω. Following a standard interpretation of quantum mechanics, this measure gives the probability distribution of the position of the particle. We will be particularly interested in the quantities ∫ 𝑏(𝑥)|𝑢𝑗 (𝑥)|2 𝑑𝑥,

𝑏 ∈ 𝐶(Ω),

(2)

Ω

which give the expected value of 𝑏(𝑥) where 𝑥 is the position of the particle. (If 𝑏 = 𝟏𝑆 is the indicator function of a set 𝑆 ⊂ Ω, then (2) is the probability of finding the particle in 𝑆. However for taking the 𝑗 → ∞ limit it is better to restrict to continuous 𝑏.) Figure 1 gives an example of Dirichlet eigenfunctions in two domains: a disk and a stadium. We observe that: NOVEMBER 2023

• The eigenfunction for the disk has a lot of geometric structure. Moreover, it is small near the center of the disk. • By contrast, the eigenfunctions for the stadium spread out evenly on the entire domain. The two eigenfunctions are different when looking closely at the pictures but they appear similar from far away. We also see that both pictures show a lot of oscillation. In fact, 𝑢𝑗 oscillates on the scale ℎ𝑗 = 1/𝜆𝑗 ,

(3)

so 𝜆𝑗 can be interpreted as the frequency of oscillation (which is why we denoted the eigenvalue by 𝜆𝑗2 and not 𝜆𝑗 ). To illustrate this, consider the case when Ω = (0, 𝜋)2 is a square, with eigenfunctions 𝑢𝑘ℓ = sin(𝑘𝑥1 ) sin(ℓ𝑥2 ) where 𝑘, ℓ ∈ ℕ. Then 𝑢𝑘ℓ oscillates at frequency 𝜆𝑘ℓ = √𝑘2 + ℓ2 . What makes eigenfunctions look so different for the disk and for the stadium? The answer lies in the behavior of the corresponding classical dynamical system. For domains with boundary, this system is the billiard ball flow, modeling a classical particle in Ω which moves in a straight line until collision with the boundary and then follows the standard law of reflection. Figure 2 shows a single longtime billiard ball trajectory in the disk and two such trajectories in the stadium. In the disk, the trajectory follows a regular pattern (perhaps reminding one of a ball of twine) and leaves out a region near the center. In the stadium, the trajectories appear chaotic, in particular covering the whole domain. In fact, they equidistribute: the amount of time the trajectory

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Figure 2. Typical billiard ball trajectories in a disk and a stadium after many bounces.

spends in a set 𝑆 tends to the ratio of the area of 𝑆 to the area of the domain, asymptotically as the length of the trajectory tends to infinity. From now on we focus on the chaotic case. The goal of this section is to formulate precisely a result known as quantum ergodicity, which informally states that If most billiard trajectories equidistribute, then most eigenfunctions equidistribute. We first explain what it means for most billiard trajectories to equidistribute, which is naturally given by the concept of ergodicity. Denote the billiard ball flow by (see Figure 3) 𝑡

1

1

𝜑 ∶Ω×𝕊 →Ω×𝕊 ,

𝑡 ∈ ℝ.

(4)

1

Here Ω × 𝕊 consists of all possible positions and (unit) velocity vectors and 𝜑𝑡 (𝑥0 , 𝜉0 ) gives the position and the velocity after time 𝑡 of the billiard ball particle starting at position 𝑥0 and velocity 𝜉0 . The billiard ball flow might be undefined for some (𝑥0 , 𝜉0 ) and 𝑡 because of various problems that can happen at the boundary, but under reasonable assumptions these form a measure 0 set and thus will not matter for the definition below—see [ZZ96]. We use the natural 𝜑𝑡 -invariant volume measure on Ω × 𝕊1 𝜇𝐿 = 𝑐 𝑑𝑥𝑑𝑆(𝜉) with the constant 𝑐 > 0 chosen so that 𝜇𝐿 is a probability measure. Definition 1. We say that the billiard ball flow 𝜑𝑡 is ergodic (with respect to 𝜇𝐿 ) if for 𝜇𝐿 -almost every (𝑥0 , 𝜉0 ), the trajectory 𝜑𝑡 (𝑥0 , 𝜉0 ) equidistributes, namely for any 𝑎 ∈ 𝐶(Ω × 𝕊1 ) we have as 𝑇 → ∞ 𝑇

1 ∫ 𝑎(𝜑𝑡 (𝑥0 , 𝜉0 )) 𝑑𝑡 → ∫ 𝑎(𝑥, 𝜉) 𝑑𝜇𝐿 . 𝑇 0 Ω×𝕊1 1594

Note that we require equidistribution in both position (𝑥) and velocity (𝜉) variables. Coming back to Figure 2, we remark that the billiard ball flow is not ergodic for the disk (in fact, it has a conserved quantity: the angle at which the trajectory intersects the boundary circle stays the same with each bounce), but it is ergodic for the stadium, as proved by Bunimovich in 1974. Next, we give a definition of equidistribution for eigenfunctions, taking the limits of expressions (2): Definition 2. Assume that 𝑢𝑗𝑘 , 𝑗 𝑘 → ∞, is a sequence of eigenfunctions from (1). We say that 𝑢𝑗𝑘 equidistributes in position if for each 𝑏 ∈ 𝐶(Ω) ∫ 𝑏(𝑥)|𝑢𝑗𝑘 (𝑥)|2 𝑑𝑥 → Ω

1 ∫ 𝑏(𝑥) 𝑑𝑥. vol(Ω) Ω

The above definition talks about the macroscopic behavior of 𝑢𝑗𝑘 since we first fix the classical observable 𝑏 and then take the limit 𝑘 → ∞. A quantum mechanical interpretation of equidistribution of eigenfunctions is as follows: in the high energy limit, the probability of observing the pure state quantum particle in a “nice” set 𝑆 ⊂ Ω becomes proportional to the area of 𝑆. We are now ready to state a version of quantum ergodicity. In the present setting it is due to Zelditch– Zworski [ZZ96], with an earlier contribution by G´erard– Leichtnam which covered the example of the stadium. In the setting of manifolds without boundary, the result goes back to the seminal works of Shnirelman, Zelditch, and Colin de Verdière in the 1970s–1980s. Theorem 1. Assume that the billiard ball flow 𝜑𝑡 is ergodic. Then there exists a density 1 increasing sequence 𝑗 𝑘 → ∞ such

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As remarked in (3) above, the eigenfunction 𝑢𝑗 oscillates on scale ℎ𝑗 = 𝜆𝑗−1 . Thus we expect 𝐷𝑥ℓ 𝑢𝑗 to be roughly of size 𝜆𝑗 . It then makes sense to multiply 𝐷 by ℎ𝑗 , which gives the semiclassical quantization procedure

x(t)

ξ(t) x0

ξ0

∂Ω

Figure 3. The billiard ball flow trajectory (𝑥(𝑡), 𝜉(𝑡)) = 𝜑𝑡 (𝑥0 , 𝜉0 ). Here 𝑡 is the distance traveled by the billiard ball. The study of such billiard ball flows is an old and subtle subject—see for instance Avila–De Simoi–Kaloshin [ADSK16] for recent progress.

that the corresponding sequence of eigenfunctions 𝑢𝑗𝑘 equidistributes in position. Here “density 1” means that #{𝑘 ∣ 𝑗 𝑘 ≤ 𝑁} →1 𝑁

as 𝑁 → ∞.

2. Semiclassical Measures We now discuss semiclassical quantization and classical/quantum correspondence, which underlie the proof of Theorem 1 and other results given below. This leads us to semiclassical measures, which are a way to capture the concentration of high-energy eigenfunctions simultaneously in position and frequency, and to a more refined version of quantum ergodicity. A quantization maps smooth functions 𝑎(𝑥, 𝜉) on ℝ2𝑛 , interpreted as classical observables, to operators 𝑎(𝑥, 𝐷𝑥 ) on 𝐶 ∞ (ℝ𝑛 ), interpreted as the corresponding quantum observables. Here the coordinate functions 𝑥ℓ should be mapped to the multiplication operators 𝑢 ↦ 𝑥ℓ 𝑢, while 𝜉ℓ should be mapped to the differentiation operators 𝐷𝑥ℓ = −𝑖𝜕𝑥ℓ . One can define a quantization procedure using the Fourier transform: ̂ 𝑑𝜉. 𝑎(𝑥, 𝐷𝑥 )𝑢(𝑥) = (2𝜋)−𝑛 ∫ 𝑒𝑖𝑥⋅𝜉 𝑎(𝑥, 𝜉)𝑢(𝜉) ℝ𝑛

By the Fourier inversion formula, if 𝑎 is a function of 𝑥 only, then 𝑎(𝑥, 𝐷𝑥 )𝑢 = 𝑎𝑢 is the corresponding multiplication operator; in particular, 1(𝑥, 𝐷𝑥 ) is the identity. More generally, if 𝑎 is a polynomial in 𝜉, then 𝑎(𝑥, 𝐷𝑥 ) is a differential operator. Since differential operators do not in general commute with each other, the map 𝑎 ↦ 𝑎(𝑥, 𝐷𝑥 ) cannot be an algebra homomorphism; however, 𝑎(𝑥, 𝐷𝑥 )𝑏(𝑥, 𝐷𝑥 ) − (𝑎𝑏)(𝑥, 𝐷𝑥 ) consists of lower-order terms. This is related to the product rule (6) below. In the theory of PDE, operators of the form 𝑎(𝑥, 𝐷𝑥 ) are called pseudodifferential operators. In mathematics they were originally motivated by singular integral operators, boundary value problems, and several complex variables. Eventually, that mathematical theory merged with the the theories of quantization from quantum mechanics— see [Zwo12] for general properties of quantization and for pointers to the vast literature on the subject. NOVEMBER 2023

Opℎ (𝑎) ∶= 𝑎(𝑥, ℎ𝐷𝑥 ).

(5)

Semiclassical quantization has several algebraic properties, such as the product rule Opℎ (𝑎) Opℎ (𝑏) = Opℎ (𝑎𝑏) + 𝒪(ℎ)

(6)

and the commutator rule [Opℎ (𝑎), Opℎ (𝑏)] = −𝑖ℎ Opℎ ({𝑎, 𝑏}) + 𝒪(ℎ2 ).

(7)

𝑛 ∑ℓ=1 (𝜕𝜉ℓ 𝑎)(𝜕𝑥ℓ 𝑏)

Here {𝑎, 𝑏} = − (𝜕𝑥ℓ 𝑎)(𝜕𝜉ℓ 𝑏) is the Poisson bracket of 𝑎 and 𝑏, and the remainders are understood in the sense of operator norm on appropriate spaces. Another key property, connecting classical and quantum dynamics, is Egorov’s theorem: 𝑈(−𝑡) Opℎ (𝑎)𝑈(𝑡) = Opℎ (𝑎 ∘ 𝜑𝑡 ) + 𝒪(ℎ)

(8)

where 𝑎 is a smooth compactly supported function on Ω × ℝ2 , 𝑈(𝑡) = 𝑒𝑖𝑡ℎ∆/2 is the Schrödinger group associated to the Dirichlet Laplacian on the domain Ω, and 𝜑𝑡 is the billiard ball flow (4) extended appropriately to (𝑥, 𝜉) ∈ Ω × ℝ2 . Note that 𝑈(𝑡) describes evolution of quantum wave functions by the Schrödinger equation and 𝜑𝑡 describes evolution of classical particles in Ω. (Some care is needed at the boundary of Ω but we omit the details here.) We now introduce semiclassical measures corresponding to eigenfunctions: Definition 3. Assume that 𝑢𝑗𝑘 is a sequence of eigenfunctions. We say that 𝑢𝑗𝑘 converges semiclassically to a Borel measure 𝜇 on Ω × ℝ2 if for each (sufficiently regular) function 𝑎(𝑥, 𝜉) on Ω × ℝ2 we have (putting ℎ𝑗𝑘 ∶= 𝜆𝑗−1 ) 𝑘 ⟨Opℎ𝑗 (𝑎)𝑢𝑗𝑘 , 𝑢𝑗𝑘 ⟩𝐿2 (Ω) → ∫ 𝑘

𝑎 𝑑𝜇.

(9)

Ω×ℝ2

We say that a measure 𝜇 on Ω × ℝ2 is a semiclassical measure if there exists a sequence of eigenfunctions converging to it. The left-hand side of (9) has a natural quantum mechanical interpretation: it is the expected value of the observable 𝑎(𝑥, 𝜉) where 𝑥 is the position and 𝜉 is the momentum of the quantum particle. Thus the limiting measure 𝜇 describes the probability distribution of the particle in position and momentum in the high energy limit along the sequence of quantum pure states 𝑢𝑗𝑘 . From a mathematical point of view, 𝜇 captures the distribution of mass of 𝑢𝑗𝑘 in position (𝑥) and frequency (𝜉). Each semiclassical measure 𝜇 has the following properties: (a) 𝜇 is a probability measure; (b) the support of 𝜇 is contained in Ω × 𝕊1 ; (c) 𝜇 is invariant under the billiard ball flow 𝜑𝑡 .

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Figure 4. An example of an anomalous, nonequidistributing, eigenfunction for the stadium (left); such eigenfunctions were numerically observed by Heller in Physical Review Letters in 1984. Its existence is related to the presence of “mildly chaotic” billiard ball trajectories which take a long time to exhibit chaotic behavior, like the one pictured on the right. A generic stadium has a sequence of nonequidistributing eigenfunctions. However, it is an open problem to show that such eigenfunctions localize precisely on the mildly chaotic trajectories.

Here (a) corresponds to the normalization ‖𝑢𝑗𝑘 ‖𝐿2 (Ω) = 1 and the fact that Opℎ (1) is the identity. The property (b) corresponds to the correct choice of the semiclassical scaling parameter ℎ𝑗𝑘 = 𝜆𝑗−1 , so that after rescaling 𝑢𝑗𝑘 oscil𝑘 lates at unit length frequency. Finally, the property (c) follows from Egorov’s theorem: indeed, pairing both sides of (8) with 𝑢𝑗𝑘 and passing to the limit we see that ∫(𝑎 ∘ 𝜑𝑡 ) 𝑑𝜇 = ∫ 𝑎 𝑑𝜇 for all 𝑎. There are many measures satisfying properties (a)–(c) above. Of particular importance is the Liouville measure 𝜇𝐿 = 𝑐 𝑑𝑥𝑑𝑆(𝜉) featured in Definition 1, which is in some sense the most “spread-out” invariant measure. The opposite, most “concentrated” case, is the delta measure on a periodic trajectory of 𝜑𝑡 . One of the central questions in quantum chaos is:

equidistribution in position without semiclassical equidistribution, see Marklof–Rudnick [MR12]. We also mention briefly the case of mixed systems, having a positive measure subset of Ω × 𝕊1 on which the billiard ball flow is ergodic. For a special class of these systems, namely generic mushroom billiards, Galkowski [Gal14] and Gomes [Gom18] showed Percival’s conjecture, giving a positive density sequence of eigenfunctions equidistributing in the ergodic region; for earlier numerics in this setting, see Barnett–Betcke [BB07].

3. QUE and Strongly Chaotic Systems A natural question to ask, known as the quantum unique ergodicity (QUE) conjecture, is whether Theorem 2 holds without passing to a density 1 subsequence:

What measures can arise as semiclassical limits of high energy eigenfunctions?

Is Liouville measure the only semiclassical measure?

This question is discussed in more detail in §4. It is not restricted to the chaotic case: even for tori it is a nontrivial question which attracted the attention of many including Jean Bourgain; see Lester–Rudnick [LR17] for a recent contribution. We can now state a stronger version of quantum ergodicity, giving equidistribution in both position and frequency. Following Definition 2, we say that a sequence of eigenfunctions semiclassically equidistributes if it converges to the Liouville measure in the sense of (9).

For general ergodic settings this can fail. In fact, this is the case for a generic stadium domain as shown by Hassell [Has10]; see Figure 4. A natural setting in which QUE is more feasible (and was explicitly conjectured by Rudnick–Sarnak in 1994) is that of strongly chaotic systems, which is a subclass of ergodic systems for which small perturbations of any trajectory lead to exponentially fast divergence from the original trajectory. More precisely, for such a system the tangent space to Ω × 𝕊1 splits into the flow, unstable, and stable subspaces, and the differential of the flow is exponentially expanding on the unstable spaces and contracting on the stable spaces as time goes to infinity. This implies that the flow has a positive Lyapunov exponent and is related to the “butterfly effect” mentioned in the opening paragraph of this article. To give an example of a strongly chaotic system, we move away from planar domains to the setting of manifolds without boundary. Let (𝑀, 𝑔) be a compact Riemannian manifold. The analog of Dirichlet eigenfunctions (1) is given by eigenfunctions of the Laplace–Beltrami

Theorem 2. Assume that the billiard ball flow 𝜑𝑡 is ergodic. Then there exists a density 1 sequence 𝑗 𝑘 → ∞ such that the corresponding sequence of eigenfunctions 𝑢𝑗𝑘 semiclassically equidistributes. Note that semiclassical equidistribution implies equidistribution in position of Definition 2, taking observables of the form 𝑎(𝑥, 𝜉) = 𝑏(𝑥) in (9); thus Theorem 2 implies Theorem 1. On the other hand, for general (not necessarily ergodic) domains one might have 1596

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Figure 5. An eigenfunction (left) and a geodesic (right) on a genus 2 hyperbolic surface obtained by gluing together the same color sides of the pictured dodecagon, embedded in the Poincare´ disk model of the hyperbolic plane. The eigenfunction is computed using the method developed by Strohmaier–Uski [SU13].

operator Δ𝑔 induced by the metric 𝑔: −Δ𝑔 𝑢𝑗 (𝑥) = 𝜆𝑗2 𝑢𝑗 (𝑥),

𝑢𝑗 ∈ 𝐶 ∞ (𝑀).

(10)

The semiclassical quantization introduced in (5) can be defined on manifolds, if we take 𝑎 to be a function on the cotangent bundle 𝑇 ∗ 𝑀. The appearance of the cotangent bundle is already evident for differential operators: if 𝑋 is a vector field on 𝑀, then the first order differential operator −𝑖ℎ𝑋 is equal to Opℎ (𝜒) + 𝒪(ℎ) where 𝜒(𝑥, 𝜉) = ⟨𝜉, 𝑋(𝑥)⟩ is the linear function on the fibers of 𝑇 ∗ 𝑀 defined by 𝑋. Note also that the Poisson bracket featured in the commutator rule (7) is well-defined on functions on 𝑇 ∗ 𝑀 since the latter has a natural symplectic form. The corresponding classical dynamical system is the geodesic flow 𝜑𝑡 ∶ 𝑆 ∗ 𝑀 → 𝑆 ∗ 𝑀 where 𝑆 ∗ 𝑀 is the unit cotangent bundle of 𝑀, consisting of pairs (𝑥, 𝜉) where 𝑥 ∈ 𝑀 and 𝜉 ∈ 𝑇𝑥∗ 𝑀 satisfies |𝜉|𝑔 = 1. Here 𝜉 is the cotangent vector dual to the velocity vector of the geodesic via the metric 𝑔. It is a result of Anosov in the 1960s that if the metric 𝑔 has negative curvature, then the geodesic flow 𝜑𝑡 is strongly chaotic. An important family of examples of negatively curved manifolds, appearing in many areas of mathematics, is given by hyperbolic surfaces which are surfaces of Gauss curvature −1; see Figure 5. In the setting of Riemannian manifolds, semiclassical measures are supported on 𝑆 ∗ 𝑀 and invariant under the flow 𝜑𝑡 , and an analog of quantum ergodicity (Theorem 2) holds. NOVEMBER 2023

Coming back to QUE, in a special setting of arithmetic hyperbolic surfaces QUE for joint eigenfunctions of the Laplacian and all Hecke operators (which are additional symmetries commuting with the Laplacian) was proved by Lindenstrauss [Lin06]. However, in general this conjecture is completely open and in fact there are toy models where it fails; the most celebrated one is described in §5 below.

4. More on Semiclassical Measures With QUE seeming out of reach, we return to the question asked in §2, now in the setting of manifolds without boundary: what measures can arise as semiclassical limits of high energy eigenfunctions? We discuss two results giving restrictions on such measures. We start with the more recent result, due to the author, Jin, and Nonnenmacher [DJN22], and relying on earlier work of Bourgain and the author on the fractal uncertainty principle: Theorem 3. Let 𝜇 be a semiclassical measure on a negatively curved surface. Then supp 𝜇 = 𝑆 ∗ 𝑀, that is 𝜇(𝑈) > 0 for any nonempty open set 𝑈 ⊂ 𝑆 ∗ 𝑀. Theorem 3 together with the unique continuation principle implies a lower bound on the mass of eigenfunctions: for any nonempty open set 𝑉 ⊂ 𝑀 we have ‖ 1l𝑉 𝑢𝑗 ‖𝐿2 ≥ 𝑐 𝑉 > 0 where the constant 𝑐 𝑉 is independent of the eigenvalue 𝜆𝑗 . This can be thought of as having no whitespace in Figure 5: for any given macroscopic ball, the probability of finding the quantum particle in that ball is separated away from 0.

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𝑡=0

𝑡=1

𝑡=2

𝑡=3

𝑡=4

𝑡=5

𝑡=6

2 Figure 6. Evolution of an image by the map 𝐴𝑡 , 𝑡 = 0, … , 6, where 𝐴 ∶ 𝕋2 → 𝕋2 corresponds to the matrix 𝐴 = ( 1

3 ). 2

It is an open question whether Theorem 3 holds in dimensions ≥ 3. Theorem 3 also implies that the delta measure on a closed geodesic cannot be a semiclassical measure. However, the latter fact (conjectured by Colin de Verdière in the 1980s) was already known as a corollary of entropy bounds of Anantharaman and Nonnenmacher. These bounds are true for general strongly chaotic systems, but for simplicity we state the result of [AN07] in a special case:

cat map and can be used to show that it is ergodic. See Figure 6. We next discuss discrete microlocal analysis and semiclassical quantization. The space 𝐿2 (𝑀) of squareintegrable functions on a manifold is replaced by the finite dimensional space ℂ𝑁 . Here the semiclassical parameter is 1 ℎ= . 2𝜋𝑁

Theorem 4. Let 𝜇 be a semiclassical measure on a hyperbolic surface. Then the Kolmogorov–Sinai entropy of 𝜇 satisfies

The discrete version of the Fourier transform, ℱ 𝑁 ∶ ℂ𝑁 → ℂ𝑁 , is given by

1

hKS (𝜇) ≥ . 2

(11)

We do not give a definition of the entropy hKS (𝜇) here but remark that it measures the complexity of the flow 𝜑𝑡 with respect to the measure 𝜇. In particular, the entropy of a delta measure on a closed geodesic is equal to 0, while the entropy of the Liouville measure is equal to 1, so in some sense (11) excludes half of 𝜑𝑡 -invariant measures as candidates for semiclassical measures.

5. Quantum Cat Maps We finally discuss semiclassical measures in the toy model setting of quantum cat maps, where a striking counterexample to QUE is known. For quantum cat maps, the phase space 𝑇 ∗ 𝑀 is replaced by the torus 𝕋2 = ℝ2 /ℤ2 and the geodesic flow 𝜑𝑡 , by a linear map. This has the advantage that the underlying dynamics, while still strongly chaotic, is simpler to understand; moreover, it is easier to compute eigenvalues and eigenfunctions numerically. On the other hand, it is harder to explain the analog of the eigenvalue problems (1), (10). We first discuss linear maps on the torus, which in this setting are analogs of the time-one map of the geodesic flow. Let 𝐴 ∈ SL(2, ℤ) be a 2 × 2 matrix with integer entries and determinant 1. The linear map on ℝ2 induced by 𝐴 descends to a diffeomorphism of the torus 𝕋2 , which we we still denote by 𝐴. The matrix 𝐴 is called hyperbolic, and the corresponding map on 𝕋2 is called a cat map (a term coined by Arnold), if | tr 𝐴| > 2. In this case 𝐴 has two real eigenvalues 𝜔, 𝜔−1 with |𝜔| > 1; the corresponding eigenspaces give the unstable and stable directions for the 1598

(ℱ 𝑁 𝑓)𝑗 =

1 √𝑁

𝑁−1

∑ 𝑒

−

2𝜋𝑖𝑗ℓ 𝑁

𝑓ℓ .

ℓ=0

We note that this is the Fourier transform used in signal processing and FFT algorithms. One can define an analog of the quantization procedure (5), mapping a smooth function 𝑎(𝑥, 𝜉) on the torus 𝕋2 to a sequence of operators Op𝑁 (𝑎) ∶ ℂ𝑁 → ℂ𝑁 . We do not give a proper definition here but note that similarly to semiclassical quantization on ℝ𝑛 • if 𝑎 is a function of 𝑥 only, then Op𝑁 (𝑎) is a multiplication operator: (Op𝑁 (𝑎)𝑓)𝑗 = 𝑎(𝑗/𝑁)𝑓𝑗 ; • if 𝑎 is a function of 𝜉 only, then Op𝑁 (𝑎) is a Fourier multiplier: (ℱ 𝑁 Op𝑁 (𝑎)𝑓)𝑗 = 𝑎(𝑗/𝑁)(ℱ 𝑁 𝑓)𝑗 . One also has analogues of the product rule (6) and the commutator rule (7). For the latter, the Poisson bracket is defined as before and corresponds to the symplectic form 𝑑𝜉 ∧ 𝑑𝑥. Implicit in the construction below is the fact that the map 𝐴 preserves the symplectic form on 𝕋2 , just as the geodesic flow 𝜑𝑡 preserves the symplectic form on 𝑇 ∗ 𝑀. We now introduce quantizations of a linear map on 𝕋2 induced by a matrix 𝐴 ∈ SL(2, ℤ), which in this setting are analogous to the time-one map of the Schrödinger group. For technical reasons we restrict to the case of even 𝑁. Quantizations of 𝐴 are sequences of unitary operators 𝑈𝑁 ∶ ℂ𝑁 → ℂ𝑁 which satisfy the following exact version of Egorov’s theorem (8): for all 𝑎 ∈ 𝐶 ∞ (𝕋2 ) 𝑈𝑁−1 Op𝑁 (𝑎)𝑈𝑁 = Op𝑁 (𝑎 ∘ 𝐴).

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𝑡=7

𝑡=8

𝑡=9

𝑡 = 10

𝑡 = 11

𝑡 = 12

Figure 7. Continuation of Figure 6, showing the times 𝑡 = 7, … , 12. Here we took the special resolution 𝑁 = 1560 points per side of the square; the picture illustrates the fact that 𝐴12 mod 𝑁 = 𝐼. A short period of the discretized classical cat map implies that the associated quantum cat map also has a short period, which is used in the example (14) below.

One way to compute these explicitly is as follows. Consider the matrices 1 𝐴1 = ( 1

0 ), 1

𝐴2 = (

0 1 ). −1 0

Then a quantization of 𝐴1 is given by the multiplication operator (𝑈𝑁 𝑓)𝑗 = 𝑒

𝜋𝑖𝑗 2 𝑁

𝑓𝑗

(13)

and a quantization of 𝐴2 is given by the discrete Fourier transform ℱ 𝑁 . The matrices 𝐴1 , 𝐴2 generate the group SL(2, ℤ) so this gives a way to quantize every linear map on the torus. One explanation for the formula (13) is as fol𝑖𝜑(𝑥)

lows: in the continuous setting the map 𝑓(𝑥) ↦ 𝑒 ℎ 𝑓(𝑥) is a phase shift, quantizing the transformation (𝑥, 𝜉) ↦ (𝑥, 𝜉 + 𝜑′ (𝑥)) (which is most evident for the case when 𝜑(𝑥) = 𝑥𝜂 is a linear function); putting 𝜑(𝑥) ∶= 𝑥2 /2, ℎ ∶= (2𝜋𝑁)−1 , and 𝑥 ∶= 𝑗/𝑁 we get the operator (13) and the associated transformation (𝑥, 𝜉) ↦ (𝑥, 𝜉 + 𝑥) is linear with the matrix 𝐴1 . From now on, let 𝑈𝑁 ∶ ℂ𝑁 → ℂ𝑁 be a quantization of the linear map on 𝕋2 corresponding to a hyperbolic matrix 𝐴 ∈ SL(2, ℤ). We call this sequence of operators a quantum cat map. Let 𝑁 𝑘 → ∞ and 𝑢𝑁𝑘 ∈ ℂ𝑁𝑘 be a sequence of normalized eigenfunctions of the map 𝑈𝑁𝑘 : 𝑈𝑁𝑘 𝑢𝑁𝑘 = 𝑧𝑘 𝑢𝑁𝑘 ,

|𝑧𝑘 | = 1,

‖𝑢𝑁𝑘 ‖ = 1.

Similarly to (9), we say that 𝑢𝑁𝑘 converges semiclassically to a measure 𝜇 on 𝕋2 if for all 𝑎 ∈ 𝐶 ∞ (𝕋2 ) we have ⟨Op𝑁 (𝑎)𝑢𝑁𝑘 , 𝑢𝑁𝑘 ⟩ → ∫ 𝑎 𝑑𝜇. 𝑘

1

1

2

2

𝜇 = 𝛿𝛾 + 𝑑𝑥𝑑𝜉.

(14)

Note that the entropy of 𝜇 is half the entropy of 𝑑𝑥𝑑𝜉 and (14) shows that the entropy bound for quantum cat maps is sharp. See Figure 8 for a numerical illustration. The construction of (14) relies on the fact (observed by Bonechi–De Bièvre in 2000) that there exists a sequence 𝑁 𝑘 → ∞ such that the restriction of the classical cat map 𝐴 to the discrete set of points 𝑁𝑘−1 ℤ2 /ℤ2 ⊂ 𝕋2 is periodic with a short period ∼ log 𝑁 𝑘 , and correspondingly the quantum cat map 𝑈𝑁𝑘 also has a short period—see Figure 7. There is also an analogue of arithmetic QUE, due to Kurlberg–Rudnick in 2000: there exists a basis of eigenfunctions of 𝑈𝑁 which converges semiclassically to the measure 𝑑𝑥𝑑𝜉. This does not contradict the counterexample (14) since the operator 𝑈𝑁𝑘 used there has eigenvalues of high multiplicity, and the eigenfunctions used in (14) do not belong to the arithmetic basis. ACKNOWLEDGMENT. The author would like to thank Alex Barnett, Alex Strohmaier, and Jared Wunsch for numerous suggestions to improve the article. This work was supported by NSF CAREER grant DMS-1749858.

𝕋2

The resulting limiting measures are called semiclassical measures for the quantum cat map 𝑈𝑁 . It follows from the normalization and Egorov’s theorem (12) that each semiclassical measure 𝜇 is a probability measure invariant under the map 𝐴. In the setting of quantum cat maps, there are versions of quantum ergodicity (Theorem 2), due to Bouzouina– De Bièvre in 1996, the full support property (Theorem 3), due to Schwartz in 2021, and entropy bounds (Theorem 4), due to Faure–Nonnenmacher in 2004 and Brooks in 2010. NOVEMBER 2023

However, there is a remarkable counterexample to QUE due to Faure–Nonnenmacher–De Bièvre [FNDB03]. More precisely, if 𝛾 is any given closed orbit of the map 𝐴, 𝛿𝛾 is the 𝐴-invariant probability measure on 𝛾, and 𝑑𝑥𝑑𝜉 is the volume measure on 𝕋2 , then there exists a sequence of eigenfunctions 𝑢𝑁𝑘 converging semiclassically to the measure

References

[ADSK16] Artur Avila, Jacopo De Simoi, and Vadim Kaloshin, An integrable deformation of an ellipse of small eccentricity is an ellipse, Ann. of Math. (2) 184 (2016), no. 2, 527–558, DOI 10.4007/annals.2016.184.2.5. MR3548532 [AN07] Nalini Anantharaman and St´ephane Nonnenmacher, Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold (English, with English and French summaries), Ann. Inst. Fourier (Grenoble) 57 (2007), no. 7, 2465–2523. Festival Yves Colin de Verdière. MR2394549

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Figure 8. Modified Wigner functions of two eigenfunctions of the quantum cat map with the same matrix 𝐴 as in Figure 6 and 𝑁 = 780 (which is half of the special 𝑁 in Figure 7). These pictures show the concentration of the eigenfunction simultaneously in position and frequency, see e.g. [DJ23]; darker shading corresponds to larger Wigner transform. On the left is a typical eigenfunction, showing equidistribution. On the right is an anomalous eigenfunction corresponding to the semiclassical 1 1 measure (14) with 𝛾 being the fixed point ( , ). Note that the scars seen on the stable/unstable manifolds of the fixed point 2 2

carry about (log 𝑁)−1 portion of the mass of the eigenfunction and thus do not contribute to the limiting semiclassical measure.

[BB07] Alex H. Barnett and Timo Betcke, Quantum mushroom billiards, Chaos 17 (2007), no. 4, 043125, 13, DOI 10.1063/1.2816946. MR2380041 [BH14] Alex H. Barnett and Andrew Hassell, Fast computation of high-frequency Dirichlet eigenmodes via spectral flow of the interior Neumann-to-Dirichlet map, Comm. Pure Appl. Math. 67 (2014), no. 3, 351–407, DOI 10.1002/cpa.21458. MR3158571 [DJ23] Semyon Dyatlov and Malo J´ez´equel, Semiclassical measures for higher-dimensional quantum cat maps, Ann. Henri Poincar´e (2023), https://doi.org/10.1007/s00023 -023-01309-x. [DJN22] Semyon Dyatlov, Long Jin, and St´ephane Nonnenmacher, Control of eigenfunctions on surfaces of variable curvature, J. Amer. Math. Soc. 35 (2022), no. 2, 361–465, DOI 10.1090/jams/979. MR4374954 [Dya21] Semyon Dyatlov. Macroscopic limits of chaotic eigenfunctions, 2021. arXiv:2109.09053. [FNDB03] Fr´ed´eric Faure, St´ephane Nonnenmacher, and Stephan De Bièvre, Scarred eigenstates for quantum cat maps of minimal periods, Comm. Math. Phys. 239 (2003), no. 3, 449–492, DOI 10.1007/s00220-003-0888-3. MR2000926 [Gal14] Jeffrey Galkowski, Quantum ergodicity for a class of mixed systems, J. Spectr. Theory 4 (2014), no. 1, 65–85, DOI 10.4171/JST/62. MR3181386 [Gom18] Sean P. Gomes, Percival’s conjecture for the Bunimovich mushroom billiard, Nonlinearity 31 (2018), no. 9, 4108–4136, DOI 10.1088/1361-6544/aa776f. MR3833087

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[Has10] Andrew Hassell, Ergodic billiards that are not quantum unique ergodic, Ann. of Math. (2) 171 (2010), no. 1, 605– 618, DOI 10.4007/annals.2010.171.605. With an appendix by the author and Luc Hillairet. MR2630052 [Lin06] Elon Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163 (2006), no. 1, 165–219, DOI 10.4007/annals.2006.163.165. MR2195133 [LR17] Stephen Lester and Ze´ev Rudnick, Small scale equidistribution of eigenfunctions on the torus, Comm. Math. Phys. 350 (2017), no. 1, 279–300, DOI 10.1007/s00220-0162734-4. MR3606476 [MR12] Jens Marklof and Ze´ev Rudnick, Almost all eigenfunctions of a rational polygon are uniformly distributed, J. Spectr. Theory 2 (2012), no. 1, 107–113, DOI 10.4171/JST/23. MR2879311 [Rud08] Ze’ev Rudnick, What is… quantum chaos?, Notices Amer. Math. Soc. 55 (2008), no. 1, 32–34. MR2373795 [Sar11] Peter Sarnak, Recent progress on the quantum unique ergodicity conjecture, Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 2, 211–228, DOI 10.1090/S0273-0979-2011-01323-4. MR2774090 [SU13] Alexander Strohmaier and Ville Uski, An algorithm for the computation of eigenvalues, spectral zeta functions and zetadeterminants on hyperbolic surfaces, Comm. Math. Phys. 317 (2013), no. 3, 827–869, DOI 10.1007/s00220-012-1557-1. MR3009726

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[Zel19] Steve Zelditch, Mathematics of quantum chaos in 2019, Notices Amer. Math. Soc. 66 (2019), no. 9, 1412–1422. MR3967933 [Zwo12] Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012, DOI 10.1090/gsm/138. MR2952218 [ZZ96] Steven Zelditch and Maciej Zworski, Ergodicity of eigenfunctions for ergodic billiards, Comm. Math. Phys. 175 (1996), no. 3, 673–682. MR1372814 Semyon Dyatlov Credits

Opening image, Figure 1 (right), and Figure 4 (left) are courtesy of Alex Barnett. Figures 1 (left), 2, 3, 4 (right), 5 (right), and 8 are courtesy of Semyon Dyatlov. Figure 5 (left) is courtesy of Alexander Strohmaier. Figures 6 and 7 are courtesy of Semyon Dyatlov, with pictures of Marcus the cat courtesy of Jeffrey Galkowski. Photo of Semyon Dyatlov is courtesy of Xuwen Zhu.

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NEXT GENERATION Early-career AMS members take a moment

Favorite memory from an AMS event: The excitement of seeing the big mathematicians whose papers I’ve read.

Favorite color: Blue

Were you inspired by a mathematician?: I was inspired to study math just in order to make it simpler for others. This is because I was the only one that passed math in my graduating high school class and I thought this could be simpler because I believed my classmates were smart too.

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Hobby: Playing Soccer Favorite color: Green Favorite food: Biryani

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h ssor ort , University of N

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A Perspective on the Regularity Theory of Degenerate Elliptic Equations

H´ector A. Chang-Lara 1. Introduction Ellipticity is a well-studied characteristic of some partial differential equations which enforces regularity on its solutions. The prototype problem is the Laplace equation Δ𝑢 ≔ 𝜕12 𝑢 + … + 𝜕𝑛2 𝑢 = 0 whose solutions are known as the harmonic functions. To illustrate its usefulness, consider an arbitrary sequence of uniformly bounded functions over a compact set. In contrast to numerical H´ector A. Chang-Lara is investigador titular in the Department of Mathematics at el Centro de Ivestigaci´on en Matem´aticas (CIMAT), Guanajuato, Mexico. His email address is [email protected]. Communicated by Notices Associate Editor Daniela De Silva. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2800

1604

sequences, a bounded sequence of functions does not necessarily have a uniformly convergent subsequence, even if we assume that each function in such sequence is smooth. However, as soon as we assume that the functions are harmonic, then such uniform limits are always guaranteed. The property we have just described is known as compactness. We will see that it can be derived from regularity estimates for harmonic functions. It is a powerful tool that can be used in many fundamental results, such as the existence theorem for harmonic functions with prescribed boundary values or the convergence of numerical schemes. To fix some ideas, let us consider a general second-order partial differential equation (PDE) of the form 𝐹(𝐷2 𝑢, 𝐷𝑢, 𝑢, 𝑥) = 0 in Ω ⊆ ℝ𝑛 , 𝑛 where 𝐹 = 𝐹(𝑀, 𝑝, 𝑧, 𝑥) ∶ ℝ𝑛×𝑛 sym × ℝ × ℝ × Ω → ℝ determines the nonlinear operator on the left-hand side,

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and the second-order differentiable function 𝑢 ∶ Ω → ℝ is the unknown of the problem. Our focus lies on elliptic problems, which are defined by the requirement that 𝐹 is nondecreasing in the Hessian variable (𝑀).1 Uniform ellipticity arises when we further assume some quantitative control on the monotonicity of 𝐹 with respect to 𝑀. In the case that the function 𝐹 is differentiable, we say that the operator is uniformly elliptic if 𝜆𝐼 ≤ 𝐷𝑀 𝐹 ≤ Λ𝐼, for some fixed constants 0 < 𝜆 ≤ Λ. In the previous expression, 𝐼 ∈ ℝ𝑛×𝑛 is the identity matrix and 𝐷𝑀 𝐹 = (𝜕𝑚𝑖𝑗 𝐹) ∈ ℝ𝑛×𝑛 sym is the matrix of partial derivatives of 𝐹 with respect to the Hessian variable 𝑀 = (𝑚𝑖𝑗 ) ∈ ℝ𝑛×𝑛 sym . Operators that are elliptic, but not necessarily uniformly elliptic, are called degenerate elliptic. These find practical applications in diverse fields such as material sciences, fluid dynamics, finance, and image processing. Some famous examples of degenerate equations include the minimal surface equation div (

𝐷𝑢 √1 + |𝐷𝑢|2

) = 0,

the 𝑝-Laplace equation (𝑝 ≥ 1) div(|𝐷𝑢|𝑝−2 𝐷𝑢) = 0, and, in the time-dependent case, the porous medium equation (𝑚 > 1) 𝜕𝑡 𝑢 = Δ(𝑢𝑚 ). In general, it has been observed that solutions of degenerate elliptic equations are not always guaranteed to have the same regularity estimates as uniformly elliptic equations. Motivated by the observation for harmonic functions, we may wonder the following: 1. Have solutions of uniformly elliptic PDEs a compactness property, similar to the harmonic functions? 2. Which additional hypotheses could complement the degenerate ellipticity in order to recover a compactness property for the solutions? The analysis of PDEs relies significantly on bounds for the modulus of continuity of a solution and its derivatives, these estimates constitute the cornerstones of the regularity theory. The development of the regularity theory for elliptic equations has a rich history with numerous authors. This survey concerns the regularity theory of uniformly elliptic equations that originated during the 1980s and 1990s, with main contributions due to Krylov, Safonov, Evans, and Caffarelli, among many others. This regularity theory is now commonly known as the Krylov–Safonov theory. 1We assume the following partial order on ℝ𝑛×𝑛 : 𝑀 ≤ 𝑁 if and only if 𝑥⋅𝑀𝑥 ≤ sym

𝑥 ⋅ 𝑁𝑥 for all 𝑥 ∈ ℝ𝑛 .

NOVEMBER 2023

Figure 1. The mean value theorem: If Δ𝑢 ≤ 0 in Ω, then for any ´ 1 𝐵𝑟 (𝑥0 ) ⊆ Ω it holds that 𝑢 ≤ 𝑢(𝑥0 ). 𝐵 (𝑥 ) |𝐵𝑟 (𝑥0 )|

𝑟

0

We aim to provide a perspective on ongoing developments in the regularity theory of degenerate elliptic problems. First, we give an overview of the regularity theory for the Laplacian and uniformly elliptic equations. This discussion will shed light on the fundamental strategies employed in classical scenarios, enabling us to appreciate better the challenges posed by degenerate equations. Afterward, we focus on the case of degenerate ellipticity, where the absence of uniform ellipticity is complemented with some additional hypotheses. These assumptions could be interpreted as some sort of alternative regularizing mechanism for the solution. It is the interplay of these phenomena what we find to be a quite attractive venue of current research. We focus on three types of degeneracies which can be loosely described in the following way: 1. Elliptic equations that hold only where the gradient is large: Either the solution obeys a uniformly elliptic equation or its gradient is bounded. 2. Small perturbations: Uniform ellipticity holds in a neighborhood of a given profile. 3. Quasi-Harnack: Uniformly ellipticity holds at macroscopic scales. Each one of the previous problems will be presented in a different section, titled accordingly. These section titles make reference to the articles [IS16], [Sav07], and [DSS21], where the respective results were originally studied.

2. The Laplacian The Laplacian Δ ≔ 𝜕12 + … + 𝜕𝑛2 is the fundamental differential operator that describes the ellipticity phenomenon in ℝ𝑛 . Solutions of the equation Δ𝑢 = 0, also known as harmonic functions, are abundant in pure and applied mathematics. Perhaps the most characteristic features of uniformly elliptic problems are the Harnack inequalities. In the case of the Laplacian, these arise as consequences of the divergence theorem through the mean value theorem (Figure 1).

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𝜃 ∈ (0, 1) such that for any 𝑟 ∈ (0, 1/3]2 osc 𝑢 ≤ (1 − 𝜃) osc 𝑢.

𝐵𝑟/3 (𝑥0 )

(2.3)

𝐵𝑟 (𝑥0 )

Indeed, (2.3) implies that the oscillation of 𝑢 has a geometric decay in triadic balls osc

𝐵3−𝑘 (𝑥0 )

Figure 2. The key observation is that for any 𝑥1 ∈ 𝐵𝑟/3 (𝑥0 ) we always get the inclusions 𝐵𝑟/3 (𝑥0 ) ⊆ 𝐵2𝑟/3 (𝑥1 ) ⊆ 𝐵𝑟 (𝑥0 ).

𝑢 ≤ (1 − 𝜃)𝑘−1 .

By conveniently fixing 𝛼 ≔ log1/3 (1 − 𝜃) and 𝐶 ≔ (1 − 𝜃)−2 we now get the desired estimate in the following form and for every 𝑟 ∈ (0, 1/3) and 𝑘 = ⌊log1/3 𝑟⌋, such that 3−(𝑘+1) < 𝑟 ≤ 3−𝑘 sup |𝑢 − 𝑢(𝑥0 )| ≤ osc 𝑢 𝐵𝑟 (𝑥0 )

𝐵𝑟 (𝑥0 )

The weak Harnack inequality states that for any 𝑢 ≥ 0 satisfying Δ𝑢 ≤ 0 in 𝐵𝑟 (𝑥0 ), and any value 𝜇 > 0, it holds that |{𝑢 ≥ 𝜇} ∩ 𝐵𝑟/3 (𝑥0 )| ≤ 2𝑛 𝜇−1 inf 𝑢. |𝐵𝑟/3 (𝑥0 )| 𝐵𝑟/3 (𝑥0 )

(2.2)

for some 𝛼 ∈ (0, 1) and 𝐶 > 0 depending only on the dimension. Although we have assumed that the solution is 𝐶 2 regular, the point of the estimate is that its own continuity gets controlled by its size rather than its derivatives. Notably, a uniformly bounded family of harmonic functions is automatically equicontinuous on any compact subset. By the Arzel´a–Ascoli theorem, this family always contains a sequence that converges locally uniformly to a limit. Moreover, it can be shown that the limit is also a harmonic function. 2.1. The diminish of oscillation. The Estimate (2.2) can be derived through a strategy known as diminish of oscillation. This elegant argument exemplifies the geometric approach in elliptic PDEs. Let us explain it in detail: 1. Assume that 𝑢 ∈ 𝐶 2 (𝐵1 ) is a harmonic function taking values between 0 and 1. For 𝑥0 ∈ 𝐵1/2 and 𝑟 ∈ (0, 1/3], we aim to show that sup |𝑢 − 𝑢(𝑥0 )| ≤ 𝐶𝑟𝛼 .

𝐵𝑟 (𝑥0 )

osc

𝐵3−𝑘 (𝑥0 )

𝑢

≤ (1 − 𝜃)𝑘−1 = 𝐶3−𝛼(𝑘+1)

(2.1)

Figure 2 illustrates a geometric argument from where to establish this estimate from the mean value theorem. This control on the distribution of the solution is quite powerful. In particular, it can be used to prove an interior Hölder estimate for harmonic functions in the following form 𝑟𝛼 [𝑢]𝐶 𝛼 (𝐵𝑟/3 (𝑥0 )) ≤ 𝐶‖𝑢‖𝐿∞ (𝐵𝑟 (𝑥0 )) ,

≤

≤ 𝐶𝑟𝛼 . 2. To get the diminish of oscillation (2.3), we apply the weak Harnack inequality to a given translation of 𝑢. Keep in mind that 𝑢 oscillates between 𝑚𝑟 ≔ inf𝐵𝑟 (𝑥0 ) 𝑢 and 𝑀𝑟 ≔ sup𝐵𝑟 (𝑥0 ) 𝑢 over 𝐵𝑟 (𝑥0 ), and consider as well 𝜇𝑟 ≔ (𝑚𝑟 + 𝑀𝑟 )/2, the level set that sits just in the middle. Hence, at least one of the following alternatives must be true: |{𝑢 ≥ 𝜇𝑟 } ∩ 𝐵𝑟/3 (𝑥0 )| 1 ≥ 2 |𝐵𝑟/3 (𝑥0 )| or |{𝑢 ≤ 𝜇𝑟 } ∩ 𝐵𝑟/3 (𝑥0 )| 1 ≥ . 2 |𝐵𝑟/3 (𝑥0 )| In the first case we apply the weak Harnack inequality (2.1) to the positive harmonic function (𝑢 − 𝑚𝑟 ) in 𝐵𝑟 (𝑥0 ) to get that inf (𝑢 − 𝑚𝑟 ) ≥ 2−(𝑛+1) (𝜇𝑟 − 𝑚𝑟 ).

𝐵𝑟/3 (𝑥0 )

This estimate raises the lower bound on 𝑢 from 𝑚𝑟 over the ball 𝐵𝑟 (𝑥0 ), to 𝑚𝑟 + 2−(𝑛+1) (𝜇𝑟 − 𝑚𝑟 ) = 𝑚𝑟 + 2−(𝑛+2) osc 𝑢 𝐵𝑟 (𝑥0 )

over 𝐵𝑟/3 (𝑥0 ) (Figure 3). For the other alternative, we apply the weak Harnack inequality to (𝑀𝑟 − 𝑢) to get a similar improvement on the upper bound instead. In conclusion, either option implies the diminish of oscillation Estimate (2.3) for 𝜃 ≔ 2−(𝑛+2) . □ 2The oscillation of a function measures the variation of the values that it takes

in a given set: osc 𝑢 ≔ sup |𝑢(𝑥) − 𝑢(𝑦)|. 𝐸

𝑥,𝑦∈𝐸

This can be proved recursively if there exists some small 1606

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By considering the extreme cases in (3.2) given under these assumptions, we obtain that 𝒫 − (𝐷2 𝑢) − Λ|𝐷𝑢| ≤ 0, { 𝜆,Λ + 𝒫𝜆,Λ (𝐷2 𝑢) + Λ|𝐷𝑢| ≥ 0,

(3.3)

where − 𝒫𝜆,Λ (𝑀) ≔

min 𝐴 ∶ 𝑀

𝜆𝐼≤𝐴≤Λ𝐼

+ 𝒫𝜆,Λ (𝑀) ≔ max 𝐴 ∶ 𝑀. 𝜆𝐼≤𝐴≤Λ𝐼

Figure 3. Diminish of oscillation: The lower bound of the solution improves if the measure of {𝑢 ≥ 𝜇𝑟 } ∩ 𝐵𝑟/3 (𝑥0 ) is at least half of the measure of 𝐵𝑟/3 (𝑥0 ).

3. Uniformly Elliptic Equations Let us now consider the second-order equation 𝐹(𝐷2 𝑢, 𝐷𝑢, 𝑥) = 0.

(3.1)

Assuming that 𝐹 is differentiable and 𝐹(0, 0, 𝑥) = 0, we get by the fundamental theorem of calculus that 𝑢 satisfies a homogeneous linear equation of the following form3 𝐴 ∶ 𝐷2 𝑢 + 𝑏 ⋅ 𝐷𝑢 = 0.

(3.2)

Indeed, we just need to integrate the derivative (𝑑/𝑑𝑡)𝐹(𝑡𝐷2 𝑢, 𝑡𝐷𝑢, 𝑥) from 𝑡 = 0 to 𝑡 = 1 to notice that 𝑛 the coefficients 𝐴 = (𝑎𝑖𝑗 (𝑥)) ∈ ℝ𝑛×𝑛 sym and 𝑏 = (𝑏𝑖 (𝑥)) ∈ ℝ are given by ˆ 1 𝑎𝑖𝑗 (𝑥) ≔ 𝜕𝑚𝑖𝑗 𝐹(𝑡𝐷2 𝑢(𝑥), 𝑡𝐷𝑢(𝑥), 𝑥)𝑑𝑡, 0

ˆ 𝑏𝑖 (𝑥) ≔

0

1

𝜕𝑝𝑖 𝐹(𝑡𝐷2 𝑢(𝑥), 𝑡𝐷𝑢(𝑥), 𝑥)𝑑𝑡.

Even though these coefficients depend on the solution as well, under suitable hypotheses on the derivatives of 𝐹 we can overlook this dependence and understand the equation in a broad sense. To start, we can just assume that the coefficients are uniformly bounded. In particular, |𝑏| ≤ Λ, which would follow from a Lipschitz assumption on 𝐹. Uniform ellipticity requires that for some constants 0 < 𝜆 ≤ Λ, it holds that 𝜆𝐼 ≤ 𝐷𝑀 𝐹 ≤ Λ𝐼, which means that 𝜆𝐼 ≤ 𝐴 ≤ Λ𝐼. We summarize our hypotheses on 𝐹 as

𝒫 − (𝐷2 𝑢) − Λ|𝐷𝑢| ≤ 0 in 𝐵𝑟 (𝑥0 ), { 𝜆,Λ + 𝒫𝜆,Λ (𝐷2 𝑢) + Λ|𝐷𝑢| ≥ 0 in 𝐵𝑟 (𝑥0 ). Then 𝑟𝛼 [𝑢]𝐶 𝛼 (𝐵𝑟/3 (𝑥0 )) ≤ 𝐶 osc 𝑢. 𝐵𝑟 (𝑥0 )

In the same way as for the Laplacian, this estimate can be deduced by a diminish of oscillation argument from the weak Harnack inequality, also known in this case as the 𝐿𝜀 estimate. The only difference is that, in the general setting, the bound on the distribution becomes of order 𝜇−𝜀 , for some exponent 𝜀 > 0, perhaps small and depending on the parameters of uniform ellipticity and the dimension. From now on, and to simplify the statements of the following lemmas and theorems, we will assume that any constants mentioned in these statements depend by default on the parameters of uniform ellipticity and the dimension. Lemma 3.2 (Weak Harnack inequality). There exist 𝜀, 𝐶 > 0 such that the following holds: Let 𝑢 ∈ 𝐶 2 (𝐵𝑟 (𝑥0 )) be nonnegative and satisfy − 𝒫𝜆,Λ (𝐷2 𝑢) − Λ|𝐷𝑢| ≤ 0 in 𝐵𝑟 (𝑥0 ).

𝜀

|{𝑢 ≥ 𝜇} ∩ 𝐵𝑟/2 (𝑥0 )| ≤ 𝐶 (𝜇−1 inf 𝑢) . |𝐵𝑟/2 (𝑥0 )| 𝐵𝑟/2 (𝑥0 )

3We assume the following inner product in ℝ𝑛×𝑛 𝑛

𝐴 ∶ 𝐵 ≔ tr(𝐴𝑇 𝐵) = ∑ 𝑎𝑖𝑗 𝑏𝑖𝑗 .

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Theorem 3.1 (Interior Hölder estimate). Given the parameters of uniform ellipticity 0 < 𝜆 ≤ Λ and the dimension 𝑛, there exist 𝛼 ∈ (0, 1) and 𝐶 > 0 such that the following holds: Let 𝑢 ∈ 𝐶 2 (𝐵𝑟 (𝑥0 )) satisfy

Then, for any 𝜇 > 0

⎧𝜆𝐼 ≤ 𝐷𝑀 𝐹 ≤ Λ𝐼, |𝐷 𝐹| ≤ Λ, ⎨ 𝑝 ⎩𝐹(0, 0, 𝑥) = 0.

𝑖,𝑗=1

are known as the Pucci extremal operators. The main result of the Krylov–Safonov regularity theory established in [KS79], states that solutions of the uniformly elliptic problem (3.3) have an interior Hölder estimate as in (2.2).

For a long time, the challenge to demonstrate this type of result was to find some connection between pointwise and measure quantities on the solution. For the Laplacian, this connection is naturally suggested by the divergence theorem (keep in mind that Δ = div 𝐷). In the general case, this link was eventually established by the

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below by4 −|𝐷2 𝜑𝑦0 (𝑥0 )|𝑜𝑝 . On the other hand, if we now assume that 𝑢 satisfies − 𝒫𝜆,Λ (𝐷2 𝑢) − Λ|𝐷𝑢| ≤ 0,

then we can also bound the eigenvalues of 𝐷2 𝑢(𝑥0 ) from above. To see this, we notice first that5 − 𝒫𝜆,Λ (𝑀) =

∑

(𝜆𝑒 + − Λ𝑒− ).

𝑒∈eig(𝐷2 ᵆ(𝑥0 ))

Figure 4. The contact set for a function 𝑢 is the set of points in the domain that admit a supporting graph of the form 𝜑𝑦0 + 𝑐 from below.

Alexandrov–Bakelman–Pucci maximum principle, often abbreviated as the ABP lemma. 3.1. The ABP lemma. Before stating the main result of this section we will need some preliminary notions. This presentation showcases constructions due to Cabr´e [Cab97] and Savin [Sav07]. Consider the family of functions given by translations of a fixed profile 𝜑𝑦0 (𝑥) ≔ 𝜑(𝑥 − 𝑦0 ). For a given function 𝑢, we say that a vertical translation of 𝜑𝑦0 touches 𝑢 from below at 𝑥0 if and only if 𝑥0 ∈ argmin(𝑢−𝜑𝑦0 ). We define the lower contact set as 𝑦0 varies in some set 𝐵 in the following way (Figure 4) 𝐴𝐵 ≔

⋃

argmin(𝑢 − 𝜑𝑦0 ).

𝑦0 ∈𝐵

The set 𝐴𝐵 may be designed to capture important information about 𝑢. For instance, if 𝜑(𝑥) = −|𝑥|2 then 𝐴𝐵𝑟 ⊆ {𝑢 ≤ 𝑢(0)+𝑟2 }. Indeed, for any 𝑥0 ∈ argmin(𝑢−𝜑𝑦0 ) 𝑢(𝑥0 ) ≤ 𝜑𝑦0 (𝑥0 ) + 𝑢(0) − 𝜑𝑦0 (0) ≤ 𝑢(0) + 𝑟2 . Consider now the mapping 𝑇 ∶ 𝐴𝐵 → 𝐵, such that 𝑇(𝑥0 ) = 𝑦0 if 𝑥0 ∈ argmin(𝑢 − 𝜑𝑦0 ). This transformation can be computed by solving for 𝑦0 in the expression 𝐷𝑢(𝑥0 ) = 𝐷𝜑(𝑥0 − 𝑦0 ). If 𝑇 is surjective, we get by the change of variable formula that ˆ |𝐵| ≤

𝜆𝑒 ≤ Λ(𝑛 − 1)|𝐷2 𝜑𝑦0 (𝑥0 )|𝑜𝑝 + Λ|𝐷𝜑𝑦0 (𝑥0 )|. The next lemma gives a concrete implementation of this construction. In this result, we fix the family of concave paraboloids 𝜑𝑦 (𝑥) ≔ 𝑝(|𝑥 − 𝑦0 |) − 𝑝(5𝑟/6), { 0 1 𝑝(𝜌) ≔ − 2 𝜌2 . 2𝑟

Lemma 3.3 (ABP). There exists 𝜂 ∈ (0, 1), such that the following holds: Let 𝐵 = 𝐵𝑟/6 (𝑥0 ) and 𝑢 ∈ 𝐶 2 (𝐵𝑟 (𝑥0 )) satisfy 𝒫 − (𝐷2 𝑢) − Λ|𝐷𝑢| ≤ 0 in 𝐵𝑟 (𝑥0 ), { 𝜆,Λ inf𝐵𝑟 (𝑥0 ) (𝑢 − 𝜑𝑦0 ) ≤ 0 for every 𝑦0 ∈ 𝐵. Then, |𝐴𝐵 | ≥ 𝜂𝑟𝑛 . This previous lemma can be used to bound the distribution of 𝑢 over 𝐵𝑟 (𝑥0 ), in terms of its infimum over 𝐵𝑟/2 (𝑥0 ). Notice first that the particular choice of 𝜑 and 𝐵 implies that for every 𝑦0 ∈ 𝐵 {

𝜑𝑦0 ≥ 𝑝(5𝑟/6) − 𝑝(2𝑟/3) = 1/8 in 𝐵𝑟/2 (𝑥0 ), 𝜑𝑦0 < 0 in ℝ𝑛 ⧵ 𝐵𝑟 (𝑥0 ).

Then, a hypothesis of the form inf𝐵𝑟/2 (𝑥0 ) 𝑢 ≤ 1/8 gives us that inf𝐵𝑟 (𝑥0 ) (𝑢 − 𝜑𝑦0 ) ≤ 0 for every 𝑦0 ∈ 𝐵 (Figure 5). On the other hand, 𝐴𝐵 ⊆ {𝑢 ≤ 1}, hence the conclusion from Lemma 3.3 let us recover a nontrivial upper bound on the density of the set {𝑢 ≥ 1} with respect to 𝐵𝑟 (𝑥0 ). In conclusion, under the hypotheses of the Lemma 3.2, we recover the following preliminary measure estimate: If inf 𝑢 ≤ 1/8,

𝐵𝑟/2 (𝑥0 )

| det(𝐷𝑇)|. 𝐴𝐵

then

The Jacobian is given by 𝐷𝑇(𝑥) = 𝐼 − [𝐷2 𝜑(𝑥 − 𝑇(𝑥))]−1 𝐷2 𝑢(𝑥). Hence, the measure of the contact set 𝐴𝐵 can be compared with the measure of 𝐵, provided some bound on the Hessian of 𝑢 over 𝐴𝐵 . At every contact point 𝑥0 ∈ 𝐴𝐵 , we can use the secondderivative test to bound the eigenvalues of 𝐷2 𝑢(𝑥0 ) from 1608

Thanks to the lower bound on the eigenvalues of 𝐷2 𝑢(𝑥0 ) and the first-derivative test, we get that for any 𝑒 ∈ eig(𝐷2 𝑢(𝑥0 ))

|{𝑢 ≥ 1} ∩ 𝐵𝑟 (𝑥0 )| ≤ (1 − 𝜂|𝐵1 |−1 ). |𝐵𝑟 (𝑥0 )|

4For 𝑀 ∈ ℝ𝑛×𝑛 , we denote |𝑀| ≔ sup op |𝑥|=1 |𝑀𝑥|. If 𝑀 is also symmetric

then we also have that |𝑀|op = max{|𝑒| | 𝑒 ∈ eig(𝑀)}. 5For 𝑒 ∈ ℝ, we denote 𝑒 ≔ max(±𝑒, 0) the positive and negative parts of 𝑒 = ± 𝑒 + − 𝑒− .

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bounded by 𝛾. The problem is to understand the behavior of the solution at the interface between these two regimes. The following interior Hölder estimate due to Imbert and Silvestre gives a positive answer to the previous question [IS16]. Theorem 4.1. Given 𝛾 > 0 there exists 𝛼 ∈ (0, 1) and 𝐶 > 0 such that the following estimate holds: Let 𝑢 ∈ 𝐶 2 (𝐵𝑟 (𝑥0 )) and Ω𝛾 ≔ {|𝐷𝑢| > 𝛾} satisfy

Figure 5. Each paraboloid 𝜑𝑦0 with 𝑦0 ∈ 𝐵𝑟/6 (𝑥0 ) must be crossed by 𝑢 if 𝑢 is less or equal than 1/8 at some point in 𝐵𝑟/2 (𝑥0 ).

𝒫 − (𝐷2 𝑢) − Λ|𝐷𝑢| ≤ 0 in Ω𝛾 ∩ 𝐵𝑟 (𝑥0 ), { 𝜆,Λ + 𝒫𝜆,Λ (𝐷2 𝑢) + Λ|𝐷𝑢| ≥ 0 in Ω𝛾 ∩ 𝐵𝑟 (𝑥0 ). Then, 𝑟𝛼 [𝑢]𝐶 𝛼 (𝐵𝑟/3 (𝑥0 )) ≤ 𝐶 osc 𝑢. 𝐵𝑟 (𝑥0 )

3.2. Summary. Here is a quick summary of the Krylov– Safonov theory revisited in this survey, before moving to the degenerate problems in the next sections: 1. Uniform ellipticity ⇒ Mean value theorem/ABP lemma: For the Laplacian it follows from the divergence theorem. In the general setting, we used instead the change of variable formula over a contact set for the solution. 2. Mean value theorem/ABP lemma ⇒ Weak Harnack: For the Laplacian it is a geometric observation (Figure 2). In general, it follows by an iterative diminish of the distribution. We did not offer any details in this presentation; however, they can be found in [CC95]. 3. Weak Harnack ⇒ Hölder estimate: In either case it follows by the iterative diminish of the oscillation as discussed for harmonic functions.

4. Elliptic Equations That Hold Only Where the Gradient is Large There are numerous scenarios in which the ellipticity parameter of a given operator depends on the gradient of the solution. This is the case for quasi-linear operators of divergence form, which emerged from problems in the calculus of variations. Among the most widely known problems, we find the minimal surface equation and the 𝑝-Laplacian. Another example is the very degenerate equation recently explored in [CLP21] max(1 − |𝐷𝑢|, Δ𝑢 + 1) = 0. This problem arises as the Hamilton–Jacobi equation of a zero-sum game. Notice that in this case, Δ𝑢 = −1 holds in the region where |𝐷𝑢| > 1. These examples raise a natural question. Can we obtain some regularity for the solutions of an elliptic equation for which uniform ellipticity only holds over the set {|𝐷𝑢| > 𝛾}, for some 𝛾 ≥ 0? In compact subsets of {|𝐷𝑢| > 𝛾} we may just invoke the classical estimates, meanwhile in the complementary region {|𝐷𝑢| ≤ 𝛾}, the Lipschitz semi-norm is automatically NOVEMBER 2023

As in the uniformly elliptic setting, the proof relies on the weak Harnack inequality and an ABP-type lemma. The idea consists of using a different family of functions for the ABP lemma, namely 𝜑(𝑥) = −𝐶|𝑥|1/2 . The advantage is that the family of functions from this profile can be arranged such that they only have contact with the solution in the region where the gradient is large and the uniform ellipticity is present. A year later in [Moo15], Mooney offered a second proof of this result which extended the analysis to equations of the form {

− (𝐷2 𝑢) − 𝑏|𝐷𝑢| ≤ 0 in Ω𝛾 ∩ 𝐵𝑟 (𝑥0 ), 𝒫𝜆,Λ + 𝒫𝜆,Λ (𝐷2 𝑢) + 𝑏|𝐷𝑢| ≥ 0 in Ω𝛾 ∩ 𝐵𝑟 (𝑥0 ).

with 𝑏 ∈ 𝐿𝑛 (𝐵𝑟 (𝑥0 )), possibly unbounded. This answered one proposed open problem in [IS16]. Notice that without imposing an equation in the region {|𝐷𝑢| ≤ 𝛾}, we have that arbitrary functions 𝑢 with |𝐷𝑢| ≤ 𝛾, are trivial solutions of these equations. By doing so, we prevent the possibility of deriving any continuity estimate on the gradient. In this sense, the previous theorem is quite optimal in terms of the expected regularity. 4.1. Some further developments. The methods in [IS16, Moo15] have proven to be quite flexible to treat other equations as well. The envelope from [IS16] was employed by Silvestre and Schwab to extend regularity estimates for parabolic integro-differential equations in [SS16]. Pimentel, Santos, and Teixeira also used this idea recently to obtain higher-order fractional estimates in [PST22]. In collaboration with Santos, we revisited the regularity theory for the porous medium equation in [CLS23] by adapting Mooney’s argument to a particular parabolic setting. 4.2. An open problem in the parabolic setting. Analogous estimates as in Theorem 4.1 for parabolic equations remain unknown. Notice that functions that depend only on time, 𝑢 = 𝑢(𝑡), are automatically solutions of 𝜕𝑡 𝑢 = 𝐴 ∶ 𝐷2 𝑢 + 𝑏 ⋅ 𝐷𝑢 in {|𝐷𝑢| > 𝛾}.

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This indicates that the corresponding estimate should only address the continuity of the solution in the spatial variable (𝑥). This problem was originally proposed in [IS16].

5. Small Perturbations Caffarelli developed in [Caf89] a perturbative approach to higher regularity estimates for solutions of uniformly elliptic equations, sometimes referred to as regularity by compactness or the improvement of flatness. By flatness we mean that a solution is uniformly close to a prescribed profile. The general strategy was inspired by De Giorgi’s regularity theorem for minimal surfaces. The idea can be roughly described by saying that if the solution is uniformly close to a smooth solution, then it inherits the estimates of the corresponding linearization. This approach quickly provided alternative proofs to the regularity estimates for some of the canonical degenerate equations. Caffarelli and Cordoba treated in [CC93] the minimal surface equation, while Wang studied in [Wan94] the estimates for the 𝑝-Laplace equation. In [Sav07], Savin demonstrated that these estimates could be extended to allow operators 𝐹 = 𝐹(𝑀, 𝑝) which are only required to be uniformly elliptic in a neighborhood of a given profile (𝑀0 , 𝑝0 ) ∈ {𝐹 = 0}. In particular, his result allows us to treat equations that become degenerate as (𝑀, 𝑝) is large, complementing the ideas in the previous section. In this case, the alternative mechanism that supplements the equation is a flatness hypothesis on the solution. For simplicity, we state the following result for (𝑀0 , 𝑝0 ) = (0, 0) and over the unit ball. Theorem 5.1. Given 𝛾 > 0 and 𝛼 ∈ (0, 1), there exist 𝛿0 , 𝐶 > 0 such that the following estimate holds: 𝑛 Let 𝐹 = 𝐹(𝑀, 𝑝) ∈ 𝐶 2 (ℝ𝑛×𝑛 sym × ℝ ) satisfy ⎧𝜆𝐼 ≤ 𝐷𝑀 𝐹 ≤ Λ𝐼, |𝐷2 𝐹| ≤ Λ, ⎨ ⎩𝐹(0, 0) = 0.

(5.1)

Let 𝑢 ∈ 𝐶 2 (𝐵1 ) and Ω𝛾 = {|𝐷2 𝑢| + |𝐷𝑢| ≤ 𝛾} satisfy 𝐹(𝐷2 𝑢, 𝐷𝑢) = 0 in Ω𝛾 ∩ 𝐵1 , { ‖𝑢‖𝐿∞ (𝐵1 ) ≤ 𝛿0 .

.

1. Given 𝑥0 ∈ 𝐵1/2 , the goal is to build a multiple scale approximation of 𝑢 around 𝑥0 of the following form and for some 𝜇 ∈ (0, 1/2) to be chosen sufficiently small 2+𝛼 𝑃 (𝜇−1 𝑥) + … + 𝜇(2+𝛼)𝑘 𝑃 (𝜇−𝑘 𝑥) + … 𝑢(𝑥) = 𝑃 0 (𝑥) + 𝜇 1 𝑘 ⏟⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⏟⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⎵⏟ ∶=𝑄𝑘 (𝑥)

We require that this approximation satisfies: (1) 𝑃0 = 𝑄0 = 0, (2) 𝑃𝑖 are quadratic polynomials with ‖𝑃𝑖 ‖𝐿∞ (𝐵1 ) ≤ 1, (3) ‖𝑢 − 𝑄 𝑘 ‖𝐿∞ (𝐵 𝑘 (𝑥0 )) ≤ 𝜇(2+𝛼)𝑘 , 𝜇

(4) Each 𝑄 𝑘 satisfies 𝐹(𝐷2 𝑄, 𝐷𝑄(𝑥0 )) = 0. From the second item we obtain that ‖𝑄 𝑘+1 − 𝑄 𝑘 ‖𝐿∞ (𝐵1 ) ≤ 𝐶𝜇𝛼𝑘 . Hence, 𝑄∗ ≔ lim 𝑄 𝑘 is well defined. Combining now the third item we get ‖𝑢 − 𝑄∗ ‖𝐿∞ (𝐵

(𝑥0 )) 𝜇𝑘

≤ 𝐶𝜇(2+𝛼)𝑘 .

This is an equivalent way to state the desired 𝐶 2,𝛼 estimate (Figure 6). 2. The goal now is to find a suitable correction 𝜇(2+𝛼)(𝑘+1) 𝑃𝑘+1 (𝜇−(𝑘+1) 𝑥) to the quadratic polynomial 𝑄 𝑘 approximating 𝑢 over 𝐵𝜇𝑘 (𝑥0 ). This correction must satisfy all the items above. However, in this sketch, we will mainly focus on the error estimate given by the third item. Assume that all the hypotheses are satisfied up to some scale 𝑟 ≔ 𝜇𝑘 , for some 𝑘 ≥ 1, now fixed. Then we consider the small perturbation 𝑣 ∈ 𝐶 2 (𝐵1 ) such that for 𝑄 = 𝑄 𝑘 , 𝑢(𝑥) = 𝑄(𝑥) + 𝑟2+𝛼 𝑣(𝑟−1 (𝑥 − 𝑥0 )).

Then 𝑢 ∈ 𝐶 2,𝛼 (𝐵1/2 ) with ‖𝑢‖𝐶 2,𝛼 (𝐵1/2 ) ≤ 𝐶‖𝑢‖𝐿∞ (𝐵1 ) . The proof of the previous theorem provides another important use of the compactness property derived from the Krylov–Safonov estimate. For this reason, we would like to offer a sketch of the argument, at least in the uniformly elliptic case, that is 𝛾 = ∞ and Ω𝛾 = 𝐵1 . Later on, we will give a few comments on the degenerate case 𝛾 < ∞. The interested reader may refer to [Sav07] for a complete proof. 1610

Figure 6. Improvement of flatness: 𝑢 ∼ 𝑄 𝑘 ∼ 𝑄∗ , both with errors of order 𝜇(2+𝛼)𝑘 over 𝐵𝜇𝑘 (𝑥0 ); then also 𝑢 ∼ 𝑄∗ with the same order of approximation.

Notice that by the inductive hypothesis ‖𝑣‖𝐿∞ (𝐵1 ) ≤ 1. Under the uniform ellipticity hypothesis we get that 𝑣 satisfies 𝐺(𝐷2 𝑣, 𝐷𝑣) = 0 in 𝐵1 , where 𝐺(𝑀, 𝑝) ≔ 𝑟−𝛼 𝐹(𝑟𝛼 𝑀 + 𝐷2 𝑄, 𝑟1+𝛼 𝑝 + 𝐷𝑄(𝑥0 )) satisfies the same hypotheses as 𝐹 in (5.1). Therefore, 𝑣 has an interior Hölder estimate that depends exclusively on the parameters of uniform ellipticity and the dimension.

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3. If we assume by contradiction that the correcting polynomial can not be found for any 𝜇 → 0+ , we then extract a pair of converging sequences 𝑣 𝑟 → 𝑣 0 and 𝑄𝑟 → 𝑄0 , locally uniformly in 𝐵1 . Using that 𝐹(𝐷2 𝑄𝑟 , 𝐷𝑄𝑟 ) = 0 and |𝐷2 𝐹| ≤ Λ, we also get that the sequence 𝑄𝑟 determines a converging sequence of operators 𝐺𝑟 (𝑀, 𝑝) ≔ 𝑟−𝛼 𝐹(𝑟𝛼 𝑀 + 𝐷2 𝑄𝑟 , 𝑟1+𝛼 𝑝 + 𝐷𝑄𝑟 (𝑥0 )) = 𝐹(𝑟𝛼 𝑀 + 𝐷2 𝑄𝑟 , 𝑟1+𝛼 𝑝 + 𝐷𝑄𝑟 (𝑥0 )) − 𝐹(𝐷2 𝑄0 , 𝐷𝑄0 (𝑥0 )) , 𝑟𝛼 such that 𝐺𝑟 → 𝐺 0 = 𝐺 0 (𝑀), the linear operator with constant coefficients given by 𝐺 0 (𝑀) ≔ 𝐷𝑀 𝐹(𝐷2 𝑄0 , 𝐷𝑄0 (𝑥0 )) ∶ 𝑀. It turns out that the limit function 𝑣 0 also satisfies the linear equation 𝐺 0 (𝐷2 𝑣 0 ) = 0. 4. As a final step, we notice that by the 𝐶 3 estimates for linear equations with constant coefficients, 𝑣 0 can be approximated by a quadratic polynomial around the origin. This leads to a contradiction of the assumed fact that the corrections did not exist for any small value of 𝜇. □ For the degenerate case when 𝛾 < ∞, Savin’s remarkable observation is that under the flatness hypothesis, it is possible to reconstruct most of the Krylov–Safonov regularity theory. However, there is a caveat related to the diminish of oscillation argument. As one rescales the equation, the degeneracy becomes more and more pervasive and the argument leading to the improvement on the oscillation eventually breaks down. This means that from an initial 𝛿-flatness hypothesis on the solution, with 𝛿 ∈ (0, 𝛿0 ), one can only get a truncated modulus of continuity for the solution. Nevertheless, the radius on the truncation also vanishes as the flatness parameter 𝛿 goes to zero. This means that the compactness of solutions in the previous argument still holds by a Cantor diagonal argument. 5.1. Some further developments. Within the scope of this presentation, it is not possible to cite the numerous articles that rely on these techniques. The original idea was developed to answer a celebrated conjecture by De Giorgi about level sets in Ginzburg–Landau phase transition models in [Sav09]. The approach has been extended by De Silva to establish regularity estimates for the Bernoulli free boundary problem starting in [DS11]. Regularity estimates for nonlocal minimal surfaces were established by Caffarelli, Roquejoffre, and Savin in [CRS10]. Armstrong, Silvestre, and Smart also utilized this approach to develop partial regularity results for fully nonlinear equations in [ASS12]. Colombo and Figalli developed regularity estimates for degenerate equations from traffic congestion models in [CF14]. Finally, in collaboration with Pimentel, we demonstrated in [CLP21] the continuity NOVEMBER 2023

of |𝐷𝑢|, where 𝑢 solves the gradient-constrained problem max(1 − |𝐷𝑢|, Δ𝑢 + 1) = 0.

6. Quasi-Harnack Degeneracy can also manifest itself across scales. For example, when modeling a PDE using finite difference schemes, the continuous formulation of uniform ellipticity breaks down at the level of the discretization. However, if the numerical scheme approximates a uniformly elliptic equation, we expect that the discrete solution will approximate the continuous solution over large scales, inheriting with it the classical manifestations of uniform ellipticity. A recent work by De Silva and Savin in [DSS21] proposes a weak notion of solution for equations where the uniform ellipticity manifests from a given scale onward. The next definition of solutions relies on the following geometric configuration: Given 𝑢 ∈ 𝐶(Ω), and 𝜑 ∈ 𝐶(𝐵𝑟 (𝑥0 )) with 𝐵𝑟 (𝑥0 ) ⊆ Ω, we say that 𝜑 touches 𝑢 from below (above) at 𝑥0 and over 𝐵𝑟 (𝑥0 ) if {

𝜑 ≤ (≥) 𝑢 in 𝐵𝑟 (𝑥0 ), 𝜑(𝑥0 ) = 𝑢(𝑥0 ).

Definition 6.1. Let 𝐹 = 𝐹(𝑀) ∈ 𝐶(ℝ𝑛×𝑛 sym ) be nondecreasing. We say that 𝑢 ∈ 𝐶(Ω) satisfies 𝐹(𝐷2 𝑢) ≤𝑟 (≥𝑟 ) 0 in Ω if for every quadratic polynomial 𝜑 that touches 𝑢 from below at 𝑥0 and over 𝐵𝑟 (𝑥0 ) ⊆ Ω, it holds that 𝐹(𝐷2 𝜑) ≤ (≥) 0. The equality =𝑟 holds when both inequalities are simultaneously satisfied. This definition is consistent with the classical notion of the inequality 𝐹(𝐷2 𝑢) ≤ 0 for 𝑢 ∈ 𝐶 2 (Ω). This follows by the second-derivative test and the monotonicity hypothesis on 𝐹. Notice also that if 𝑟1 < 𝑟2 and 𝐹(𝐷2 𝑢) ≤𝑟1 0, then also 𝐹(𝐷2 𝑢) ≤𝑟2 0, meaning that this notion of solution is more relaxed as 𝑟 becomes larger. To give a concrete example, let us consider a twodimensional numerical approximation of the Laplacian over the two-dimensional lattice 𝜀ℤ2 . In this scenario, we will work with a continuous function 𝑢, but the relevant values will be given on the lattice as 𝑢𝑖𝑗 ≔ 𝑢(𝜀𝑖, 𝜀𝑗) for (𝑖, 𝑗) ∈ ℤ2 . We can then extend 𝑢 to each square 𝑄 𝑖𝑗 ≔ [𝜀𝑖, 𝜀(𝑖 + 1)) × [𝜀𝑗, 𝜀(𝑗 + 1)) in a continuous manner, ensuring that the maximum and minimum of 𝑢 over 𝑄 𝑖𝑗 are reached at the corners of 𝑄 𝑖𝑗 (Figure 7). A classical discretization of the Laplace equation over the lattice is formulated by 𝑢𝑖𝑗 =

𝑢𝑖−1,𝑗 + 𝑢𝑖+1,𝑗 + 𝑢𝑖,𝑗−1 + 𝑢𝑖,𝑗+1 . 4

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(6.1)

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Assume without loss of generality that 𝜆1 ≤ 𝜆2 and that the angle subtended by the vectors 𝜉2 and 𝑒 2 = (0, 1) is between −𝜋/4 and 𝜋/4. By hypothesis, 2 − 𝒫1,Λ (𝐷2 𝜑) = ∑ (𝜆𝑖 )+ − Λ(𝜆𝑖 )− > 0, 𝑖=1

so that 𝜆2 > 0 and 𝜆1 ∈ (−𝜆2 /Λ, 𝜆2 ]. 2. We get in this way that for 𝜆 ≔ 𝜆2 /Λ, ̄ 𝑃(𝑥) ≔ −(𝜉1 ⋅ (𝑥 − 𝑥0 ))2 + Λ(𝜉2 ⋅ (𝑥 − 𝑥0 ))2 , Figure 7. An interpolation of a discrete function such that over each square 𝑄 𝑖𝑗 , the extremal values are attained over the corners.

In particular, it follows from a straightforward computation, that a quadratic polynomial is harmonic if and only if satisfies (6.1) at one point. Assume 𝑢 satisfies (6.1) for (𝑖, 𝑗) ∈ 𝜀−1 Ω ∩ ℤ2 , and consider a quadratic polynomial 𝜑 touching 𝑢 from below at 𝑥0 and over 𝐵𝑟 (𝑥0 ). If 𝑟 < 𝜀, it is not difficult to come up with interpolations for 𝑢 for which 𝐷2 𝜑 could be arbitrary. On the other hand, if 𝑟 ≥ 𝜀 and 𝑥0 = 𝜀(𝑖0 , 𝑗0 ) ∈ 𝜀ℤ2 is a lattice point, then 𝜑𝑖0 ,𝑗0 = 𝑢𝑖0 ,𝑗0 , 𝑢𝑖 −1,𝑗0 + 𝑢𝑖0 +1,𝑗0 + 𝑢𝑖0 ,𝑗0 −1 + 𝑢𝑖0 ,𝑗0 +1 = 0 , 4 𝜑𝑖 −1,𝑗0 + 𝜑𝑖0 +1,𝑗0 + 𝜑𝑖0 ,𝑗0 −1 + 𝜑𝑖0 ,𝑗0 +1 ≥ 0 . 4 This implies that Δ𝜑 ≤ 0, as would be required by the definition of Δ𝑢 ≤𝑟 0. Would it be possible to get a similar result in the general case, when 𝑥0 is not necessarily a lattice point? The answer to this question is affirmative. However, it is necessary to modify the operator at hand. Indeed, let us see that if 𝑢𝑖𝑗 satisfies (6.1) for (𝑖, 𝑗) ∈ 𝜀−1 Ω ∩ ℤ2 , then 𝑢 satisfies 𝒫 − (𝐷2 𝑢) ≤3𝜀 0 in Ω, { 1,Λ + 𝒫1,Λ (𝐷2 𝑢) ≥3𝜀 0 in Ω.

(6.2)

The parameter Λ will be conveniently fixed as a large constant by the end of the argument. − We will show the inequality for 𝒫1,Λ , as the one corre+ sponding to 𝒫1,Λ has a similar analysis. 1. Let 𝜑 be a quadratic polynomial touching 𝑢 from below at 𝑥0 over 𝐵3𝜀 (𝑥0 ) ⊆ Ω. Assume by contradiction − that 𝒫1,Λ (𝐷2 𝜑) > 0. This test function can be written as ⎧𝜑 = 𝑃 + 𝐿 𝑃(𝑥) = 𝜆1 (𝜉1 ⋅ (𝑥 − 𝑥0 ))2 + 𝜆2 (𝜉2 ⋅ (𝑥 − 𝑥0 ))2 ⎨ ⎩𝐿 is affine, with {𝜉1 , 𝜉2 } orthonormal. 1612

≤ 𝑃(𝑥)/𝜆, touches (𝑢 − 𝐿)/𝜆 from below at 𝑥0 and over 𝐵3𝜀 (𝑥0 ). By computing the infimum of 𝑃 ̄ over 𝐵3𝜀 (𝑥0 ), we get that (𝑢 − 𝐿) ≥ −9𝜆𝜀2 in 𝐵3𝜀 (𝑥0 ).

(6.3)

3. Let (𝑖0 , 𝑗0 ) ∈ ℤ2 such that 𝑥0 ∈ 𝑄 𝑖0 𝑗0 . We will see now that in at least one corner 𝑦 = (𝜀𝑖1 , 𝜀𝑗1 ) ∈ 𝜕𝑄 𝑖0 𝑗0 ∩ 𝜀ℤ2 we must have (𝑢 − 𝐿)(𝑦) ≥ 𝑐𝜆2 𝜀2 − 9𝜆𝜀2 ,

(6.4)

for some constant 𝑐 > 0 to be fixed. By considering 𝑦− ≔ (𝜀𝑖0 , 𝜀𝑗0 ), 𝑦+ ≔ (𝜀𝑖0 , 𝜀(𝑗0 + 1)), and the angle assumption on 𝜉2 , we get that |𝜉2 ⋅ (𝑦− − 𝑥0 ))| + |𝜉2 ⋅ (𝑦+ − 𝑥0 )| ≥ 𝜀(𝜉2 ⋅ 𝑒 2 ) ≥ 𝜀/√2. Hence, in at least one of these two corners we must have that |𝜉2 ⋅(𝑦−𝑥0 ))| ≥ 𝜀/(2√2), and the desired bound follows for 𝑐 = 1/8 by using that (𝑢 − 𝐿) ≥ 𝜆𝑃.̄ 4. By the contact given by (𝑢 − 𝐿) and 𝜆𝑃 ̄ at 𝑥0 we get ̄ 0 ) = 0. The way in which we conthat (𝑢 − 𝐿)(𝑥0 ) = 𝜆𝑃(𝑥 sidered the continuous extension of 𝑢 also implies that the minimum of (𝑢 − 𝐿) over the four corners of 𝑄 𝑖0 𝑗0 must be nonpositive. In this final step, we will see how to get a contradiction from this fact, together with (6.3) and (6.4). The choice on the scale 𝑟 = 3𝜀 was made such that the nine closed squares of the form 𝑄 𝑖𝑗 with |𝑖 − 𝑖0 | ≤ 1 and |𝑗 − 𝑗0 | ≤ 1 are also contained in 𝐵3𝜀 (𝑥0 ) (Figure 8). Let 𝑄 ≔ (𝑖0 −1, 𝑖0 +2)×(𝑗0 −1, 𝑗0 +2) and 𝑤 𝑖𝑗 be defined for (𝑖, 𝑗) ∈ 𝑄 ∩ ℤ2 such that the following holds: For the three interior nodes (𝑖, 𝑗) ∈ (𝑄 ∩ ℤ2 ) ⧵ {(𝑖1 , 𝑗1 )} 𝑤 𝑖−1,𝑗 + 𝑤 𝑖+1,𝑗 + 𝑤 𝑖,𝑗−1 + 𝑤 𝑖,𝑗+1 𝑤 𝑖𝑗 = . 4 Meanwhile, it also satisfies the boundary conditions {

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Besides the already discussed applications to numerical schemes, the previous theorem can also be applied in the homogenization of elliptic problems with degeneracies, as was studied in [AS14]. Finally, it was shown in [DSS21] that integro-differential uniformly elliptic equations of order 𝜎 close to two, also fit in the framework of the previous theorem. In this way, it provides a new proof to the Harnack inequality of Caffarelli and Silvestre [CS09].

7. Concluding Remarks Figure 8. The ball 𝐵3𝜀 (𝑥0 ) contains the 9 squares surrounding the square 𝑄 𝑖0 𝑗0 in which falls the center of the ball.

A simple computation determines that 𝑤 𝑖𝑗 = 2/7 in the interior nodes of 𝑄 ∩ ℤ2 adjacent to (𝑖1 , 𝑗1 ), and 𝑤 𝑖𝑗 = 1/7 in the opposite one. By the discrete comparison principle we get that for any of the four interior nodes (𝑖, 𝑗) ∈ 𝑄 ∩ ℤ2 (𝑢 − 𝐿 + 9𝜆𝜀2 )𝑖𝑗 ≥ 𝜆2 𝜀2 𝑤 𝑖𝑗 /8, ≥ 𝜆2 𝜀2 /48. Nevertheless, this contradicts that the minimum of (𝑢 − 𝐿) over these interior nodes is nonpositive once we choose Λ > 432. □ In the recent article of De Silva and Savin [DSS21], the authors get a weak Harnack inequality for the general degenerate problem (6.2). The alternative mechanism used in this case is a measure estimate over any ball of radius 𝜌 = 𝐶𝑟. We say that 𝑢 ∈ 𝐶(Ω) satisfies the measure estimate with respect to the parameters 𝜌, 𝑀 > 0 and 𝛿 ∈ (0, 1), if for every 𝐵𝜌 (𝑥0 ) ⊆ Ω, and 𝜇 > 0, we get that 𝑢 − inf 𝑢 ≤ 𝜇 in 𝐵𝜌/2 (𝑥0 ), implies |𝐵𝜌/2 (𝑥0 )|

≤ 𝛿.

Theorem 6.1. There exists a small fraction 𝛿 ∈ (0, 1) depending on the dimension for which the following statement is true: Given Λ, 𝑀 ≥ 1, there exist 𝑟0 , 𝐶, 𝑐 > 0 and 𝛼 ∈ (0, 1) such that the following holds: Let 𝑢 ∈ 𝐶(𝐵1 ) satisfy for some 𝑟 ∈ (0, 𝑟0 ) {

− 𝒫1,Λ (𝐷2 𝑢) ≤𝑟 0 in 𝐵1 , + 𝒫1,Λ (𝐷2 𝑢) ≥𝑟 0 in 𝐵1 ,

and ±𝑢 satisfy the measure estimate with respect to 𝐶𝑟, 𝑀, and 𝛿. Then 𝑢 has a truncated Hölder estimate of the form sup 𝑥0 ∈𝐵1/2 𝜌∈(𝑐𝑟,1/2)

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𝜌−𝛼 osc 𝑢 ≤ 𝐶‖𝑢‖𝐿∞ (𝐵1 ) . 𝐵𝜌 (𝑥0 )

ACKNOWLEDGMENT. The author would like to thank his collaborators N´estor Guill´en, Edgard Pimentel, and Alberto Saldana; ˜ his colleagues at CIMAT, Octavio Arizmendi and Luis Nu´ nez; ˜ and the anonymous referees for their helpful feedback on this manuscript. The author was supported by CONACyT-MEXICO grant A1-S48577. References

𝐵𝜌 (𝑥0 )

|{𝑢 − inf𝐵𝜌 (𝑥0 ) 𝑢 ≥ 𝑀𝜇} ∩ 𝐵𝜌/2 (𝑥0 )|

In this note, we have revisited the regularity theory of uniformly elliptic equations under the perspective of degenerate ellipticity. It is our hope to have conveyed some of the beautiful geometric insights of the theory. While we did not go deeper into the models that have brought up these particular degeneracies, it is important to emphasize that a careful understanding of such natural phenomena has been instrumental in the analysis of the solutions. It can be easily the topic of just one survey to uncover each one of these models in detail, such as minimal surfaces, the 𝑝-Laplacian, the porous medium equation, etc. For the same reason, many important references have been unfortunately left out. This article is dedicated to Luis Caffarelli with gratitude and admiration.

[AS14] Scott N. Armstrong and Charles K. Smart, Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity, Ann. Probab. 42 (2014), no. 6, 2558– 2594, DOI 10.1214/13-AOP833. MR3265174 [ASS12] Scott N. Armstrong, Luis E. Silvestre, and Charles K. Smart, Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Comm. Pure Appl. Math. 65 (2012), no. 8, 1169–1184, DOI 10.1002/cpa.21394. MR2928094 [Cab97] Xavier Cabr´e, Nondivergent elliptic equations on manifolds with nonnegative curvature, Comm. Pure Appl. Math. 50 (1997), no. 7, 623–665, DOI 10.1002/(SICI)10970312(199707)50:73.3.CO;2-B. MR1447056 [Caf89] Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213, DOI 10.2307/1971480. MR1005611 [CC93] Luis A. Caffarelli and Antonio Cordoba, ´ An elementary regularity theory of minimal surfaces, Differential Integral Equations 6 (1993), no. 1, 1–13. MR1190161

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[CC95] Luis A. Caffarelli and Xavier Cabr´e, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995, DOI 10.1090/coll/043. MR1351007 [CF14] Maria Colombo and Alessio Figalli, Regularity results for very degenerate elliptic equations (English, with English and French summaries), J. Math. Pures Appl. (9) 101 (2014), no. 1, 94–117, DOI 10.1016/j.matpur.2013.05.005. MR3133426 [CLP21] H´ector A. Chang-Lara and Edgard A. Pimentel, Nonconvex Hamilton-Jacobi equations with gradient constraints, Nonlinear Anal. 210 (2021), Paper No. 112362, 17, DOI 10.1016/j.na.2021.112362. MR4249793 [CLS23] H´ector A. Chang-Lara and Makson S. Santos, Hölder regularity for non-variational porous media type equations, J. Differential Equations 360 (2023), 347–372, DOI 10.1016/j.jde.2023.02.055. MR4557323 [CRS10] L. Caffarelli, J.-M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144, DOI 10.1002/cpa.20331. MR2675483 [CS09] Luis Caffarelli and Luis Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), no. 5, 597–638, DOI 10.1002/cpa.20274. MR2494809 [DS11] D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound. 13 (2011), no. 2, 223–238, DOI 10.4171/IFB/255. MR2813524 [DSS21] D. De Silva and O. Savin, Quasi-Harnack inequality, Amer. J. Math. 143 (2021), no. 1, 307–331, DOI 10.1353/ajm.2021.0001. MR4201786 [IS16] Cyril Imbert and Luis Silvestre, Estimates on elliptic equations that hold only where the gradient is large, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 6, 1321–1338, DOI 10.4171/JEMS/614. MR3500837 [KS79] N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure (Russian), Dokl. Akad. Nauk SSSR 245 (1979), no. 1, 18–20. MR525227 [Moo15] Connor Mooney, Harnack inequality for degenerate and singular elliptic equations with unbounded drift, J. Differential Equations 258 (2015), no. 5, 1577–1591, DOI 10.1016/j.jde.2014.11.006. MR3295593 [PST22] Edgard A. Pimentel, Makson S. Santos, and Eduardo V. Teixeira, Fractional Sobolev regularity for fully nonlinear elliptic equations, Comm. Partial Differential Equations 47 (2022), no. 8, 1539–1558, DOI 10.1080/03605302.2022.2059676. MR4462186 [Sav07] Ovidiu Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations 32 (2007), no. 4-6, 557–578, DOI 10.1080/03605300500394405. MR2334822 [Sav09] Ovidiu Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2) 169 (2009), no. 1, 41–78, DOI 10.4007/annals.2009.169.41. MR2480601 [SS16] Russell W. Schwab and Luis Silvestre, Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE 9 (2016), no. 3, 727–772, DOI 10.2140/apde.2016.9.727. MR3518535

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Hector ´ A. Chang-Lara Credits

All figures and the opener are courtesy of H´ector A. ChangLara. Photo of H´ector A. Chang-Lara is courtesy of CIMAT.

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Threshold Phenomena for Random Discrete Structures

Jinyoung Park 1. Erd˝ os–Renyi ´ Model To begin, we briefly introduce a model of random graphs. Recall that a graph is a mathematical structure that consists of vertices (nodes) and edges. Jinyoung Park is an assistant professor of mathematics at Courant Institute of Mathematical Sciences, NYU. Her email address is [email protected]. Her research is supported by NSF grant DMS-2153844.

Figure 1. A graph.

Communicated by Notices Associate Editor Emilie Purvine. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2802

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Roughly speaking, a random graph in this article means that, given a vertex set, the existence of each potential edge is decided at random. We will specifically focus on the NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

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Erd˝os–R´enyi random graph (denoted by 𝐺𝑛,𝑝 ), which is defined as follows. Consider 𝑛 vertices that are labelled from 1 to 𝑛.

Figure 3. 𝐺 3,0.01 .

Observe that on those 𝑛 vertices, there are potentially (𝑛) edges, that is, the edges labelled {1, 2}, {1, 3}, … , {𝑛−1, 𝑛}. 2

Given a probability 𝑝 ∈ [0, 1], include each of the (𝑛) po2 tential edges with probability 𝑝, where the choice of each edge is made independently from the choices of the other edges. Example 1.1. As a toy example of the Erdos–R´ ˝ enyi random graph, let’s think about what 𝐺𝑛,𝑝 looks like when 𝑛 = 3 and the value of 𝑝 varies. First, if 𝑝 = 1/2, then 𝐺𝑛,𝑝 has the probability distribution as in Figure 2, defined on the collection of eight graphs. Observe that each graph is equally likely (since each potential edge is present with probability 1/2 independently).

and 𝑝. Assuming 𝑛 is given, the structure of 𝐺𝑛,𝑝 changes as the value of 𝑝 changes, and in order to understand 𝐺𝑛,𝑝 , we ask questions about the structure of 𝐺𝑛,𝑝 such as What’s the probability that 𝐺𝑛,𝑝 is connected? or What’s the probability that 𝐺𝑛,𝑝 is planar? Basically, for any property ℱ(= ℱ𝑛 ) of interest, we can ask What’s the probability that 𝐺𝑛,𝑝 satisfies property ℱ? In those questions, usually we are interested in understanding the typical structure/behavior of 𝐺𝑛,𝑝 . Observe that, unless 𝑝 = 0 or 1, there is always a positive probability that all of the edges in 𝐺𝑛,𝑝 are absent, or all of them are present (see Examples 1.2, 1.3). But in this article, we would rather ignore such extreme events that happen with a tiny probability, and focus on properties that 𝐺𝑛,𝑝 possesses with a probability close to 1. We often use languages and tools from probability theory to describe/understand behaviors of 𝐺𝑛,𝑝 . Below we discuss some very basic examples. 𝑓(𝑛) We will write 𝑓(𝑛) ≪ 𝑔(𝑛) if → 0 as 𝑛 → ∞. 𝑔(𝑛)

Example 1.2. One important object in graph theory is the complete graph, a graph with all the potential edges present. The complete graph on 𝑛 vertices is denoted by 𝐾𝑛 .

Figure 2. 𝐺 3,1/2 .

Of course, we will have a different probability distribution if we change the value of 𝑝. For example, if 𝑝 is closer to 0, say 0.01, then 𝐺𝑛,𝑝 has the distribution as in Figure 3, where sparser graphs are more likely (as expected). On the other hand, if 𝑝 is closer to 1, then denser graphs will be more likely. In reality, when we consider 𝐺𝑛,𝑝 , 𝑛 is a large (yet finite) number that tends to infinity, and 𝑝 = 𝑝(𝑛) is usually a function of 𝑛 that tends to zero as 𝑛 → ∞. For example, 𝑝 = 1/𝑛, 𝑝 = log 𝑛/𝑛, etc. As we saw in Example 1.1, a random graph is a random variable with a certain probability distribution (as opposed to a fixed graph) that depends on the values of 𝑛 1616

We can easily imagine that, unless 𝑝 is very close to 1, it is extremely unlikely that 𝐺𝑛,𝑝 is complete. Indeed, ℙ(𝐺𝑛,𝑝 = 𝐾𝑛 ) = 𝑝

(𝑛) 2

(since we want all the edges present), which tends to 0 unless 1 − 𝑝 is of order at most 𝑛−2 . Example 1.3. Similarly, we can compute the probability that 𝐺𝑛,𝑝 is “empty” (let’s denote this by ∅) meaning that

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no edges are present.1 The probability for this event is ℙ(𝐺𝑛,𝑝 = ∅) = (1 − 𝑝)

(𝑛) 2

.

When 𝑝 is small, 1 − 𝑝 is approximately 𝑒−𝑝 , so the above computation tells us that ℙ(𝐺𝑛,𝑝

0 = ∅) → { 1

if 𝑝 ≫ 1/𝑛2 ; if 𝑝 ≪ 1/𝑛2 .

Example 1.4. How many edges does 𝐺𝑛,𝑝 typically have? The natural first step to answer this question is computing the expected number of edges in 𝐺𝑛,𝑝 . Using linearity of expectation,

in 𝐺𝑛,𝑝 is a very challenging task, and this became a motivation for the Kahn–Kalai conjecture that we will discuss in the latter sections.

2. Threshold Phenomena One striking thing about 𝐺𝑛,𝑝 is that appearance and disappearance of certain properties are abrupt. Probably one of the most well-known examples that exhibit threshold phenomena of 𝐺𝑛,𝑝 is the appearance of the giant component. A component of a graph is a maximal connected subgraph. For example, the graph in Figure 4 consists of four components, and the size (the number of vertices) of each component is 1, 2, 6, and 8.

𝔼[number of edges in 𝐺𝑛,𝑝 ] = ∑ ℙ(edge {𝑖, 𝑗} is present in 𝐺𝑛,𝑝 ) 𝑖 1 − 𝜀). 1

So around the value 𝑝 = , the giant component “sud𝑛 denly” appears, and therefore the structure of 𝐺𝑛,𝑝 also drasitically changes. This is one example of the threshold phe1 nomena that 𝐺𝑛,𝑝 exhibits, and the value 𝑝 = is a thresh𝑛 old function for 𝐺𝑛,𝑝 of having the giant component. (The formal definition of a threshold function is given in Definition 2.3. See also the definition of the threshold in Section 5.) The abrupt appearance of the giant component of 𝐺𝑛,𝑝 is just one instance of vast threshold phenomena for random discrete structures. In this article, we will mostly deal with 𝐺𝑛,𝑝 for the sake of concreteness, but there will be a brief discussion about a more general setting in Section 5. Now we introduce the formal definition of a threshold function due to Erdos ˝ and R´enyi. Recall that, in

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Now it immediately follows from the above theorem that all the properties that we have mentioned so far— connectivity, planarity,3 having the giant component, etc.—have a threshold function (thus exhibit a threshold phenomenon). How fascinating it is! On the other hand, knowing that a property ℱ has a threshold function 𝑝0 = 𝑝0 (ℱ) does not tell us anything about the value of 𝑝0 . So it naturally became a central interest in the study of random graphs to find a threshold function for various increasing properties. One of the most studied classes of increasing properties is subgraph containment, i.e., the question of for what 𝑝 = 𝑝(𝑛), 𝐺𝑛,𝑝 is likely/unlikely to contain a copy of the given graph. Figure 8 shows some of the well-known threshold functions for various subgraph containments (and that for connectivity).

2By the definition, a threshold function is determined up to a constant factor

thus not unique, but conventionally people also call this the threshold function. In this article, we will separately define the threshold in Section 5, which is distinguished from a threshold function. 3We can apply the theorem for nonplanarity, which is an increasing property.

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the difficulty of this problem immensely changes if we require 𝐺𝑛,𝑝 to contain a specific (up to isomorphisms) spanning tree (or more broadly, a spanning graph.5) For example, one of the biggest open questions in this area back in 1960s was finding a threshold function for a Hamiltonian cycle (a cycle that contains all of the vertices).

Figure 8. Some well-known thresholds. 1

Example 2.5. Figure 8 says that 𝑝 = is a threshold func𝑛 tion for the property ℱ = {contains a triangle}. Recall from the definition of a threshold that this means (i) if 𝑝 ≪ (ii) if 𝑝 ≫

1 𝑛 1 𝑛

then ℙ(𝐺𝑛,𝑝 contains a triangle) → 0; and then ℙ(𝐺𝑛,𝑝 contains a triangle) → 1.

We have already justified (i) in Example 1.2 by showing that 1 𝔼[number of triangles in 𝐺𝑛,𝑝 ] → 0 if 𝑝 ≪ . 𝑛 However, showing (ii) is an entirely different story. As discussed in Remark 1.7, the fact that

Figure 10. A graph and a Hamiltonian cycle in it.

This problem was famously solved by Posa ´ in 1976. Theorem 2.6 (Posa ´ [16]). A threshold function for 𝐺𝑛,𝑝 to contain a Hamiltonian cycle is 𝑝0 (𝑛) =

𝔼[number of triangles in 𝐺𝑛,𝑝 ] → ∞ does not necessarily imply that 𝐺𝑛,𝑝 typically contains many triangles. Here we briefly describe one technique, which is called the second moment method, that we can use to show (ii): let 𝑋 be the number of triangles in 𝐺𝑛,𝑝 , noting that then 𝑋 is a random variable. By showing that the variance of 𝑋 is very small, which implies that 𝑋 is “concentrated around” 𝔼𝑋, we can derive (from the fact that 𝔼𝑋 is huge) that typically the number of triangles in 𝐺𝑛,𝑝 is huge. We remark that the second moment method is only a tiny part of the much broader topic of concentration of a probability measure. We stress that, in general, finding a threshold function for a given increasing property is a very hard task. To illustrate this point, let’s consider one of the most basic objects in graph theory, a spanning tree—a tree that contains all of the vertices.

Figure 9. A connected graph and a spanning tree in it.

The question of finding a threshold function for 𝐺𝑛,𝑝 of containing a spanning tree4 was one of the first questions studied by Erdos ˝ and R´enyi. Already in their seminal paper [6], Erdos ˝ and R´enyi showed that a threshold funclog 𝑛 . However, tion for containing a spanning tree is 𝑝0 = 𝑛

4This is equivalent to 𝐺

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log 𝑛 . 𝑛

Note that both threshold functions for {contain any spanning tree} and {contain a Hamiltonian cycle} are of orlog 𝑛 , even though the latter is a stronger requirement. der 𝑛 Later we will see (in the discussion that follows Examlog 𝑛 ple 4.6) that is actually an easy lower bound on both 𝑛 threshold functions. It has long been conjectured that for any spanning tree6 with a constant maximum degree, its log 𝑛 threshold function is of order . This conjecture was 𝑛 only very recently proved by Montgomery [14].

3. The Kahn–Kalai Conjecture: A Preview Henceforth, ℱ always denotes an increasing property. In 2006, Jeff Kahn and Gil Kalai [12] posed an extremely bold conjecture that captures the location of threshold functions for any increasing properties. Its formal statement will be given in Conjecture 4.11 (graph version) and Theorem 5.7 (abstract version), and in this section we will give an informal description of this conjecture first. All of the terms not defined here will be discussed in the forthcoming sections. Given an ℱ, we are interested in locating its threshold function, 𝑝0 (ℱ).7 But again, this is in general a very hard task. Kahn and Kalai introduced another quantity which they named the expectation threshold and denoted by 𝑝𝔼 (ℱ), which is associated with some sort of expectation 5A spanning graph means a graph that contains all of the vertices 6More precisely, for any sequence of spanning trees {𝑇 } 𝑛

7We switch the notation from 𝑝 (𝑛) to 𝑝 (ℱ) to emphasize its dependence on 0 0 𝑛,𝑝

is connected.

ℱ.

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calculations as its name indicates. By its definition (Definition 4.5), 𝑝𝔼 (ℱ) ≤ 𝑝0 (ℱ) for any ℱ,

As we did in Examples 1.4 and 1.6, 𝔼[number of 𝐻’s in 𝐺𝑛,𝑝 ] = (number of (labelled) 𝐻’s in 𝐾𝑛 )× ℙ(each (labelled) copy of 𝐻 is present in 𝐺𝑛,𝑝 ) (†)

and, in particular, 𝑝𝔼 (ℱ) is easy to compute for many interesting increasing properties ℱ. So 𝑝𝔼 (ℱ) provides an “easy” lower bound on the hard parameter 𝑝0 (ℱ). A really fascinating part is that then Kahn and Kalai conjectured that 𝑝0 (ℱ) is, in fact, bounded above by 𝑝𝔼 (ℱ) multiplied by some tiny quantity!

≍ 𝑛 4 𝑝5 ,

where (†) is because the number of 𝐻’s in 𝐾𝑛 is of order 𝑛4 (since 𝐻 has four vertices), and ℙ(each copy of 𝐻 is present) is precisely 𝑝5 (since 𝐻 has five edges). So we have 0 𝔼[number of 𝐻’s in 𝐺𝑛,𝑝 ] → { ∞

if 𝑝 ≪ 𝑛−4/5 ; if 𝑝 ≫ 𝑛−4/5 ,

(1)

and let’s (informally) call the value 𝑝 = 𝑛−4/5 “the threshold for the expectation of 𝐻.” So this conjecture asserts that, for any ℱ, 𝑝0 (ℱ) is actually well-predicted by (much) easier 𝑝𝔼 (ℱ)! The graph version of this conjecture (Conjecture 4.11) is still open, but the abstract version (Theorem 5.7) is recently proved in [15].

This name makes sense since 𝑝 = 𝑛−4/5 is where the expected number of 𝐻’s drastically changes. Note that (1) tells us that

4. Motivating Examples

so, by the definition of a threshold, we have

The conjecture of Kahn and Kalai is very strong, and even the authors of the conjecture wrote in their paper [12] that “it would probably be more sensible to conjecture that it is not true.” The fundamental question that motivated this conjecture was:

ℙ(𝐺𝑛,𝑝 ⊇ 𝐻) → 0

𝑛−4/5 ≲ 𝑝0 (𝐻). This way, we can always easily find a lower bound on 𝑝0 (𝐹) for any graph 𝐹. What is interesting here is that, for 𝐻 in Figure 11, we can actually show that ℙ(𝐺𝑛,𝑝 ⊇ 𝐻) → 1

Question 4.1. What drives thresholds? All of the examples in this section are carefully chosen to show the motivation behind the conjecture. Recall that the definition of a threshold (Definition 2.3) doesn’t distinguish constant factors. So in this section, we will use the convenient notation ≳, ≲, and ≍ to mean (respectively) ≥, ≤, and = up to constant factors. Finally, write 𝑝0 (𝐻) for a threshold function for 𝐺𝑛,𝑝 of containing a copy of 𝐻, for notational simplicity. Example 4.2. Let 𝐻 be the graph in Figure 11. Let’s find 𝑝0 (𝐻).

if 𝑝 ≪ 𝑛−4/5 ,

if 𝑝 ≫ 𝑛−4/5

using the second moment method (discussed in Example 2.5). This tells us a rather surprising fact that 𝑝0 (𝐻) is actually equal to the threshold for the expectation of 𝐻. Dream. Maybe 𝑝0 (𝐹) is always equal to the threshold for the expectation of 𝐹 for any graph 𝐹? The next example shows that the above dream is too dreamy to be true. Example 4.3. Consider 𝐻̃ in Figure 12 this time. Notice that 𝐻̃ is 𝐻 in Figure 11 with a “tail.”

Figure 12. Graph 𝐻.̃ Figure 11. Graph 𝐻.

By repeating a similar computation as before, we have

In Example 2.5, we observed that there is a connection between a threshold function and computing expectations. 1620

̃ in 𝐺𝑛,𝑝 ] → { 𝔼[number of 𝐻’s

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if 𝑝 ≪ 𝑛−5/6 ; if 𝑝 ≫ 𝑛−5/6 , VOLUME 70, NUMBER 10

so the threshold for the expectation of 𝐻̃ is 𝑛−5/6 . Again, this gives that ℙ(𝐺𝑛,𝑝 ⊇ 𝐻)̃ → 0

if 𝑝 ≪ 𝑛−5/6 ,

̃ However, the actual threshold so we have 𝑛−5/6 ≲ 𝑝0 (𝐻). 𝑝0 (𝐻)̃ is 𝑛−4/5 , which is much larger than the lower bound.

Theorem 4.4 (Bollob´as [4]). For any fixed graph 𝐹, 𝑝0 (𝐹) is equal to the threshold for the expectation of the densest subgraph of 𝐹. For example, in Example 4.2, the densest subgraph of 𝐻 is 𝐻 itself, so 𝑝0 (𝐻) is determined by the expectation of 𝐻. This also determines 𝑝0 (𝐻)̃ in Example 4.3, since the densest subgraph of 𝐻̃ is again 𝐻. Motivated by the preceding examples and Theorem 4.4, we give a formal definition of the expectation threshold. Definition 4.5 (Expectation threshold). For any graph 𝐹, the expectation threshold for 𝐹 is

Figure 13. Gap between 𝑝0 (𝐻)̃ and the expectational lower bound.

𝑝𝔼 (𝐹) = min{𝑝 ∶ 𝔼[number of 𝐹 ′ in 𝐺𝑛,𝑝 ] ≥ 1

∀𝐹 ′ ⊆ 𝐹}.

Observe that 𝑝𝔼 (𝐹) ≲ 𝑝0 (𝐹) for any 𝐹,

This is interesting, because Figure 13 tells us that when 𝑛−5/6 ≪ 𝑝 ≪ 𝑛−4/5 , 𝐺𝑛,𝑝 contains a huge number of 𝐻̃ “on average,” but still it is very unlikely that 𝐺𝑛,𝑝 actually contains 𝐻.̃ What happens in this inverval? Here is an explanation. Recall from Example 4.2 that if 𝑝 ≪ 𝑛−4/5 , then 𝐺𝑛,𝑝 is unlikely to contain 𝐻. But the absence of 𝐻 implies the absence of 𝐻,̃ because 𝐻 is a subgraph of 𝐻!̃ So when 𝑛−5/6 ≪ 𝑝 ≪ 𝑛−4/5 , it is highly unlikely that 𝐺𝑛,𝑝 contains 𝐻̃ because it is already unlikely that 𝐺𝑛,𝑝 contains 𝐻. However, if 𝐺𝑛,𝑝 happens to contain 𝐻, then that copy of 𝐻 typically has lots of “tails” as in Figure 14. This ̃ in 𝐺𝑛,𝑝 . produces a huge number of copies of 𝐻’s

(2)

and in particular, Theorem 4.4 gives that 𝑝𝔼 (𝐹) ≍ 𝑝0 (𝐹) for any fixed 𝐹. Note that this gives a beautiful answer to Question 4.1 whenever ℱ is a property of containing a fixed graph. Example 4.6. Theorem 4.4 characterizes threshold functions for any fixed graphs. To extend our exploration, in this example we consider a graph that grows as 𝑛 grows. We say a graph 𝑀 is a matching if 𝑀 is a disjoint union of edges. 𝑀 is a perfect matching if 𝑀 is a matching that contains all the vertices. Write PM for perfect matching.

Figure 14. 𝐻 with many “tails.”

Figure 15. A matching (above) and a perfect matching (below).

Maybe you have noticed the similarity between this example and the example of a lottery in Remark 1.7.

Keeping Question 4.1 in mind, let’s first check the validity of Theorem 4.4 to a perfect matching,9 which is not a fixed graph. By repeating a similar computation as before, we obtain that

In Example 4.3, 𝑝0 (𝐻)̃ is not predicted by the expected number of 𝐻,̃ thus the Dream is broken. However, it still shows that 𝑝0 (𝐻)̃ is predicted by the expected number of some subgraph of 𝐻,̃ and, intriguingly, this holds true in general. To provide its formal statement, define the density of a graph 𝐹 by density(𝐹) =

(the number of edges of 𝐹) . (the number of vertices of 𝐹)

The next theorem tells us the exciting fact that we can find 𝑝0 (𝐹) by just looking at its densest subgraph, as long as 𝐹 is fixed.8 8For example, a Hamiltonian cycle is not a fixed graph, since it grows as 𝑛 grows.

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𝔼[number of PM’s in 𝐺𝑛,𝑝 ] ≍ (𝑛/𝑒)𝑛/2 𝑝𝑛/2 , which tends to 0 if 𝑝 ≪ 1/𝑛. In fact, it is easy to compute (by considering all subgraphs of a perfect matching) that 𝑝𝔼 (PM) ≍ 1/𝑛, so by (2), 𝑝0 (PM) ≳ 1/𝑛. However, unlike threshold functions for fixed graphs, 𝑝0 (PM) is not equal to 𝑝𝔼 (PM); it was proved by Erdos ˝ and R´enyi that log 𝑛 𝑝0 (PM) ≍ (3) (≫ 𝑝𝔼 (PM)). 𝑛 9We assume 2|𝑛 to avoid a trivial obstruction from having a perfect matching.

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Threshold

Lower bounds Expectation 𝑝0 ≳ 𝑝𝔼 threshold Coupon 𝑝0 ≳ 𝑝𝔼 log 𝑛 collector Figure 16. Gap between 𝑝0 (PM) and 𝑝𝔼 (PM).

Notice that, in Figure 16, what happens in the gap is fundamentally different from that in Figure 13. When 1 log 𝑛 ≪𝑝≪ , 𝐺𝑛,𝑝 contains huge numbers of PMs and 𝑛 𝑛 all its subgraphs “on average.” This means the absence of a subgraph of a PM is not the obstruction for 𝐺𝑛,𝑝 from having a PM when 𝑝 ≫ 1/𝑛. Then what happens here, and what’s the real obstruction? It turned out, we have log 𝑛 𝑝0 (PM) ≳ 𝑛 for a very simple reason: the existence of an isolated verlog 𝑛 tex10 in 𝐺𝑛,𝑝 . It is well-known that if 𝑝 ≪ , then 𝑛 𝐺𝑛,𝑝 contains an isolated vertex with high probability (this phenomenon is elaborated in Example 4.7). Of course, if there is an isolated vertex in a graph, then this graph cannot contain a perfect matching. So (3) says the very compelling fact that once 𝑝 is large enough that 𝐺𝑛,𝑝 avoids isolated vertices, 𝐺𝑛,𝑝 contains a perfect matching! The existence of an isolated vertex in 𝐺𝑛,𝑝 is essentially equivalent to the coupon collector’s problem: Example 4.7 (Coupon collector’s problem). Each box of cereal contains a random coupon, and there are 𝑛 different types of coupons. If all coupons are equally likely, then how many boxes of cereal do we (typically) need to buy to collect all 𝑛 coupons? The well-known answer to this question is that we need to buy ≳ 𝑛 log 𝑛 boxes of cereal. This phenomenon can be translated to 𝐺𝑛,𝑝 in the following way: in 𝐺𝑛,𝑝 , the 𝑛 vertices are regarded as coupons. If a vertex is contained in a (random) edge in 𝐺𝑛,𝑝 , then that is regarded as being “collected.” Note that if 𝑝 ≪

log 𝑛 𝑛

, then typically the number

of edges in 𝐺𝑛,𝑝 is (𝑛)𝑝 ≪ 𝑛 log 𝑛, and then the coupon col2 lector’s problem says that there is typically an “uncollected coupon,” which is an isolated vertex. Observe that, in Example 4.6, the “coupon-collector behavior” of 𝐺𝑛,𝑝 provides another lower bound on 𝑝0 (PM), pushing up the first lower bound, 𝑝𝔼 (PM), by log 𝑛. And it turned out that this second (better) lower bound is equal to the threshold. 10a vertex not contained in any edges

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𝑝0 ≍ 𝑝𝔼 log 𝑛

Figure 17. Bounds on 𝑝0 (PM).

Hitting time. Again, the existence of an isolated vertex in a graph trivially blocks this graph from containing any spanning subgraphs. In Example 4.6, we observed the compelling phenomenon that if 𝑝 is large enough that 𝐺𝑛,𝑝 typically avoids isolated vertices, then for those 𝑝, 𝐺𝑛,𝑝 contains a perfect matching with high probability. Would this mean that, for 𝐺𝑛,𝑝 , isolated vertices are the only barriers to the existence of spanning subgraphs? To investigate this question, we consider a random process defined as below. Consider a sequence of graphs on 𝑛 vertices 𝐺 0 = ∅, 𝐺 1 , 𝐺 2 , … , 𝐺(𝑛) = 𝐾𝑛 , 2

where 𝐺𝑚+1 is obtained from 𝐺𝑚 by adding a random edge.

Figure 18. Random process.

Then 𝐺𝑚 , the 𝑚-th graph in this sequence, is the random variable that is uniformly distributed among all the graphs on 𝑛 vertices with 𝑚 edges. The next theorem tells us that, indeed, isolated vertices are the obstructions for a random graph to having a perfect matching. Theorem 4.8 (Erdos–R´ ˝ enyi [7]). Let 𝑚0 denote the first time that 𝐺𝑚 contains no isolated vertices. Then, with high probability, 𝐺𝑚0 contains a perfect matching. We remark that Theorem 4.8 gives much more precise information about 𝑝0 (PM) (back in 𝐺𝑛,𝑝 setting). For example, Theorem 4.8 implies: Theorem 4.9. Let 𝑝 =

log 𝑛+𝑐𝑛 𝑛

. Then

⎧0 −𝑐 lim ℙ(𝐺𝑛,𝑝 ⊇ PM) = 𝑒−𝑒 𝑛→∞ ⎨ ⎩1

if 𝑐𝑛 → −∞ . if 𝑐𝑛 → 𝑐 if 𝑐𝑛 → ∞

We observe a similar phenomenon for Hamiltonian cycles. Notice that in order for a graph to contain a Hamiltonian cycle, a minimum requirement is that every vertex is contained in at least two edges. The next theorem tells us that, again, this naive requirement is essentially the only barrier.

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Theorem 4.10 (Ajtai-Komlos-Szemer´ ´ edi [1], Bollob´as [3]). Let 𝑚1 denote the first time that every vertex in 𝐺𝑚 is contained in at least two edges. Then, with high probability, 𝐺𝑚1 contains a Hamiltonian cycle. Returning to Question 4.1, so far we have established that there are two factors that affect threshold functions. We first observed that 𝑝𝔼 always gives a lower bound on 𝑝0 . We then observed that, when it applies, the couponcollector behavior of 𝐺𝑛,𝑝 pushes up this expectational lower bound by log 𝑛. Conjecture 4.11 below dauntingly proposes that there are no other factors that affect thresholds. Conjecture 4.11 (Kahn–Kalai [12]). For any graph 𝐹, 𝑝0 (𝐹) ≲ 𝑝𝔼 (𝐹) log 𝑛. Conjecture 4.11 is still wide open even after the “abstract version” of this conjecture is proved. We close this section with a very interesting example in which 𝑝0 lies strictly in between 𝑝𝔼 and 𝑝𝔼 log 𝑛. A triangle factor is a (vertex-) disjoint union of triangles that contains all the vertices.

In other words, 𝑋𝑝 is a “𝑝-random subset” of 𝑋, which means 𝑋𝑝 contains each element of 𝑋 with probability 𝑝 independently. Example 5.1. If 𝑋 = ([𝑛]), then 2

𝑋𝑝 = 𝐺𝑛,𝑝 . So 𝐺𝑛,𝑝 is a special case of the random model 𝑋𝑝 . We redefine increasing property in our new setup. A property is a subset of 2𝑋 , and ℱ ⊆ 2𝑋 is an increasing property if 𝐵 ⊇ 𝐴 ∈ ℱ ⇒ 𝐵 ∈ ℱ. Informally, a property is increasing if we cannot “destroy” this property by adding elements. Note that in this new definition, ℱ is not required to possess strong symmetry as in increasing graph properties; for example, there is no longer a requirement “invariant under graph isomorphisms.” Observe that 𝜇𝑝 (ℱ)(≔ ∑𝐴∈ℱ 𝜇𝑝 (𝐴) = ℙ(𝑋𝑝 ∈ ℱ)) is a polynomial in 𝑝, thus continuous. Furthermore, it is a well-known fact that 𝜇𝑝 (ℱ) is strictly increasing in 𝑝 unless ℱ = ∅, 2𝑋 (see Figure 20). For the rest of this section, we always assume ℱ ≠ ∅, 2𝑋 .

Figure 19. A triangle factor.

The question of a threshold function for a trianglefactor11 was famously solved by Johansson, Kahn, and Vu [10]. Observe that an obvious obstruction for a graph from having a triangle factor is the existence of a vertex that is not contained in any triangles. The result below is the hitting time version of [10], which is obtained by combining [11] and [9]. Theorem 4.12. Let 𝑚2 denote the first time that every vertex in 𝐺𝑚 is contained in at least one triangle. Then, with high probability, 𝐺𝑚2 contains a triangle factor. The above theorem implies that 𝑝0 (triangle factor) ≍ 𝑝𝔼 (triangle factor) ⋅ (log 𝑛)1/3 .

5. The Expectation Threshold Theorem The abstract version of the Kahn–Kalai conjecture, which is the main content of this section, is recently proved in [15]. We remark that the discussion in this section is not restricted by the languages in graph theory anymore. We introduce some more definitions for this general setting. Given a finite set 𝑋, the 𝑝-biased product probability measure, 𝜇𝑝 , on 2𝑋 is defined by 𝜇𝑝 (𝐴) = 𝑝|𝐴| (1 − 𝑝)|𝑋⧵𝐴|

∀𝐴 ⊆ 𝑋.

We use 𝑋𝑝 for the random variable whose law is ℙ(𝑋𝑝 = 𝐴) = 𝜇𝑝 (𝐴) ∀𝐴 ⊆ 𝑋. 11or, more generally, a 𝐾 -factor for any fixed 𝑟 𝑟

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Figure 20. 𝜇𝑝 (ℱ) for 𝑝 ∈ [0, 1], and 𝑝𝑐 (ℱ).

Because 𝜇𝑝 (ℱ) is continuous and strictly increasing in 𝑝, there exists a unique 𝑝𝑐 (ℱ) for which 𝜇𝑝𝑐 (ℱ) = 1/2. This 𝑝𝑐 (ℱ) is called the threshold for ℱ. Remark 5.2. The definition of 𝑝𝑐 (ℱ) does not require sequences. Incidentally, by Theorem 2.4, for any increasing graph property ℱ(= ℱ𝑛 ), 𝑝𝑐 (ℱ) is an (Erdos–R´ ˝ enyi) threshold function for ℱ. For a general increasing property ℱ ⊆ 2𝑋 , the definition of 𝑝𝔼 is not applicable anymore. Kahn and Kalai introduced the following generalized notion of the expectation threshold, which is also introduced by Talagrand [17]. Definition 5.3. Given a finite set 𝑋 and an increasing property ℱ ⊆ 2𝑋 , 𝑞(ℱ) is the maximum of 𝑞 ∈ [0, 1] for which there exists 𝒢 ⊆ 2𝑋 satisfying the following two properties. (a) Each 𝐴 ∈ ℱ contains some member of 𝒢. (b) ∑𝑆∈𝒢 𝑞|𝑆| ≤ 1/2. A family 𝒢 ⊆ 2𝑋 that satisfies (a) is called a cover of ℱ.

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Remark 5.4. The definition of 𝑞(ℱ) eliminates the “symmetry” requirement—which seems very natural (and seemingly easier to deal with) in the context of thresholds for subgraph containments—from the definition of 𝑝𝔼 . It is worth noting that this flexibility is crucially used in the proof of Theorem 5.7 in [15]. The next proposition says that 𝑞(ℱ) still provides a lower bound on the threshold. Proposition 5.5. For any finite set 𝑋 and increasing property ℱ ⊆ 2𝑋 , 𝑞(ℱ) ≤ 𝑝𝑐 (ℱ). Justification. Write 𝑞 = 𝑞(ℱ). By the definition of 𝑝𝑐 (ℱ), it suffices to show that 𝜇𝑞 (ℱ) ≤ 1/2. We have 𝜇𝑞 (ℱ) ≤ ∑

∑ 𝜇𝑞 (𝐴) ≤ ∑ ∑ 𝜇𝑞 (𝐵)

𝑆∈𝒢 𝑆⊆𝐴∈ℱ

𝑆∈𝒢 𝐵⊇𝑆

= ∑ 𝑞|𝑆| ≤ 1/2, 𝑆∈𝒢

where the first inequality uses the fact that 𝒢 covers ℱ.

□

For a graph 𝐹, write ℱ 𝐹 for the increasing graph property of containing a copy of 𝐹. The example below illustrates the relationship between 𝑝𝔼 (𝐹) and 𝑞(ℱ 𝐹 ). Example 5.6 (Example 4.3 revisited). For 𝑋 = ([𝑛]) (so 2 ̃ 𝑋𝑝 = 𝐺𝑛,𝑝 ) and the increasing property ℱ = {contain 𝐻}(⊆ 𝑋 2 ), 𝒢1 ≔ {all the (labelled) copies of 𝐻̃ in 𝐾𝑛 } is a cover of ℱ. The left-hand side of Definition 5.3 (b) is ̃ in 𝐾𝑛 ) ∑ 𝑞|𝑆| =(number of 𝐻’s 𝑆∈𝒢1

× ℙ(each copy of 𝐻̃ is present in 𝐺𝑛,𝑝 ), ̃ in 𝐺𝑛,𝑝 . which is precisely the expected number of 𝐻’s Combined with Proposition 5.5, the above computation gives that 𝑛−5/6 ≲ 𝑝𝑐 (ℱ). On the other hand, we have (implicitly) discussed in Example 4.3 that there is another cover that gives a lower bound better than that of 𝒢1 ; if we take 𝒢2 ≔ {all the (labelled) copies of 𝐻 in 𝐾𝑛 }, then the computation in Definition 5.3 (b) gives that 𝑛−4/5 ≲ 𝑝𝑐 (ℱ). The above discussion shows that, for any (not necessarily fixed) graph 𝐹, 𝑝𝔼 (𝐹) ≲ 𝑞(ℱ 𝐹 ) (whether 𝑝𝔼 (𝐹) ≍ 𝑞(ℱ 𝐹 ) is unknown). The abstract version of the Kahn–Kalai conjecture is similar to its graph version, with 𝑝𝔼 replaced by 𝑞(ℱ). This is what’s proved in [15]. 1624

Theorem 5.7 (Park–Pham [15], conjectured in [12, 17]). There exists a constant 𝐾 such that for any finite set 𝑋 and increasing property ℱ ⊆ 2𝑋 , 𝑝𝑐 (ℱ) ≤ 𝐾𝑞(ℱ) log ℓ(ℱ) where ℓ(ℱ) is the size of a largest minimal element of ℱ. Theorem 5.7 is extremely powerful; for instance, its immediate consequences include historically difficult results such as the resolutions of Shamir’s problem [10] and the “tree conjecture” [14]. Here we mention one smaller consequence: Example 5.8. If 𝐹 is a fixed graph, then ℓ(ℱ 𝐹 ) is the number of edges in 𝐹, thus a constant. So in this case Theorem 5.7 says 𝑝𝑐 (ℱ) ≍ 𝑞(ℱ), which recovers Theorem 4.4. The sunflower conjecture, and “fractional” Kahn–Kalai. The proof of Theorem 5.7 is strikingly easy given its powerful consequences. The approach is inspired by remarkable work of Alweiss, Lovett, Wu, and Zhang [2] on the Erd˝os–Rado sunflower conjecture, which seemingly has no connection to threshold phenomena. This totally unexpected connection was first exploited by Frankston, Kahn, Nayaranan, and the author in [8], where a “fractional” version of the Kahn–Kalai conjecture (conjectured by Talagrand [17]) was proved, illustrating how two seemingly unrelated fields of mathematics can be nicely connected! Note that 𝑞(ℱ) is in theory hard to compute. For instance, in Example 4.3, we can estimate 𝑝𝔼 (𝐻)̃ by finding 𝐹 ⊆ 𝐻̃ with the maximum 𝑒(𝐹)/𝑣(𝐹). On the other hand, to compute 𝑞(ℱ𝐻̃ ), we should in principle consider all possible covers of ℱ𝐻̃ , which is typically not feasible. The good news is that there is a convenient way to find an upper bound on 𝑞(ℱ), which is often of the correct order. Namely, Talagrand [17] introduced a notion of fractional expectation threshold, 𝑞𝑓 (ℱ), satisfying 𝑞(ℱ) ≤ 𝑞𝑓 (ℱ) ≤ 𝑝𝑐 (ℱ) for any increasing property ℱ. He conjectured (and it was proved in [8]) that the (abstract) Kahn–Kalai conjecture (now Theorem 5.7) holds with 𝑞𝑓 (ℱ) in place of 𝑞(ℱ). This puts us in linear programming territory: by LP duality, a bound 𝑞𝑓 (ℱ) ≤ 𝛼 (𝛼 ∈ [0, 1]) is essentially equivalent to existence of an “𝛼-spread” probability measure on ℱ. In all applications of Theorem 5.7 to date, what is actually used to upper bound 𝑞(ℱ) is an appropriately spread measure.12 So all these applications actually follow from the weaker Talagrand version. We close this article with a very interesting conjecture of Talagrand [17] that would imply the equivalence of Theorem 5.7 and its fractional version: 12The problem of constructing well-spread measure is getting growing attention

now; see, e.g., [13] for a start.

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Conjecture 5.9. There exists a constant 𝐾 such that for any finite set 𝑋 and increasing property ℱ ⊆ 2𝑋 , 𝑞𝑓 (ℱ) ≤ 𝐾𝑞(ℱ). ACKNOWLEDGMENT. The author is grateful to Jeff Kahn for his helpful comments. References

[1] M. Ajtai, J. Komlos, ´ and E. Szemer´edi, First occurrence of Hamilton cycles in random graphs, Cycles in graphs (Burnaby, B.C., 1982), North-Holland Math. Stud., vol. 115, North-Holland, Amsterdam, 1985, pp. 173–178, DOI 10.1016/S0304-0208(08)73007-X. MR821516 [2] R. Alweiss, S. Lovett, K. Wu, and J. Zhang, Improved bounds for the sunflower lemma, Ann. of Math. (2) 194 (2021), no. 3, 795–815, DOI 10.4007/annals.2021.194.3.5. MR4334977 [3] B. Bollob´as, The evolution of sparse graphs, Graph theory and combinatorics (Cambridge, 1983), Academic Press, London, 1984, pp. 35–57. MR777163 [4] B. Bollob´as, Random graphs, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 73, Cambridge University Press, Cambridge, 2001, DOI 10.1017/CBO9780511814068. MR1864966 [5] B. Bollob´as and A. Thomason, Threshold functions, Combinatorica 7 (1987), no. 1, 35–38, DOI 10.1007/BF02579198. MR905149 [6] P. Erdos ˝ and A. R´enyi, On the evolution of random graphs (English, with Russian summary), Magyar Tud. Akad. Mat. Kutato´ Int. Közl. 5 (1960), 17–61. MR125031 [7] P. Erdos ˝ and A. R´enyi, On the existence of a factor of degree one of a connected random graph, Acta Math. Acad. Sci. Hungar. 17 (1966), 359–368, DOI 10.1007/BF01894879. MR200186 [8] K. Frankston, J. Kahn, B. Narayanan, and J. Park, Thresholds versus fractional expectation-thresholds, Ann. of Math. (2) 194 (2021), no. 2, 475–495, DOI 10.4007/annals.2021.194.2.2. MR4298747 [9] A. Heckel, M. Kaufmann, N. Müller, and M. Pasch, The hitting time of clique factors, Preprint, arXiv 2302.08340 [10] A. Johansson, J. Kahn, and V. Vu, Factors in random graphs, Random Structures Algorithms 33 (2008), no. 1, 1–28, DOI 10.1002/rsa.20224. MR2428975

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[11] J. Kahn, Hitting times for Shamir’s problem, Trans. Amer. Math. Soc. 375 (2022), no. 1, 627–668, DOI 10.1090/tran/8508. MR4358678 [12] J. Kahn and G. Kalai, Thresholds and expectation thresholds, Combin. Probab. Comput. 16 (2007), no. 3, 495–502, DOI 10.1017/S0963548307008474. MR2312440 [13] D. Kang, T. Kelly, D. Kühn, A. Methuku, and D. Osthus, Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor, Transactions of the American Mathematical Society, to appear. [14] R. Montgomery, Spanning trees in random graphs, Adv. Math. 356 (2019), 106793, 92, DOI 10.1016/j.aim.2019.106793. MR3998769 [15] J. Park and H. T. Pham, A proof of the Kahn–Kalai conjecture, J. Amer. Math. Soc., electronically published on August 7, 2023, DOI: https://doi.org/10.1090/jams /1028 (to appear in print). [16] L. Posa, ´ Hamiltonian circuits in random graphs, Discrete Math. 14 (1976), no. 4, 359–364, DOI 10.1016/0012365X(76)90068-6. MR389666 [17] M. Talagrand, Are many small sets explicitly small?, STOC’10—Proceedings of the 2010 ACM International Symposium on Theory of Computing, ACM, New York, 2010, pp. 13–35. MR2743011

Jinyoung Park Credits

Opening graphic is courtesy of enjoynz via Getty. Figures 1–20 and photo of Jinyoung Park are courtesy of Jinyoung Park.

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Random Phenomena with Fractal-like Features

Patricia Alonso Ruiz It is summer. My brother and I are walking along the seashore and arrive at a stony formation: The salty water that passes through the rocks from time to time creates tiny lagoons, where some kind of seaweed seems to find its ecosystem. We are captivated by these “microlakes” and my fascination grows when we discover how the seaweed seems to randomly climb and dry on the rock, creating a Patricia Alonso Ruiz is an associate professor of mathematics at Texas A&M University. Her email address is [email protected]. Communicated by Notices Associate Editor Scott Sheffield. For permission to reprint this article, please contact: [email protected].

fractal pattern; see Figure 1. I draw my brother’s attention: “Look, this is a fractal!” Excitement rises, we take out our phones and capture yet another instance of random phenomena in nature exhibiting fractal features. An artist/mural painter himself, my brother is as curious an observer as I am and our discovery is going to initiate an interesting conversation. “What is, again, a fractal? How do you mathematicians know that something is fractal?”

1. What Makes It Fractal? “How does a mathematician describe the notion of fractal?” My brother’s question puts me in an unusual situation because here we find a mathematical concept

DOI: https://doi.org/10.1090/noti2803

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Hausdorff measure. The latter generalizes the concept of length (𝑑 = 1), area (𝑑 = 2) and volume (𝑑 = 3) to any 𝑑 ∈ [0, ∞) and goes back to ideas of Carath´eodory: Given a set 𝐹 ⊆ ℝ𝑛 and 𝛿 > 0 one first looks at all covers of 𝐹 by sets of diameter at most 𝛿 and tries to minimize the sum of the 𝑑-power of the diameters of the cover, that is ℋ𝛿𝑑 (𝐹) ∶= inf { ∑ diam(𝑈 𝑖 )𝑑 ∶ {𝑈 𝑖 }𝑖 𝛿-cover of 𝐹}. 𝑖

Decreasing 𝛿 (the “degree of refinement” of the cover) reduces the possible coverings to take, and thus increases the value of ℋ𝛿𝑑 (𝐹). The 𝑑-dimensional Hausdorff measure of 𝐹 is then defined as ℋ 𝑑 (𝐹) ≔ lim+ ℋ𝛿𝑑 (𝐹) 𝛿→0

Figure 1. Pointing to salty seaweed on a rock.

without a unique or generally established characterization. The term fractal was coined by Mandelbrot in [Man75] in analogy to the Latin word fractus, which means broken. He would use it to describe objects whose geometry was too “broken” to fit into a traditional setting. His original mathematical definition, which he would later realize was too restrictive, was that a set is fractal when its Hausdorff dimension is strictly larger than its topological dimension. The topological dimension of a nonempty set is a nonnegative integer defined recursively: it is 0 if the set is either a singleton or totally disconnected, it is 1 if each point has arbitrarily small neighborhoods whose boundary has dimension 0, it is 2 if each point has neighborhoods with boundary of dimension 1 etc. For instance, the topological dimension of an interval [𝑎, 𝑏] ⊂ ℝ is 1; see Figure 2. The Hausdorff dimension of an interval is also 1—Mandelbrot would thus say that [𝑎, 𝑏] is not fractal.

and it may take any nonnegative value including 0 and infinity. The Hausdorff dimension of 𝐹 is now defined as the critical value 𝑑 where its 𝑑-dimensional Hausdorff measure ℋ 𝑑 (𝐹) “jumps” from infinity to 0, i.e., dimH 𝐹 ≔ inf{𝑑 ≥ 0 ∶ ℋ 𝑑 (𝐹) = 0}.

(1)

In other words, the Hausdorff dimension determines which Hausdorff measure may be able to actually “see” the set: Looking at an interval with “2-dimensional” eyes we will see nothing (the area of an interval is 0!), while looking at it with “1-dimensional” eyes we will be able to measure its (finite, nonzero) length. To determine the Hausdorff dimension of a set is in general difficult. However, in the presence of certain similarity properties, exact computations are possible. Figure 3 shows a set 𝐾 ⊂ ℝ2 created by the Swedish mathematician Helge von Koch in the early 1900s. This kind of set is called self-similar because it is made up of smaller copies of itself. In this case there are four copies, each scaled by a factor of 1/3.

Figure 2. Any point 𝑥 ∈ [𝑎, 𝑏] has an 𝜀-neighborhood whose boundary, two totally disconnected points, has topological dimension 0.

What about the Hausdorff dimension of a more intricate set? In contrast to the topological dimension, the Hausdorff dimension may be any nonnegative real number. Broadly speaking, this dimension captures the proportion between the diameter and the mass of an object, where the diameter is usually considered with respect to the Euclidean length and the mass refers to the Euclidean NOVEMBER 2023

Figure 3. The von Koch curve is a self-similar set.

Thus, the “Euclidean mass” of each copy scales by a factor of 1/4 while its “Euclidean diameter” scales by a factor of 1/3. And since the Hausdorff dimension expresses the mass-to-diameter ratio, we expect dimH 𝐾 to be the number 𝑑 such that 1/4 = (1/3)𝑑 . In general, a self-similar set

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consisting of 𝑁 scaled copies of itself with scaling factors 𝑟1 , 𝑟2 , … , 𝑟𝑁 respectively, has Hausdorff dimension 𝑑, where 𝑑 solves the equation

One example of a random analogue to the von Koch curve is shown in Figure 4.

𝑁

∑ 𝑟𝑖𝑑 = 1;

(2)

𝑖=1

see, e.g., [Fal14, Theorem 9.3]. For the von Koch curve, 𝑁 = 4 and 𝑟 𝑖 = 1/3, and indeed dim𝐻 𝐾 = ln 4/ ln 3. As for the topological dimension of 𝐾, it equals 1 in the same way as for the interval [𝑎, 𝑏]. The topological dimension of the von Koch curve is thus strictly smaller than the Hausdorff dimension! Mandelbrot would therefore say that the von Koch curve is indeed fractal ,. A common rule of thumb to decide about the “fractality” of a set is to check whether its Hausdorff dimension is not an integer. This criterion ought to be taken with a grain of salt because it excludes objects one would consider fractal, for instance the paths of a Wiener process in three dimensions (more about this later!). My personal viewpoint follows K. Falconer in that the term “fractal” may be regarded similarly to how biologists may regard the term “life”: Most objects we would classify as fractal share certain properties, which we may call “fractal features,” and yet there will be interesting fractal objects lacking some of them. The properties listed by Falconer in [Fal14] are: (i) having a fine structure, that is, detail at arbitrarily small scales. (ii) being too irregular to be described in traditional geometrical language, both locally and globally. (iii) being self-similar in some way, perhaps approximate or statistical. (iv) having “fractal” (usually Hausdorff) dimension greater than the topological dimension. (v) being often defined in a simple way, perhaps recursively. “Alright, to me the pattern in the rocks would classify as fractal.” My brother is checking items in the previous list. His curiosity does not stop here and he brings into the conversation another intrinsic aspect of nature: “I wonder whether the seaweed has deposited there somewhat randomly. Would that make the pattern a random fractal? How do you mathematicians describe such an object?”

2. Random Fractals The mathematician in me feels compelled to first clarify that, technically speaking, the pattern we have found is a realization, that is, one of the many possible patterns created by the random process of seaweed deposition. Introducing randomness in the construction of fractals allows us to reproduce irregular patterns closer to what we see in nature. In fact, many computer-drawn landscapes are constructed following some kind of random procedure. 1628

Figure 4. A random von Koch curve. Each generation consists of four random choices of the patterns above.

A set like the random von Koch curve displayed is called statistically self-similar. Besides consisting of random variations of the whole, that randomness appears at all scales: each random variation consists again of random variations of the copy itself. Thus, enlarging a smaller part we shall see a set with the same distribution as the original one. What about the Hausdorff dimension? Dealing with random objects, their properties are stated in probabilistic terms. In this case, with probability one, the Hausdorff dimension of a statistically self-similar fractal constructed analogously to the random von Koch curve from Figure 4 is the solution to 𝑁

𝔼[ ∑ 𝑅𝑑𝑖 ] = 1.

(3)

𝑖=1

Here, 𝑅1 , … , 𝑅𝑁 are now random variables that determine the scale of the 𝑁 random copies the whole is made of; see, e.g., [Fal14, Theorem 15.2]. Indeed, my brother notices how (3) resembles (2)! As in the deterministic case, one can usually hope at least to obtain certain bounds for the Hausdorff dimension or the Hausdorff measure of a random fractal. Although the biological laws that govern a phenomenon like our concrete seaweed deposition may still be far too complex to fit a tractable mathematical model, there are random processes that do lead to random fractal sets. Significant examples are diffusion-limited aggregation (DLA) processes that model particle accumulation processes as seen in crystal growth or forked lightning, or random percolation models that simulate particle clustering. My brother finds another rock that also displays a fractal pattern. “I see,” he says, “the fractal pattern is a product of the random process of seaweed deposition. That makes the process a random phenomenon with fractal-like features.” I couldn’t have said it any better ,. He is now scratching his forehead. “I wonder though: are there random phenomena one would consider fractal even without actually producing a fractal output?” Great question! This conversation is getting pretty interesting.

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3. Fractals Behind Random Scenes As my brother points out, our discussion has thus far focused on physical fractal objects produced by a random phenomenon. The world of fractals, however, expands rapidly the moment one starts searching (and finding!) the properties listed at the end of Section 1, not just in actual observables such as plants, snowflakes or metal particles, but also in relationships, which mathematicians encode in functions. Processes modeled by Brownian motion, also known as (the) Wiener process among mathematicians, constitute an important class of random phenomena that exhibit these kinds of fractal features. The model became popular through the work of the botanist Robert Brown in 1827, who investigated it to describe the random movement of pollen particles suspended in water. The increasing interest by physicists in the model led to the work of Einstein in 1905, where he successfully applied Brownian motion to test the validity of the molecular-kinetic theory of heat. As he states in [Ein56]: “bodies of microscopically visible size suspended in a liquid will perform movements of such magnitude that they can be easily observed in a microscope, on account of the molecular motions of heat.” The rigorous mathematical construction of Brownian motion as a random function was presented by Norbert Wiener in 1923, hence its namesake process. Nowadays Brownian motion is the basis to model not only the movement of particles suspended in a fluid, but also many other diverse natural phenomena. Changes in stock prices, medical imaging, topographical surfaces, and decision making algorithms are just a few of the fields where it plays a vital role. Mathematically speaking, the standard 1-dimensional Wiener process {𝑊𝑡 }𝑡≥0 is a random function of time. For each 𝑡 ≥ 0, 𝑊𝑡 is a random variable, and each element 𝜔 in the sample space Ω defines a sample path 𝑡 ↦ 𝑊𝑡 (𝜔). The graph of a possible outcome for a sample path is displayed in Figure 5. The main properties that characterize standard Brownian motion are: (i) with probability 1, the process starts at the origin, i.e., ℙ(𝑊0 = 0) = 1;

(ii) with probability 1, the process has continuous sample paths, i.e., ℙ(𝑡 ↦ 𝑊𝑡 is continuous) = 1; (iii) the process has stationary increments, i.e., for any 𝑡, 𝑠 ≥ 0, 𝑊𝑡+𝑠 − 𝑊𝑡 is normally distributed with mean 0 and variance 𝑠; (iv) the process has independent increments, i.e., for any 0 ≤ 𝑡1 < 𝑡2 < … < 𝑡𝑛 , the increments 𝑊𝑡𝑛 − 𝑊𝑡𝑛−1 , … , 𝑊𝑡2 − 𝑊𝑡1 are independent random variables. A characteristic fractal feature of the Wiener process in ℝ appears in its space-time scaling, which we can analyze by looking at the graph from Figure 5, {(𝑡, 𝑊𝑡 ) ∶ 𝑡 ∈ [0, 1]} ⊂ ℝ2 . Let us scale time by a factor 𝑐 and ask: How does space need to scale so that (𝑐𝑡, 𝑊𝑐𝑡 ) has the same statistical distribution as (𝑡, 𝑊𝑡 )? The definition of the process and the properties of the normal distribution imply that 𝑊𝑐𝑡 is normally distributed 𝑁(0, 𝑐𝑡) and the same happens with √𝑐𝑊𝑡 . We may thus write 𝑑

(𝑐𝑡, 𝑊𝑐𝑡 ) = (𝑐𝑡, √𝑐𝑊𝑡 ),

(4)

𝑑

where = means that the left- and right-hand side are equal in distribution. So time and space scale differently! The mapping (𝑥, 𝑦) ↦ (𝑐𝑥, √𝑐𝑦) belongs to the class of affine transformations, which are linear transformations that contract with a different ratio in each direction. Because of (4), one says that the graph of 𝑊𝑡 ∶ [0, 1] → ℝ is statistically self-affine. What about its Hausdorff dimension? While there is no general method to compute it for an arbitrary statistically self-affine random set, one can show that the Hausdorff dimension of {(𝑡, 𝑊𝑡 ) ∶ 𝑡 ∈ [0, 1]} is 1.5 with probability 1. For Mandelbrot, a genuine fractal ,. In the computation of the latter Hausdorff dimension, a fundamental property of the Wiener process plays a crucial role: sample paths are continuous and yet “rough.” This roughness is described by the fact that paths are almost surely Hölder continuous of order 𝛾 for any 0 < 𝛾 < 1/2 and they just fail to be so for 𝛾 = 1/2. That means, for 0 < 𝛾 < 1/2, there is a random constant 𝐶𝛾 for which ℙ(|𝑊𝑡+ℎ − 𝑊𝑡 | ≤ 𝐶𝛾 |ℎ|𝛾 ) = 1

(5)

for any 𝑡, 𝑡 + ℎ ∈ [0, 1], and the latter statement fails when 𝛾 = 1/2. In fact, one can show that |𝑊𝑡+ℎ − 𝑊𝑡 | ≤ 𝐶|ℎ log(1/ℎ)|1/2

Figure 5. Graph of a simulated Brownian sample path {(𝑡, 𝑊𝑡 ) ∶ 𝑡 ∈ [0, 1]}.

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with probability 1 for 𝑡, 𝑡 + ℎ ∈ [0, 1]. Loosely speaking, this means that a spatial increment of the Wiener process along a period of time ℎ is almost surely no larger than √ℎ. Since √ℎ is the standard deviation of such a spatial increment, the increments are often roughly of that size. The Hölder regularity of the paths remains true in higher dimensions. For example, a sample path

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𝑊𝑡 ∶ [0, 1] → ℝ2 like the one displayed in Figure 6 also satisfies (5), cf. [Fal14, Proposition 16.4].

Figure 6. A sample path of 2-dimensional Brownian motion.

These paths were Robert Brown’s initial object of analysis and they generally model the random trajectory of a heated particle. However, in contrast to the 1-dimensional case, the trail {𝑊𝑡 ∶ 𝑡 ∈ [0, 1]} ⊂ ℝ2 (and in fact any path 𝑊𝑡 ∶ [0, 1] → ℝ𝑛 with 𝑛 ≥ 2) has almost surely Hausdorff dimension 2. This is an example of a fractal object whose Hausdorff dimension is in fact a whole natural number ,. My brother has been listening to my talk with a mixture of amusement and awe. “You sounded really excited giving that explanation! May I try to put it in my own words?” Of course, I am all ears! He says: “The wind brings a grain of pollen to the surface of one of the minilakes we observe here between the rocks. The movement of that grain follows a random path which mathematicians describe as Brownian motion or Wiener process. And that movement is fractal because its space-time ratio is a fraction, namely 1/2.” Wow, he has really captured the essence of my talk! “Now I wonder,” my brother’s observation and curiosity again at work: “Is the square root special? What if the pollen is suspended in a medium that is itself highly irregular, something fractal?” “Would the way the pollen moves tell anything about the geometric fractal features of the medium?”

4. Brownian Motion “Feels” Spatial Fractality A natural yet challenging question! The mathematicians Kusuoka [Kus87] and Goldstein [Gol87] thought about such a situation in the 1980s and were able to give a rigorous construction of the analogue to Brownian motion on a fractal called the Sierpinski ´ gasket (SG), approximated in Figure 7.

Just as classical Brownian motion can be obtained as the scaled limit of random walks on lattices (the scaling ratio is again 1/2, think of time as number of steps), Kusuoka and Goldstein considered a sequence of random walks on the graph approximations of S𝐺 displayed in Figure 7. They found that the space-time ratio, instead of 1/2, is log 2/ log 5. Which means that, in the long run, the motion is on average slower than standard Brownian motion in ℝ2 . This kind of behavior is called by physicists subdiffusive. Why would that be so? Intuitively, the physical structure of the Sierpinski ´ gasket features many holes which the particle must “circumvent” to get from one place to another. The particle’s movement is thus constrained and slowed down by the intrinsic structure of the fractal medium in which it moves. If the medium is “wide enough” for the particle to move in it for a “long enough” period of time without reaching its boundary, the Hausdorff dimension of the particle’s trajectory (a sample path) will correspond to the inverse of the space-time scaling ratio. Remember how we had wondered that the Hausdorff dimension of a Brownian path was 2? Well, that is precisely the inverse of the ratio 1/2 ,. This intuitive idea is one of the reasons for physicists to often call the Hausdorff dimension of such a path the “dimension of the walk,” or walk dimension. Kusuoka and Goldstein’s work thus tell us that the walk dimension of the Sierpinksi ´ gasket is 𝑑𝑤 = 𝑑𝑤 (SG) =

log 5 , log 2

which means that Brownian motion on SG moves in time 𝑡 a distance comparable to 𝑡1/𝑑𝑤 . “Wait a minute.” My brother has closed his eyes and I can feel he is trying to connect the dots. “Robert Brown’s grain of pollen moved in time 𝑡 a distance comparable to 𝑡1/2 which corresponds to 𝑡1/𝑑𝑤 because the walk dimension is 𝑑𝑤 = 2. Earlier on you mentioned that the spacetime scaling 1/2 was related to the (was it Hölder?) continuity of the paths.” Opening his eyes he points to expression (5), that is written on the sand. “Does the same happen with the Sierpinski ´ gasket?” Good question! The mathematicians Barlow and Perkins presented in [BP88] a detailed analysis of Brownian motion on SG. In particular, they proved that for any 0 < 𝛾 < log 2/ log 5 = 1/𝑑𝑤 there is a random variable 𝐶𝛾 such that ℙ(|𝑊𝑡+ℎ − 𝑊𝑡 | ≤ 𝐶𝛾 |ℎ|𝛾 ) = 1

⋯ Figure 7. Graphs approximating SG.

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(6)

for any 𝑡, 𝑡 + ℎ ∈ [0, 1], and the latter fails when 𝛾 = 1/𝑑𝑤 . One of the main ingredients to obtain (6) was to estimate the probability with which a particle performing Brownian motion on SG moves away from its starting point.

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“By the way,” my brother’s observational skills are blossoming today, “I notice that we often speak of Hölder continuity as an indicator for fractal features. Are there other Hölder continuous functions associated with Brownian motion?” His inquiry actually touches on an aspect that, in the case of diffusion processes on fractals, presents many challenges and open questions. When existent, one of the functions encoding Brownian motion is the heat kernel, which basically describes the probability that a particle performing Brownian motion moves from 𝑥 to 𝑦 in a period of time 𝑡. In the case of ℝ𝑛 there is an explicit expression for that function, known as the Gaussian heat kernel, which reads 2

|𝑥−𝑦| 1 − 4𝑡 , 𝑒 (7) (4𝜋𝑡)𝑛/2 where 𝑥, 𝑦 ∈ ℝ𝑛 and 𝑡 > 0. For example, the probability that a particle starting at a point 𝑥 ∈ ℝ𝑛 lies in a region 𝐵 ⊂ ℝ𝑛 at time 𝑡 may be computed as

𝑝𝑡 (𝑥, 𝑦) =

∫ 𝟏𝐵 (𝑦)𝑝𝑡 (𝑥, 𝑦) 𝑑𝑦 = 𝑃𝑡 𝟏𝐵 (𝑥).

(8)

ℝ𝑛

Here, {𝑃𝑡 }𝑡≥0 denotes an operator called the heat semigroup. The existence of a neat explicit formula like (7) is nearly a miracle. In general, one can at most hope to understand the behavior of the heat kernel in terms of approximate relationships between the distance between two points 𝑥, 𝑦 and the time 𝑡 that is being considered. In the case of the Sierpinski ´ gasket, Barlow and Perkins found that the heat kernel behaves up to (different) constants like 𝑑(𝑥,𝑦)

𝑝𝑡 (𝑥, 𝑦) ≍

𝑐1 −𝑐 ( 𝑒 2 𝑡1/𝑑𝑤 𝑑 𝑡 𝐻 /𝑑𝑤

)

𝑑𝑤 𝑑𝑤 −1

,

(9)

where 𝑑(𝑥, 𝑦) is the Euclidean distance between 𝑥 and 𝑦 and 𝑑𝐻 denotes the Hausdorff dimension of SG, cf. [BP88, Theorem 1.5]. They were also able to show the existence of a constant 𝐶 > 0 such that 𝐶 |𝑝𝑡 (𝑥, 𝑦) − 𝑝𝑡 (𝑥, 𝑧)| ≤ 𝑑(𝑦, 𝑧)𝑑𝑤 −𝑑𝐻 (10) 𝑡 for any 𝑡 > 0 and distinct points 𝑥, 𝑦, 𝑧 in SG. In other words, the heat kernel on SG is Hölder continuous with exponent 𝑑𝑤 −𝑑𝐻 . Estimates like (9) and (10) are by now well-established results in many nonsmooth settings such as so-called fractional diffusion processes; see, e.g., [Bar98, Definition 3.5]. The latter include the Sierpinski ´ gasket and the Sierpinski ´ carpet; see Figure 8. “Oh, the carpet is also an artful object!” My brother goes back and forth between the pictures of the carpet and the gasket. “They look somewhat related and yet different.... For instance, I could disconnect the gasket by removing just the three points connecting the three largest triangles. To disconnect the carpet one would need to remove much more than a few points.... Is the carpet more challenging when you study Hölder regularity of functions?” NOVEMBER 2023

Figure 8. The standard Sierpinski ´ carpet (SC) and the gasket (SG).

Oh brother, sure it is! In fact, a major achievement of Barlow and Bass in [BB92] was precisely to obtain the corresponding estimates (9) of the heat kernel for Brownian motion on SC. And challenges continue today! A few lines of computations, invoking also available estimates for the time-derivative 𝜕𝑡 𝑝𝑡 (𝑥, 𝑦), made it possible in [ARBC+ 21, Theorem 3.7] to derive from (10) the following Hölder regularity of the corresponding heat semigroup 𝑑 −𝑑

𝑑(𝑥, 𝑦) 𝑤 𝐻 ‖𝑓‖∞ (11) ) 𝑡1/𝑑𝑤 for 𝑡 ∈ (0, 1) and 𝑓 bounded and measurable. The lat´ ter estimate has been named the weak Bakry-Emery curvature condition BE(𝑑𝑤 −𝑑𝐻 ) in [ARBC+ 21] because of its con´ nection with the Bakry-Emery curvature dimension inequality that may be characterized as |𝑃𝑡 𝑓(𝑥) − 𝑃𝑡 𝑓(𝑦)| ≤ 𝐶(

‖∇(𝑃𝑡 𝑓)‖∞ ≤ 𝑒−𝐾𝑡 ‖∇𝑓‖∞

(12)

for any compactly supported smooth function 𝑓. In the case of a smooth Riemannian manifold, the parameter 𝐾 is the lower bound on the Ricci curvature of the manifold, cf. [vRS05, Theorem 1.3]. In the context of fractals, the gradient operator ∇ appearing in (12) is less straightforward to conceptualize and to analyze. A natural generalization of gradient in the metric measure space setting is the carr´e du champ operator Γ. The name of the operator has its origins in the mathematical theory of electrostatics and means “square (norm) of the (electric) field.” In ℝ𝑛 , Γ(𝑓) = ‖∇𝑓‖2 and its connection with the (Brownian) heat semigroup {𝑃𝑡 }𝑡≥0 is a fundamental result in the theory of Dirichlet forms, namely lim

𝑡→0+

1 1 ∫ (𝑓 − 𝑃𝑡 𝑓)𝑓 𝑑𝑥 = ∫ Γ(𝑓)𝑑𝑥 = ℰ(𝑓), 𝑡 ℝ𝑛 2 ℝ𝑛

cf. [FOT11, Lemma 1.3.4 and Lemma 3.2.3]. Due to its interpretation in Physics, ℰ(𝑓) is usually referred to as the “energy” of the system associated with 𝑓. In this way, the carr´e du champ can be regarded as the Radon Nikodym derivative of the energy measure of 𝑓 with respect to the reference measure of the underlying space. However, there are many fractals, including SG, for which the energy measure is singular with respect to the

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standard reference measure! Therefore, instead of the gradient, we decided to analyze in (11) the difference |𝑃𝑡 𝑓(𝑥)− 𝑃𝑡 𝑓(𝑦)| as a weak replacement of (12). Besides the reasonable desire to extend these to larger classes of fractal spaces, even in the case of the Sierpinski ´ carpet it is still an open question whether estimates like (10) or (11) are optimal. In fact, we conjectured in [ARBC+ 21] that one may improve upon (11) in the Sierpinski ´ carpet because it possibly satisfies the weak BE(𝑑𝑤 − 𝑑𝐻 + 𝑑𝑡𝐻 − 1) condition, that is 𝑑𝑤 −𝑑𝐻 +𝑑𝑡𝐻 −1

|𝑃𝑡 𝑓(𝑥) − 𝑃𝑡 𝑓(𝑦)| ≤ 𝐶(

𝑑(𝑥, 𝑦) ) 𝑡1/𝑑𝑤

‖𝑓‖∞ .

(13) Figure 9. Brownian motion crossing the boundary.

Here, 𝑑𝑡𝐻 denotes the topological Hausdorff dimension of the space, which roughly speaking corresponds to the largest possible Hausdorff dimension of the boundary of an open cover of the space. While our discussion is warming my mathematical thoughts, it is actually getting colder on the shore and we decide to head back home. “One last question, though.” My brother is about to ask something almost philosophical: “Where does a conjecture come from? What brings you and your collaborators to ‘guess’ that (13) may be true?” His inquiry could fill another chapter of connections between probability, geometry, and analysis in the world of fractals—what an amazing observer an artist can be! Walking back I try to convey a reason to conjecture the estimate (13). The Hölder regularity of the heat semigroup {𝑃𝑡 }𝑡≥0 has a geometric implication that I personally find truly beautiful. In the 1990s, Ledoux discovered that the heat semigroup could be used to describe the perimeter of a ball 𝐵 ⊆ ℝ𝑛 by Per(𝐵) = lim+ 𝑡→0

√𝜋 2√𝑡

∫ 𝑃𝑡 (|𝟏𝐵 − 𝟏𝐵 (𝑥)|)(𝑥) 𝑑𝑥

(14)

ℝ𝑛

cf. [Led94]. (By now, the time scaling √𝑡 is no surprise!) And what does (14) actually mean? Let us visualize it using Brownian motion. The expression inside the integral in (14) can be rewritten as

(𝑋, 𝑑, 𝜇) by studying lim inf + 𝑡→0

1 ∫𝑃 (|𝟏 − 𝟏𝐵 (𝑥)|)(𝑥)𝑑𝜇(𝑥) 𝑡𝛼 𝑋 𝑡 𝐵

(15)

with 𝛼 > 0 and 𝐵 ⊆ 𝑋 Borel. In principle, this quantity may or not be finite depending on the value of 𝛼. And ´ where is the connection to the weak Bakry-Emery condition? If the space satisfies BE(𝑑𝑤 (1 − 𝛼)), then (15) is finite when the so-called upper (𝛼𝑑𝑤 )-Minkowski content of 𝐵 is finite cf. [ARBC+ 21, Theorem 4.9 and Lemma 2.7]. On the Sierpinski ´ carpet it is possible to find open sets whose (𝑑𝐻 + 𝑑𝑡𝐻 − 1)-Minkowski content is finite, whence 𝛼𝑑𝑤 = 𝑑𝐻 − 𝑑𝑡𝐻 + 1 ⇒ 𝑑𝑤 (1 − 𝛼) = 𝑑𝑤 − 𝑑𝐻 + 𝑑𝑡𝐻 − 1 et voilà the guess ,,. My brother can feel my excitement telling this story when we arrive at his place. Opening the door he asks: “Could you make a picture of one of those open sets in the Sierpinski ´ gasket?” I smile: Do you have a couple of colors to paint with here? My brother’s face brightens. Let the artistic evening begin! ACKNOWLEDGMENT. I am grateful to my colleagues Roger Smith and Harold P. Boas, as well as to two anonymous referees. Their comments and feedback on an early draft version of this article greatly helped to improve it.

𝔼𝑥 [|𝟏𝐵 (𝑊𝑡 ) − 𝟏𝐵 (𝑥)|], which represents the probability that by time 𝑡 a particle following the Brownian motion 𝑊𝑡 lies on the other side of the boundary than where it started. So the perimeter happens to be expressible in terms of Brownian motion crossing the boundary within a properly scaled short time period! The characterization (14) was later extended to any Borel set 𝐸 of a Riemannian manifold in [MPPP07] and it partly motivated our project [ARBC+ 21] to search for a definition of perimeter for sets in a metric measure space 1632

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[BB92] Martin T. Barlow and Richard F. Bass, Transition densities for Brownian motion on the Sierpinski ´ carpet, Probab. Theory Related Fields 91 (1992), no. 3-4, 307–330, DOI 10.1007/BF01192060. MR1151799 [BP88] Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpinski ´ gasket, Probab. Theory Related Fields 79 (1988), no. 4, 543–623, DOI 10.1007/BF00318785. MR966175 [Ein56] Albert Einstein, Investigations on the theory of the Brownian movement, Dover Publications, Inc., New York, 1956. Edited with notes by R. Fürth; Translated by A. D. Cowper. MR0077443 [Fal14] Kenneth Falconer, Fractal geometry: Mathematical foundations and applications, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. MR3236784 [FOT11] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, Second revised and extended edition, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011. MR2778606 [Gol87] Sheldon Goldstein, Random walks and diffusions on fractals, Percolation theory and ergodic theory of infinite particle systems (Minneapolis, Minn., 1984), IMA Vol. Math. Appl., vol. 8, Springer, New York, 1987, pp. 121– 129, DOI 10.1007/978-1-4613-8734-3_8. MR894545 [Kus87] Shigeo Kusuoka, A diffusion process on a fractal, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), Academic Press, Boston, MA, 1987, pp. 251–274. MR933827 [Led94] Michel Ledoux, Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space, Bull. Sci. Math. 118 (1994), no. 6, 485–510. MR1309086

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[Man75] Benoit Mandelbrot, Les objets fractals (French), ´ Nouvelle Bibliothèque Scientifique, Flammarion, Editeur, Paris, 1975. Forme, hasard et dimension. MR0462040 [MPPP07] M. Miranda Jr., D. Pallara, F. Paronetto, and M. Preunkert, Heat semigroup and functions of bounded variation on Riemannian manifolds, J. Reine Angew. Math. 613 (2007), 99–119, DOI 10.1515/CRELLE.2007.093. MR2377131 [vRS05] Max-K. von Renesse and Karl-Theodor Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), no. 7, 923–940, DOI 10.1002/cpa.20060. MR2142879

Patricia Alonso Ruiz Credits

Opening graphic is courtesy of Bruno Alonso Ruiz. Figures 1–9 and photo of Patricia Alonso Ruiz are courtesy of Patricia Alonso Ruiz.

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EARLY CAREER The Early Career Section offers information and suggestions for graduate students, job seekers, early career academics of all types, and those who mentor them. Krystal Taylor and Ben Jaye serve as the editors of this section. Next month’s theme will be Math Institutes.

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Math and the Real World BIG Article Jesse Berwald For the first years after my PhD I held positions as a postdoc. As many people reading this can relate to, these roles involved major upheavals with moves across the country. With a third transition looming and my spouse starting a new job in the Twin Cities I made the decision to pull the plug on an academic career. In this article I would like to chronicle some insights gleaned in the ensuing decade. In For permission to reprint this article, please contact: [email protected].

Jesse Berwald is principal quantum systems architect at Quantum Computing, Inc. His email address is [email protected]. DOI: https://doi.org/10.1090/noti2795

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particular, I will focus on comparisons between academia and industry to help the reader form a cost/benefit matrix when considering a career switch. It is approaching ten years since I left academia to take a job as a data scientist at Target. In that time I’ve grown intellectually and professionally and my career trajectory has taken me from retail analytics to quantum computing. The engineering skills I learned early on at Target have been applied to create efficient cloud infrastructure and algorithms for quantum computing. As my decade in industry approaches, I find myself equating my current work with my academic work more often than I would have predicted. At some point I realized that nearly every job requires the employee to produce some sort of product. By focusing on “products” we can more easily compare some of the features of academic and industrial jobs for mathematicians.

Research The peer-reviewed paper is arguably the main product of academic mathematicians. At a research institution it is often the only product that truly matters. It is difficult to find similar products in technical roles in industry that hold such sway. Research articles have two unique features: permanence and novelty. Once written they expand the field by creating or improving upon the current state of knowledge (theorems, ideas, connections between ideas); and they are not meant to be replaced, but extended or improved upon, a concept I will touch on below. If we rephrase “knowledge” as “mathematical tools,” then it becomes more clear that there are many products in industry that expand an existing toolbox. Take recommendation algorithms, where a retailer is trying to improve the click-through and purchase rate by recommending items to customers. Often these recommendations are made by finding items that are close to each other in a vector space, a task for which neural networks shine. For example, a team I was on built a clothing recommendation model (a “fashion model”) using a large neural network. These models typically embed data into two hundred or more dimensions. After an initial training stage to acquire a general visual model, projecting the arrangement of the data into two dimensions for visualization showed obvious collections of clothing such as sweaters, athletic shoes, and dress pants all separated into distinct clusters. The next step in the modeling process was to use past customer purchasing patterns to associate styles across clusters. For instance,

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Early Career the extended model learned from customer preferences the visual attributes of sweaters that are preferred with a certain style of pants. With the extended training, the model would rearrange the clusters and one could soon see highheels near dresses or warm hats near gloves in the twodimensional projection. Recommendations could then be made based on these nearby associations. Updating an algorithm might be an easy improvement to an old one, or it might take months to get an entirely new data pipeline and code written. Algorithm updates are one way that novel work expands the current knowledge base in an industrial field. In the fashion recommendation example, we built our own small GPU computing cluster plus acquired new data sources due to the novelty of the approach. The algorithm and coding development stretched the project over more than a year, and the outcome was a new method of clustering that (to our knowledge) had never been used before. Even within industry, one aspect of knowledge creation that is underappreciated is internal documentation. When you put many highly educated individuals in a group working on similar problems, you are likely to see very nice mathematics. This does not often make it to the public realm, but is recorded in the internal documentation outlining the theory behind the algorithms that are implemented, as well as insightful justifications for the business as a whole to support their development. In all organizations where I have worked, years of reference documentation exist that effectively constitutes large literature reviews for the algorithms implemented over time, as well as starting points for future research and algorithms. The permanence of academic papers contrasts sharply with the ephemeral nature of most tools in industry, which are assumed to have a finite lifespan. While a new algorithm may provide a novel improvement over a past algorithm, changes in customer behavior (think yearly fashion trends or shopping habits after Covid-19), hardware improvements, or updates in the underlying software libraries may necessitate modifications to the underlying code or algorithms themselves. In this way, industry software tools have a semipermanent character.

Service Faculty service, such as curriculum development, hiring, and advising, is an often-overlooked component of a professor’s time. Therefore, it is important to include this as an aspect of an academic career. The work done in service facilitates the maintenance and growth of the department. It is hard to argue that a tangible product is produced, and there are parallels in industry. This type of work is not performed by individuals at all levels in industry. Typically, such organizational work is aggregated up to director (manager) positions or higher.

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In healthy organizations, decisions about technical direction originate from individual members of technical teams, those who best understand the problem space. Nevertheless, decisions surrounding hiring, resource allocation, and prioritization of projects most often flow topdown. There can be a lot of noise while such decisions get figured out, especially in a large corporation with pressures coming from many angles. The benefit of aggregating these service-level roles at a managerial level is that it frees up individual contributors to focus more on the research and code development. Starting a career at a large company can give a researcher time to perform familiar tasks while learning the ropes of the corporate environment. For instance, with Target’s well-defined organization structure, I was able to take my time to develop algorithms with help and advice from more experienced team members, while gradually accumulating knowledge about the best way to make larger impacts. Smaller organizations or startups will often require a much larger industry knowledge base or at least broader software experience for those just starting out. This required breadth and depth of experience may be an ideal fit for some, but may stretch others’ capabilities.

Teaching and Speaking In many academic institutions teaching takes the place of the research paper as the principal product. A good instructor introduces novelty in their teaching and leaves a lasting impression. So giving teaching equal weight with journal articles as academic products is justified, modulo the institution and its value system concerning research versus teaching. For simplicity, I equate teaching and presenting here, since success at both requires clear exposition of the underlying material, even though the audiences differ. We can find numerous parallels in industrial activities, and I would argue that those with prior teaching experience are well positioned to flourish when these situations arise. For instance, selling novel software and hardware requires demonstrating one’s products to customers. A “demo” is nothing more than a polished talk or lecture, very similar in content to what academic mathematicians prepare regularly. Many people in industry have spent little time speaking in public, so prior teaching experience often provides a huge benefit and can boost one’s profile as an effective presenter. In every industry role I’ve worked in, I have found myself presenting results internally, demo’ing results or products for customers externally, or speaking at a conference. The presentation skills I gained in academia have been invaluable in these situations. Another example is when higher-level management requires updates on a project’s status. Being able to efficiently relay information, either formally in a short presentation, or informally at your desk, is crucial. Equally

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Early Career important, is understanding the technical level of the person asking the questions, and presenting answers clearly at that level. Teaching hones this skill. It is possible for incredibly intelligent and curious people to attain high positions in corporations and still have never heard the word “vector.” That’s where the patience and expository skills gained from teaching shine and create useful connections across diverse groups and organizations.

Conclusion Hopefully in this short article I have helped graduate students and early-career academics understand some of the similarities and differences between academia and industry. By viewing these realms in terms of contributions or products, it becomes easier to see across the chasm that sometimes separates academic mathematics from industrial mathematics. In many ways, roles in industry leverage very similar skills, and it’s simply a matter of adjusting a viewpoint or translating a previously well-honed skill in order to make a significant contribution. Personally, I have enjoyed the focus on software algorithms and the complexities inherent in working with diverse business partners more fulfilling and engaging than similar work I could be doing in academia. If Early Career readers are interested in exploring a career in industry, data science provides a smooth entry ramp and comes in many forms, from retail analytics to banking and finance. Don’t be afraid to ask acquaintances you may know for informational meetings over coffee. Talking to people about what they do will help you determine if that’s an area you would want to work in. Another thing to remember is it is unlikely that there is a job in industry requiring your specific mathematical skill set. This means that focusing on a more general skill set plus your computer skills on a resume, with publications secondary, is often the best approach.

Teaching Mathematical Finance John Holmes Although the launch of my academic career in mathematics was not auspicious, it has, without a doubt, impacted both my teaching and my appreciation of students. I began my undergraduate studies at Wabash College majoring in economics, earned a C in the required freshman calculus class, and vowed to never enroll in another math course. This intention was contravened by my decision to learn finance in order to become an investment banker or hedge fund manager. My goal was clear and motivation obvious: make a lot of money. An economics professor, Frank Howland, suggested that I would be more marketable with stronger quantitative skills if I had a minor, if not major, in math. Despite my misgivings, I followed his suggestion and enrolled in my next math classes—probability and interest theory—which are typically taught to students interested in actuarial science. Finding these classes much more tolerable than calculus, I decided that a minor or major in math was doable. It was upon enrolling in my first real math course, abstract algebra, taught by Mike Axtell (now at U. St. Thomas), that I fell in love with math. Soon after I knew that I wanted to be a mathematician. My passion for mathematics and the beauty in finding connections between different areas has only grown since then. Although I no longer have ambitions to be an investment banker or hedge fund manager, my interest in mathematical finance has endured. In this essay, I will describe what mathematical finance is, how teaching math-finance differs from other math courses, and the challenges and rewards of teaching mathematical finance. Of particular importance are the types of assignments I use to promote learning, how to think about the topics we discuss, and how to think critically about the assumptions underlying mathematical finance models.

What is Mathematical Finance?

Jesse Berwald Credits

Photo of Jesse Berwald is courtesy of Jesse Berwald.

Mathematical finance, for the unfamiliar, can be summarized as risk modeling. We build models to understand sources of risk, how to mitigate it, and the fair price to pay to another investor to assume our risk. In some traditional business, say a bakery, the costs of the building, tools, and ingredients for the production of bread are known before the bread is baked, then sold and profits are calculated. Other businesses, such are airlines, may face uncertain costs after the sale of their product. For example, airlines John Holmes is an assistant professor of mathematics at Ohio State University. His email address is [email protected]. DOI: https://doi.org/10.1090/noti2811

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Early Career may sell tickets for a flight months in advance, before the jet fuel is purchased and therefore, they face an additional risk the simple baker avoided. Likewise, costs on the sale of home insurance can only be calculated in the future after a house is either damaged or not. The return on an investment can only be calculated after the investor takes a financial position, often with uncertain future liabilities. Understanding and finding ways to mitigate these risks form interesting problems within mathematical finance. Within finance, there are several subareas which assume and/or reject particular hypothesis, such as rational expectations or the efficient market hypothesis. Rational expectations is the assumption that, on average, investors are rational and accurately predict the distribution of random variables. In models, this manifests sometimes as a representative agent, who “knows” the true distribution of a stochastic process or random variable, and uses that information to make decisions. The efficient market hypothesis says that all available information has already been incorporated into the prices of assets. Investors are constantly reading balance sheets and SEC reports to get a small edge, and therefore, whatever small advantage an investor can find will quickly close. This manifests itself frequently in models by requiring that there are no arbitrage opportunities. Both of these assumptions are contentious, and the area of behavioral finance began as an alternative to these assumptions. Nevertheless, these (and other) assumptions form the backbone of the area of mathematical finance today. I think it is important for students to think about these assumptions and come to their own conclusions on whether or not these assumptions are sound. Perhaps by thinking about these assumptions we will cultivate more careful practitioners. Therefore, unlike when teaching more traditional math classes, I believe that there is an enormous value to assigning research papers in these courses. Our students are very good at using quantitative skills, however, they may not yet have developed their critical thinking skills to the same level as a math major. I have found that while some students think deeply about these assumptions and their practical implications, others don’t quite have those skills, yet.

How is Teaching Mathematical Finance Different from Other Applied Mathematics? In many applied mathematics courses at the undergraduate level, we build and analyze models. For example, a course on mathematical biology might focus on developing the susceptible–infected–recovered (SIR) model for the spread of infectious diseases. We, too, build and analyze models in the area of mathematical finance. I think a major difference is that when teaching mathematical finance, you will encounter students who are using what you teach, and sometimes more sophisticated models, to

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trade assets using an online brokerage firm on their phone. They will often ask you practical investing related questions which range from extremely naive to very advanced and sophisticated. This is scary. I am not a financial advisor, and I am confident most professors are not financial advisors. When we build models, for say, the price of a put or call option, we ignore an enormous amount of real world sources of risk. This may leave students with the erroneous belief that they understand something about investing in options, when the reality is that they don’t know much, and we are only conducting an academic exercise in modeling. Hence, I think that it is enormously important to emphasize our lack of understanding and share anecdotes of how the real world differs from our model. For example, when I was a graduate student I read in the Wall Street Journal (WSJ) the mistake a naive investor made which bankrupted his family. In our models, we frequently assume that markets are liquid; investors can buy or sell (or short sell) any number of assets at any time. As reported in the WSJ, the investor in question would find companies on the brink of bankruptcy. They would have large balloon debt payments due, with little or no income, and therefore, he was sure they would go under. In order to take advantage of this, the investor would short sell their stock, or trade in derivatives whose payoff was large in the event the value of the stock decreased in the future. However, these positions opened the investor to an unlimited liability (which sounds bad because it is bad). This worked for him several times until one day the value of the stock he shorted jumped from pennies to several dollars per share. His brokerage firm attempted to close his position before things went so badly for him, however, because of illiquidity in this market, they were unable to. Even though his investments were only on the order of several tens of thousands of dollars, his liability was much larger. Even after selling the family house and liquidating his children’s college funds, he was unable to pay this debt. I often caution my students with this and other anecdotes. There is a lot in the real world that is different than the models we teach in mathematical finance, and students should be wary of using what we teach without learning much, much, more.

The More Challenging Aspects of Teaching Mathematical Finance What I find most challenging about teaching a mathematical finance course is the variety of backgrounds my students have. Whether you teach a 200 level introductory class, or a graduate class on mathematical finance, you will find that some of your students will have already taken several courses through the business school or economics department on the subject. Other students have pursued

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Early Career their interest in the subject by reading books, such as Graham’s The Intelligent Investor or by following social media influencers on Youtube or similar platforms. For other students, this is their first time discussing topics such as stocks and bonds in an academic setting, and they have had no real interest in learning about issues related to finance on their own. This past semester, I had a student write in my evaluations that the course was challenging because they didn’t figure out what it meant to short sell a share of stock until just before the final exam. It is difficult to dial in the level of the course when your students have such a widely varied background on the fundamental topics being discussed. In typical math classes, we have prerequisites in order to ensure students have a similar background, and ideally are capable of learning material at a similar pace. This isn’t possible when teaching a graduate-level course on the subject in a mathematics department. What’s more; I have found that, in the departments in which I have taught, there are few mathematicians who are interested in teaching such a course. Therefore, I frequently try to open courses to advanced undergraduates as well as graduate students. The result is a variety of backgrounds in both mathematics and finance, and it is impossible to teach the material at a level and pace suitable for all of my students. My first experience teaching mathematical finance came as a teaching assistant for my PhD advisor, Alex Himonas. Alex and his collaborator, Tom Cosimano, developed a course “Mathematical Methods in Financial Economics” at Notre Dame, which was widely popular for both graduate students and undergraduate students. The course served as a survey of the most widely used tools in the area, with interesting applications to motivate their development. One of the important aspects of the course was the inclusion of a course project or paper; something we don’t see in many theoretical math courses. I have found tremendous value in including projects and papers in my courses as well. Over the last few years, I have found that projects and papers serve a variety of purposes, and allow me to teach the subject matter in a more comprehensive and inclusive way. The addition of projects, especially for graduate students with strong mathematical backgrounds, helps the course become more flexible. I sometimes elect to leave technical details of proofs to the more mathematically advanced students to finish as projects, and ask my students with more comprehensive backgrounds in financial economics to explain interesting phenomenon appearing in the market, or applications of the topics to our class. During the spring 2020 semester, when our lives were first disrupted by Covid and online classes become the new modality for many of us, I reimagined the last quarter of my mathematical finance course. The market was in

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turmoil; the Federal Reserve seemed to be panicking and there were talks of a stimulus package to stave off a serious recession. In the first three or four weeks online, I finished most of my lectures on the derivation of the celebrated Black–Scholes option pricing formula. I could tell my students were struggling with the online format, and so, we reimagined the remainder of the course. Rather then continue with lectures, my students agreed to research previous economic recessions, stimulus packages, and policies implemented by central banks around the world and then each in turn, lead the class in discussion. They came to our online class and debated whether or not the Federal Reserve should lower interest rates, and by how much, or whether stimulus packages would be effective. They used ideas and mathematical tools we had discussed to evaluate the arguments their peers were making. They researched, read and reported on models discussed in the literature, but which we had not yet discussed in class, and argued about their assumptions and the validity of their usefulness. Our class became alive with debate over Zoom. It was in these weeks that I appreciated the value of discussing topics other than mathematics in a mathematical finance class. I learned that some of my students had amazingly strong backgrounds in economics and could add significant value to the course by sharing their knowledge in this interdisciplinary area.

The Most Enjoyable Aspect of Teaching Mathematical Finance The students! Even though some students ask you scary questions about investing, their interest, curiosity, and enthusiasm is invigorating. They are some of the best students I have taught in any math course. I have found that we can draw talented students into math by using finance as a lure. I think students like seeing the tools from analysis or differential equations used in a novel way. As mathematicians, we might sometimes forget that our students don’t always learn mathematics because they love mathematics. Sometimes the student who was quiet, and seemed uninterested in your analysis course, awakens in your mathematical finance course, even if you are still just discussing analysis. Yes, most of these students stick to their plan and go to Wall Street. However, their appreciation for math grows, and the deeper understanding of the tools they will likely use in their future will hopefully help them be more responsible practitioners. If you are new to the area and interested in getting your toes wet, I recommend teaching a course on interest theory for actuaries. The mathematics will be very familiar; mostly just tools from a second semester calculus course. You’ll learn a variety of terminology and understand topics, such as bonds, a great deal better. What’s more is that you will find the same students who despised learning about

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Early Career series in their previous course, now find them interesting and valuable.

Figure 1. Participants of the 2017 Spring School on Applied and Computational Algebraic Topology.

John Holmes Credits

Photo of John Holmes is courtesy of John Holmes.

DONUT: Creation, Development, and Opportunities of a Database Barbara Giunti, Jˉanis Lazovskis, and Bastian Rieck 1. Origin DONUT1 [GLR22] is a database of papers about practical, real-world uses of topological data analysis (TDA). Its original seed was planted in a group chat formed during the HIM Spring School on Applied and Computational Algebraic Topology in April 2017. In 2019, Barbara Giunti, at the time a PhD student at the University of Pavia, asked the group chat whether anyone had heard of applications of topology in a specific area. Jˉanis Lazovskis, then also a PhD student, at the University of Illinois at Chicago, had been collecting such papers during the spring school and later events, and shared a list of about 10 papers demonstrating TDA applications. The format of an online spreadsheet soon proved too restrictive, and in 2020, they moved to Zotero [GLR20], an application specifically designed to handle bibliographic databases. The number of applications had increased by then to around 30, and Jˉanis and Barbara started to feel Barbara Giunti was a postdoctoral researcher at Graz University of Technology and now is an assistant professor at University at Albany. Her email address is [email protected]. Jˉanis Lazovskis is a docent at the Riga Technical University and a faculty member at RTU Riga Business School. His email address is janis.lazovskis_1@rtu .lv. Bastian Rieck is a principal investigator at Helmholtz Munich and a faculty member of the School of Computation, Information, and Technology at the Technical University of Munich. His email address is [email protected]. DOI: https://doi.org/10.1090/noti2797

the need for smart planning: what if this project grows as we dream, with hundreds of entries and to be used by many? How can we make it searchable and versatile? They came up with a tags and flavors system compatible with the Zotero infrastructure and classified all the papers accordingly (more information about the system can be found in Section 3.2). In the meantime, they started advertising the database, and received immediate positive feedback from the community. Among the backers, they got the great help of Professor Mikael Vejdemo-Johansson, who provided more than one hundred papers from his personal database. Moreover, the year after, Professor VejdemoJohansson covered the cost of the annual Zotero subscription since the volume of papers had by then exceeded the threshold of free storage (the fee has since been footed by Barbara).

Figure 2. Participants of the 2022 workshop Topology of Data in Rome.

In 2022, the Zotero database had reached over 300 entries, all classified according to the tags and flavors system, and it began to show its limitations: one needed to be familiar with the app to successfully find the desired references. Luckily, in Salzburg during the biannual TDA Austrian Meeting and then at the workshop “Topology of Data in Rome,” Barbara met Bastian Rieck, who fell in love with the project and revitalized it with his contribution: the web frontend search engine DONUT. This acronym stands for “Database of Original & Non-Theoretical Uses of Topology,” and references one of the most basic shapes with nontrivial topology, the donut.

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Early Career 2. Motivation The original goal was to have a tool to retrieve all applications of TDA in a specific domain, to provide an overview not only at the specific application level but also of all the areas of applications at a higher level. This tool is useful, for example, to promote the research field, showcasing its richness and power. Having such a tool is particularly handy in preparing introductions of papers and theses and, crucially, when writing grant applications, to reference relevant uses of a particular method. It also serves as a way to attract more researchers to the field; TDA being a highly intersectional and interdisciplinary field, it is open to new contributions from different domains. DONUT is also useful to create or extend projects, for example, by applying TDA in novel domains or overcoming limitations of previous approaches. Another goal in creating this database was to organize existing knowledge, a burning necessity in an age of information overload. For this reason, the tags and flavors include not only the area of applications but also which mathematical tools are used, how the data are retrieved and pre-processed, and how novel the results are in the specific domain of application. Having the information in such a structured format not only helps the practitioners achieve their research goals, but also allows for literature and cross-sectional studies. In the absence of a structured bibliographic format for reporting such details, DONUT serves the important purpose of providing an ever-evolving, dynamic taxonomy of the field.

Figure 3. Histograms of (A) the year of publication and (B) the number of tags for each entry.

to continuously refine rules on adding this tag. Ideally, this tag should involve subtags, to ensure optimal search results. For example, an entry about epilepsy should have as area-of-applications tag medicine, as the general field, neurology as the specification of it, and, finally, the most precise tag epilepsy. Because of how DONUT works (see Section 4), searching for “epilepsy” and not for “tag:epilepsy” will result in all entries that mention the word and thus in an imprecise search output. Mathematical tools used. This class is easiest to tag, as authors are usually clear about the technical description of the analysis process and state the used tools explicitly. We aim to tag all employed tools, not just the ones from TDA, to provide a faithful summary of the context in which TDA is applied.

3. How the Cataloging Works As of May 2023, the database contains over 430 entries. 3.1. Admissibility criteria. To be included in DONUT, an entry must use a TDA technique to analyze data. We therefore exclude applications to other areas of mathematics or computer science, or employing mathematical (even topological) tools that are not part of the TDA toolbox. The entries we index must be either published or available as a preprint on a preprints server (such as arXiv or bioRxiv). We prefer open-access (OA) publications and items with a DOI. Preprints that are later published are replaced with their published version. If the later publication is not open access, a link to the public preprint is kept. Conference submissions are allowed only if published in proceedings; conference submissions that only consist of an abstract are not included. 3.2. Tags & flavors. There are three classes of tags (area of applications, mathematical tools used, and input type) and two flavor labels (innovate and confirm). Every indexed entry must have at least one tag for each class; this is a hard requirement to ensure the utility of DONUT. Area of applications. This is the most difficult tag to add. We hope to harness feedback from the community

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Figure 4. The most popular tags (left) for the type of tool used, and a histogram (right) of how many tags of this type each entry has.

Input type. The third class of tag is conceptually very simple and denotes the data type(s) used in the TDA pipeline, such as grayscale image, point cloud, time series, etc. However, in practice, we find that this tag is difficult to apply as input data usage is often not explicitly stated by the authors, or stated indirectly. Analysis should always be reproducible, and authors’ failing to provide how the raw data were preprocessed to become suitable TDA input severely hinders reproducibility. Flavors. Flavors are not mandatory, both because their classification is more delicate and because not all entries fall clearly in one or the other type. The label confirm states that the findings of the entry are aligned with findings of already-published methods. These entries perform the crucial job of reproducing results, and show that TDA

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Early Career can be used as an alternative method. The label innovate encodes all entries whose results are novel to the specific area of application. A result can be novel, for example, if it comes from a larger dataset that was too big to be handled by other methods, or because TDA extracts more information from the same data, or also because TDA can analyze data that other methods could not. Since this tag is critical in promoting TDA, we decided to be strict about it: if a result is not unquestionably novel, the flavor is not added. As of May 2023, 12 entries are labeled confirm and 59 entries are labeled innovate.

4. Technical Details DONUT is based on a database of bibliography entries that is maintained via Zotero. The advantages of using Zotero are (i) it provides a simple way of searching for publications and indexing them, and (ii) all bibliographical entries are stored as BIBTEX entries. This means that DONUT remains flexible and can be easily switched to another data source in the future, while at the same time not having to worry about issues with data entry. Thus, DONUT consists of three independent components: 1. An importer for one-way synchronization between Zotero and the database of entries. 2. A fulltext search engine for handling queries and maintaining the entries. 3. A web frontend for interaction with the fulltext search engine. The importer is realized as a stand-alone program, making use of the Zotero API via Pyzotero [Hüg19]. The result of the parse process is a sequence of BIBTEX entries. Each of these entries are then inserted into Xapian, an open-source full-text search engine. Xapian indexes bibliographic information of documents and makes them accessible via a well-defined API. Finally, a web interface based on Flask, a Python frontend for web development, interacts with the database, depicts the results, and renders all queries. We briefly comment on the choice behind the search engine and the frontend. 4.1. A full-text search engine. The benefit of a full-text search engine like Xapian is that the indexing process of structured document data is full of hidden complexities. For instance, are “high-dimensional” and “high dimensional” the same? How are simple spelling mistakes such as “simplical” instead of “simplicial” handled? How are transliterated spellings (“Pawel” instead of Paweł) or approximate spellings (“Pavel”) treated? The frontend by Zotero ignores such questions and only permits simple queries that match a given string perfectly. Xapian, by contrast, is language-aware and can be set up to permit alternative spelling suggestions for queries. Since the utility of DONUT hinges on the quality of its results for a given 1642

query string, we opted to index as much information about a bibliographic entry as possible. As a result, DONUT is able to find documents more quickly than Zotero (with query times ranging in the lower millisecond range) and provide more depth to queries. Currently, only the content of BIBTEX entries is used when searching, which includes the abstract, but not the full text. 4.2. Frontend. Users interact with databases typically through specialized query interfaces that are, ideally, as easy to use as Google. Using Flask, a Python-based web framework, we provide such an interface (in some sense, end users might perceive DONUT to be the web interface, but as outlined above, DONUT actually consists of multiple parts). The design choices behind the interface are first and foremost driven by speed and simplicity, following a minimalist design philosophy. The search interface will work well on big screens and small screens alike, and care has been taken to follow accessibility guidelines.

Figure 5. User interface with a search term (only top result shown).

DONUT does not track users by means of cookies or related technologies. General web server logs are stored in anonymized form, making it impossible to identify users. These logs are used for diagnosing problems and providing summary statistics about accesses to the database. Logs are stored encrypted and are automatically deleted on a rolling basis. DONUT is thus fully compliant with GDPR and goes well beyond the “best practices” of contemporary websites. 4.3. Code. We make the code for DONUT available using GitHub, using a BSD 3-Clause License.2 This license essentially permits anyone to use the code and modify it, 2https://github.com/Pseudomanifold/DONUT

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Early Career provided our original copyright notice remains intact. By making the code publicly available, any missing functionalities of the frontend or backend can be raised easily and addressed by the community or us. We also hope that DONUT will inspire similar initiatives, since the code is not specific to applications in topology and could easily accommodate other scientific domains as well. 4.4. Example queries. The frontend of DONUT can be used just like a regular search engine would be used. Any queries are searched for in any of the fields of the database. Thus, searching for general will find an entry called “The Euler Characteristic: A General Topological Descriptor for Complex Data,” but also all documents that have general somewhere in their abstract, for instance. This mimics the default behavior of search engines, which do not care about the location of a search term within a website. To accommodate the needs of researchers, the DONUT frontend supports more refined queries and various operators: searching for title:”euler” will return all entries that have the word “Euler” somewhere in their title. Similarly, one can search for document tags and authors, via tag: and author:, respectively. Automated normalization of queries. Concerning the aforementioned issues of different spellings, one prominent feature of the query interface is that it supports transliterated spellings of names. Hence, to stay with the original example, the search term author:pawel will show results that include both the spelling “Paweł,” as well as the spelling “Pawel.” Similar rules apply to names with umlauts and other special characters. This functionality is unique to DONUT and not provided by the Zotero query interface. Spelling suggestions. Moreover, DONUT is capable (in contrast to Zotero) of suggesting other search terms to users based on similarity. DONUT will never change the search query on its own. It will, however, suggest alternative concepts or spellings. For instance, searching for homotopy brings up homology as a potentially related query. Searching for homollogy, on the other hand, will bring up no results, prompting DONUT to suggest “Did you mean ‘homology’?” We expect to further improve this functionality over time. 4.5. Experimental features. We also use DONUT as a platform to experiment with various ways of making application papers more accessible and queryable. For instance, we provide a “landscape visualization” that shows all indexed documents, following a natural landscape metaphor [FMM10]. This provides a way to interact with documents and potentially find similar papers. Another ongoing improvement involves the indexing of the text of open-access publications. This is considerably more complicated than integrating overall bibliographic information (authors, abstracts, . . . ) since it requires

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being able to process PDF files. For entries whose text we can successfully process, we will incorporate the text in the search engine, meaning that keywords or phrases that appear only in the text of the entry will be made accessible to readers. To ensure compliance with copyright, we will only do this for open-access publications.

5. Future Opportunities We want DONUT first and foremost to be a useful tool by the community for the community. As such, we believe that the most useful opportunities consist in expanding the taxonomy, that is, expanding the tagging system. Going from “more general” to “more specific” leads to a natural hierarchy of tags. For instance, an entry whose data tag is graphs could be assigned a more specific tag of the form graphs:directed if that is its context. Over time, we expect that such a hierarchy will become more refined, allowing both unspecific and highly specific queries. To aid users in their interactions with the hierarchy, we plan on implementing a “tree visualization” of tags. When viewing an individual entry, we will make excerpts of the hierarchy visible, making it possible for users to navigate within the tree. We hope that DONUT continues to be a useful tool for our community. Everyone is warmly invited to contribute to DONUT in various ways. We are open to additional suggestions for inclusion, updates to the web interface, as well as suggestions for new functionality. ACKNOWLEDGMENTS. The authors would like to thank the Hausdorff Research Institute for Mathematics for organizing excellent mathematical events, including the one at which the idea for this database was born, and Professor Mikael Vejdemo-Johansson and Professor Nina Otter for their contributions to the database. B.R. is grateful for discussions with Lukas Hahn, Maximilian Schmahl, and Daniel Spitz. B.G. was supported by the Austrian Science Fund (FWF) P 33765-N.

References

[FMM10] Sara Irina Fabrikant, Daniel R. Montello, and David M. Mark, The natural landscape metaphor in information visualization: The role of commonsense geomorphology, Journal of the American Society for Information Science and Technology 61 (2010), no. 2, 253–270. [GLR20] Barbara Giunti, Jˉanis Lazovskis, and Bastian Rieck, Zotero database of real-world applications of Toplogical Data Analysis, 2020. https://www.zotero.org/groups /tda-applications. [GLR22] Barbara Giunti, Jˉanis Lazovskis, and Bastian Rieck, DONUT: Database of Original & Non-Theoretical Uses of Topology, 2022. https://donut.topology.rocks. [Hüg19] Stephan Hügel, Pyzotero, Zenodo, 2019. https:// doi.org/10.5281/zenodo.7057503.

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Early Career

Barbara Giunti

Janis ˉ Lazovskis

Bastian Rieck Credits

Figure 1 is courtesy of ©Hausdorff Research Institute for Mathematics (HIM), Bonn. Figure 2 is courtesy of Ryan Budney. Figures 3–5 and photo of Jˉanis Lazovskis are courtesy of Jˉanis Lazovskis. Photo of Barbara Giunti is courtesy of Barbara Giunti. Photo of Bastian Rieck is courtesy of Andreas Heddergott.

Thinking About Failure in Data Analysis and Beyond Roger D. Peng Data science is a career that has expanded greatly over the past 10 years, with data science touching almost every aspect of our daily lives. Early career statisticians and mathematicians occasionally come to me and ask about the job of the data scientist, particularly in an academic setting. This can be a challenging question to answer as the job itself is continuously evolving both inside and outside of academia. My goal in this article is to touch on a concept that I think is common to the work of all data scientists, which is the question of what it means for a data analysis to fail. As a statistician working in academia, I have a variety of jobs, including analyzing data and teaching about Roger D. Peng is a professor of statistics and data science at the University of Texas, Austin. His email address is [email protected]. DOI: https://doi.org/10.1090/noti2798

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analyzing data. In the context of data analysis, I often think about what does it mean for a data analysis to fail? Usually, this is amongst the first questions students ask me because they want to do well in the class and outside of it. And yet answering this question is not necessarily straightforward. You can imagine how much the students love hearing that! My general thinking about failure in any context is that the word “failure” is a bit overused and arguably has too many meanings. I like to split those meanings into three categories: funnies, anomalies, and (genuine) failures. What separates these three categories of outcomes are people’s expectations and the consequences. Funnies are unexpected outcomes that are interesting but don’t necessarily change what you would do in the present or future. It’s often useful to understand how these outcomes occur, but it’s not necessarily an urgent matter. For example, if I’m analyzing a large dataset and one data point appears far different from what would consider a typical value, I might continue with the analysis anyway. But I want to eventually find out what caused that data point to be corrupted. In the context of data analysis, anomalies are larger deviations from what we expect and can change how we do the analysis or how we collect future data. If in the previous example I had loaded the dataset and saw that half the observations were corrupted, I might pause and try to figure out what is going on. But that is only because my expectation was that all of the data points would be clean. Another, perhaps more experienced analyst, might know that this is what the data always look like. A person’s expectations are critical to defining what is anomalous or unexpected. In other words, one person’s anomaly can be another person’s expected outcome. When I was in graduate school, I submitted what I thought was a strong paper to a journal but the journal sent it back asking for a revision. I was so disappointed and frustrated at this outcome. But a senior professor later told me that in his entire career he had never had a paper accepted on the first submission. In this situation, the only thing that differed between me and this senior professor was our expectations. Failures are the most severe category of outcome and distinguish themselves from anomalies in that people’s expectations for what should happen are largely in agreement. If I am driving my car and notice that the brakes are no longer working, that is a failure. I would be hard-pressed to find a person that doesn’t expect a car’s brake to work all the time. The interesting thing about failures is that although they are easy to observe, their root causes may not be immediately obvious. I once had a student fail my class because he didn’t hand in any work. Later, I discovered that he thought he had dropped the class early in the

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Early Career semester, but for whatever reason did not. Was dropping the class ultimately his responsibility? Yes. Was this the kind of thing that could have happened to anyone? Also yes, which is why I subsequently added an announcement in the first lecture about how to properly drop the course if students didn’t think it suited them. On a somewhat larger scale, in the early days of the genomics revolution, a flashy series of papers was published that ultimately had to be retracted because of numerous data analysis problems. Although it was arguably straightforward to identify the people involved in making these mistakes, figuring out how to prevent something like this from occurring again was a challenging question. Ultimately, a National Academy of Medicine panel had to be convened to develop recommendations for future studies of genomics-based predictors.

Potential Outcomes Statisticians like to talk about data analysis failures because it’s a little like Monday morning quarterbacking. It’s often easy to recognize a failure after the analysis is done. But wouldn’t it be better to recognize a failure before the analysis is done? Considering the question of how do funnies, anomalies, and failures occur in a data analysis is important because thinking about them is a good way to prevent them from actually occurring. However, thinking about the ways an analysis can produce an anomaly or failure is easier said than done. First of all, it’s not always a fun activity. But more importantly, it requires an active imagination, an ability to think about what might happen or what might have happened. It requires thinking about what was not observed rather than what was observed. Catching an anomaly before it happens requires an understanding of the potential outcomes of an analysis. Most data analysis questions will admit a range of possible approaches to analyzing the data and it is up to the analyst to choose one. Given a plan for analyzing the data, there is then, we can imagine, a set of potential outcomes that this plan can produce before it is applied to the data. Once we apply the plan to the data, we will observe one of these potential outcomes. For example, if our analysis is to take the arithmetic mean of a set of numbers, the set of potential outcomes is some subset of the real line. Once we characterize the set of potential outcomes, we can divide that set into two broad regions: expected outcomes and unexpected outcomes. When we apply our analysis plan to the data, the outcome will fall into one of these two regions. Outcomes that fall into the “unexpected” region are what I am characterizing as funnies, anomalies, or failures here. Continuing our example from above, we might expect that the arithmetic mean will fall into the interval [3, 7]. If the observed mean ended up

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being 10, that would be unexpected. If it were 4 then that would be as expected.

Going from Success to Failure The traditional notions of success and failure would seem to suggest that we should favor success over failure. But in the data analysis context, what we need to consider is how an analysis can go from success to failure and vice versa. If an analysis outcome is a success and is as expected, it is important to ask “How could this have failed?” If an analysis outcome is a failure and is unexpected, it is important to ask “How could this have succeeded?” There is therefore a common task when it comes to the observed output of a data analysis, regardless of whether it could be considered a success or a failure. That task is to consider the entire potential outcome set and determine what could cause one to observe a different outcome than what was actually observed. In the case of failure, this scenario is a bit easier to understand. When I observe an unexpected outcome, usually I am highly incentivized to get to the bottom of what caused that to occur. It might have been an error in the dataset, or a problem in the data wrangling, or a misunderstanding of how the methods or software work. Finally, there might have been a misunderstanding in our expectation (i.e., in the underlying science) and that perhaps this outcome should have been expected. In the case of success, it is critical that we apply essentially the same thinking, especially in the early stages of analysis. We should be getting to the bottom of what caused this (success) to occur. In this case, it is still possible to ask whether there might have been an error in the dataset, or a problem in the wrangling, or a misunderstanding of the methods, software, or underlying science. It is sometimes valuable to ask what would happen if we induced some sort of problem, like an error in the dataset, or an outlier, or a misapplied statistical model (sometimes this is called sensitivity analysis). In both success and failure it is valuable to consider the unobserved outcomes in the potential outcome set and ask whether we actually should be observing one of those other outcomes, perhaps because there exists a better model or because the data should be processed in a different way. It is this consideration of the potential outcomes of an analysis, as well as the alternating between success and failure, that drives the iterative nature of data analysis. Ultimately, we want to come to a place where we feel we understand the data, the data generation process, and the analytic process that leads to our result.

Generalizing from Failure When I hear people (including myself) say that data analysis is learned through experience, I realize now that what we mean is that experience is what allows one to

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Early Career build up that active imagination of what could happen. Producing the observed data analytic outcome requires specific skills—fitting statistical models, data wrangling, visualization—that can largely be taught in the classroom. But building the set of potential outcomes becomes easier and faster with experience as we observe more outcomes across many data analyses. The way that we generalize our experience across different data analyses is by enriching and expanding our set of potential outcomes for use in future analyses. What was once unexpected might become somewhat as-expected. And because such outcomes are expected, we know to watch out for them and we learn the techniques for checking on them. But there is another way that we can “learn from experience” that has the potential to take a lot less time—— collaborating with other people. Other people may have more experience or they may have different experience. Combining people with different experiences on the same analysis can produce a similar effect to a single person having more experience. Each person has seen different outcomes in their past and together they can produce a potential outcome set that is much larger than they could produce on their own. In this way, people can gain “experience” by working together. And the more diverse the experiences of the individual collaborators, the richer and larger the potential outcome set will be that they can construct and imagine.

Roger D. Peng Credits

Photo of Roger D. Peng is courtesy of Roger D. Peng.

/masterblog/count-me-in-with-della-and-deanna-a -podcast.

Dear Early Career I have postdoc offers from an institution in the US as well as one overseas. I am excited about both offers and have always wanted to live abroad. Which offer should I take? —Job Applicant

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Dear Job Applicant, Congratulations on the completion of your PhD and your postdoc job offers! Your time as a postdoc is crucial in establishing your path as an independent researcher, and you will want to find an environment that encourages your ability to do this. There are several factors to consider when choosing a postdoc position. Who is your postdoc mentor? Ideally, you will find a postdoc mentor who is actively engaged in research, who you get along with mathematically, and who will make time on a regular basis to meet with you. Before taking either postdoc, can you arrange a visit or a Zoom call with your potential mentor to discuss projects that you may work on and papers to read during your appointment? There is no doubt that having a postdoc at a well-respected university will look good on your CV, but having a good mentor can steer the course of your career and have a lasting impact on your research path. Will you be provided with travel funds? The mathematical community you connect to during your time as a postdoc is of crucial importance to establishing your career as a mathematician. Your community will include the people who are at the institution you choose to work at, as well as the other postdocs who will be there at the same time as you, but it also includes the people in the workshops and conferences that you attend. For this reason, you will want to consider if the postdoc offer comes with travel funds so you may attend conferences in your area and give talks. This exposure is important to ensuring you receive further job offers. It will also fuel future collaborations, research connections, and creative endeavors. Alternatively, there are travel grants available for postdocs through, for instance, the AMS, the Simons Foundation, and mathprograms.org. You can also find online communities and talks to stream. For instance, check out the AMR website at https://amathr.org. Also see the Count Me In podcast with Professors Della Dumbaugh and Deanna Haunsperger https://www.mathvalues.org

How much teaching will be required? Having teaching experience will broaden the range of jobs available to you after your postdoc appointment. At the same time, you will want to make sure that you will have ample time to conduct your research and to attend meetings in your area. Several institutes, such as ICERM and Institut Mittag-Leffler, have semesterlong programs in specific areas. You might ask your

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prospective postdoc mentor if it would be possible to take a semester away from teaching to take advantage of such an opportunity. Are there advantages to going overseas? Taking a postdoc overseas may be a profound time for personal growth and a chance to experience a new culture. Additionally, it is an opportunity to broaden your mathematical network. You will meet new potential collaborators, see new perspectives on your field, and attend conferences that would otherwise be too far to attend. Also, if you want to spend some time overseas, doing so before you have a family or other obligations might be more convenient. Do you really have to choose? I received postdoc offers at the IMA at the University of Minnesota and at the Technion in Israel. Both were a good fit for me for different reasons, and I was torn between these two options. Lucky for me, my mentor at the IMA, Professor Fadil Santosa, suggested that I take both jobs! He suggested that I go for one year to Israel and then do two more years in Minnesota, and I accepted. I spent the summer before going to Israel in Minnesota so that I could establish a project with Fadil at the IMA, and then I started a new project in Israel with my mentor there, Amos Nevo. This worked well for me, and I had many opportunities to travel and collaborate. While this option might not work for everyone—there is also something to be said about staying in one place for the duration of one’s postdoc—it may be something to consider.

It’s almost time

to celebrate! Monday, November 27, 2023

—Early Career Editors Have a question that you think would fit into our Dear Early Career column? Submit it to Taylor.2952 @osu.edu or [email protected] with the subject Early Career. DOI: https://doi.org/10.1090/noti2796

Join us on Monday, November 27, 2023 as we celebrate members via “AMS Day,” a day of special offers on AMS publications, membership, and much more! Stay tuned on social media and membership communications for details about this exciting day.

Spread the word about #AMSDay today!

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In Memory of Andrew J. Majda Bjorn Engquist, Panagiotis Souganidis, Samuel N. Stechmann, and Vlad Vicol Andrew Majda passed away on March 12, 2021, at the age of 72. He was hard working until the end even though he suffered from serious health issues for quite some time. Andy was one of the great applied mathematicians of our time. He was passionate in his research and an applied mathematician in the best possible way, by using sophisticated mathematics to Figure 1. Andrew J. Majda advance our understanding 1949–2021. of science and also when needed to make fundamental contributions to mathematical analysis, in particular, to the theory of partial differential equations. He advocated a philosophy for applied mathematics research that involves the interaction of math theory, asymptotic modeling, numerical modeling, and observed and experimental data, as summarized in Figure 2 and often displayed by Andy in his research presentations. Andy’s legacy lives Bjorn Engquist is a professor of mathmatics at the University of Texas at Austin. His email address is [email protected]. Panagiotis Souganidis is a professor of mathematics at the University of Chicago. His email address is [email protected]. Samuel N. Stechmann is a professor of mathematics at the University of Wisconsin-Madison. His email address is [email protected]. Vlad Vicol is a professor of mathematics at New York University. His email address is [email protected]. Communicated by Notices Associate Editor Reza Malek-Madani. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2810

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on in the mathematical science he created, but also in the many students and postdocs he so enthusiastically taught and mentored. He was the son of Polish immigrants and grew up in a working-class environment in East Chicago, where he worked in the steel mills during summer breaks and played on his high school’s football team. He got his bachelor of science degree from Purdue University as a mathematics major in 1970, and his PhD in mathematics from Stanford University in 1973. He wrote his thesis “Coercive Inequalities for Nonelliptic Symmetric Systems” with Ralph Philips as an advisor. Stanford was very important for Andy, not only because of his PhD, but above all that is where he met Gerta Keller. They married and stayed together for the rest of his life. She is a highly accomplished scientist herself and is now professor emeritus in the Geosciences Department at Princeton University. At Stanford, he also got to know the Ralph Phillips’ collaborator Peter Lax, who became a mentor for Andy and encouraged him to join the Courant Institute as an instructor. This postdoc period exposed Andy to the Courant style of applied mathematics and nonlinear PDEs. After three years, he joined the faculty in the UCLA Mathematics Department in 1976 where he revitalized the applied and computational mathematics group. The period at UCLA was followed by five years at Berkeley, 1979–1984. During this productive time, he developed “Majda’s model” for combustion in reactive flows [Maj81], and together with Tosio Kato and Tom Beale derived the so-called “Beale-Kato-Majda criterion,” which characterizes a putative incompressible Euler singularity in terms of the accumulation of vorticity [BKM84]. From Berkeley, Andy moved to Princeton and stayed there from 1984–1995, where he briefly served as director for the Program in Applied and Computational Mathematics. During this time Andy continued his work in applied analysis, obtaining fundamental results concerning, e.g., finite-time singularities in ideal fluids [CLM85, CMT94],

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oscillations and concentrations in measure-valued solutions of the two-dimensional incompressible fluid equations together with Ron DiPerna [DM87,DM88], and turbulent diffusion with Marco Avellaneda [AM90, AM92a, AM92b], just to name a few. It was while working in Princeton that Andy started focusing on another area of applications: atmosphere and ocean science, which became his passion for the rest of his life and where his impact is felt most prominently. Andy returned to Courant in 1994 and founded the Center for Atmosphere Ocean Science within the Courant Institute of Mathematical Sciences.

Figure 2. Andy Majda’s modus operandi of modern applied mathematics, as a symbiotic relationship between (i) rigorous mathematical theory, (ii) numerical analysis and numerical modeling, (iii) observed phenomena and experimental data, and (iv) qualitative and/or asymptotic modeling. From [Maj00].

At Courant, Andy shifted his research efforts to crossdisciplinary research in modern applied mathematics with climate–atmosphere–ocean science. He created a variety of novel models of physical phenomena, such as his “multicloud model” with Boualem Khouider to explain the mechanisms of convectively coupled equatorial waves (CCEWs), his “skeleton model” with Sam Stechmann to explain the mechanisms of the Madden–Julian oscillation (MJO), his multi-scale asymptotic models for the tropics with Rupert Klein and Joe Biello, and many other models. He also designed novel computational methods for data assimilation, uncertainty quantification, stochastic climate modeling, ensemble prediction, and the fluctuation–dissipation theorem, among others. Common themes of Andy’s work in these areas included a focus on intermittency and rare events, for instance in his novel data analysis technique with Dimitris Giannakis called Nonlinear Laplacian Spectral Analysis, and the use NOVEMBER 2023

of information theory and entropy in many of his computational algorithms. He used full mathematical rigor to prove theorems and bring clarity to explanations of intriguing behavior such as catastrophic filter divergence in data assimilation. While Andy’s favorite applications were always changing throughout his career, he maintained his modus operandi of modern applied mathematics (Figure 2) through it all. Andy stayed at Courant for the remainder of his life, even if it meant he had to commute weekly from his home in Princeton. In recognition of Andy’s achievements, he accumulated a grand collection of awards and honors. He was a member of the National Academy of Sciences and received numerous honors and awards including the National Academy of Science Prize in Applied Mathematics, the John von Neumann Prize of the Society of Industrial and Applied Mathematics (SIAM), the Gibbs Prize of the American Mathematical Society (AMS), the Wiener Prize of AMS/SIAM, the Lagrange Prize (awarded every four years by the International Council of Industrial and Applied Mathematics, ICIAM), and the Steele Prize of the AMS. He was also a member of the American Academy of Arts and Sciences, a Fellow of SIAM, and a Fellow of the AMS. He was awarded the Medal of the College de France, twice, and was a Fellow of the Japan Society for the Promotion of Science. He received an honorary doctorate from his undergraduate alma mater, Purdue University, as well as honorary degrees from Fudan University and China Northwestern University. He gave plenary one-hour lectures at both the International Congress of Mathematicians (ICM; Kyoto 1990) and the first ICIAM (Paris 1987). In what follows, Andy’s contributions in several research areas are described in more detail, and many of his friends and colleagues share their reminiscences of Andy.

1. Computation and Modelling At Stanford, in 1976, Bjorn Engquist asked Ralph Phillips for some advice on pseudo differential operators. Phillips wisely suggested talking to Andy who was visiting for the summer. This turned out to be fortunate. The application was computational far-field boundary conditions which Bjorn had experienced in simulating wave propagation with applications in seismology. For infinite or very large domains, an artificial boundary is introduced to limit the computational domain where a wave equation is defined. If no boundary condition is added, which would be ideal, there is however no uniqueness. With the standard Dirichlet or Neumann boundary conditions the initial boundary value problem is well-posed but strong artificial reflections are generated at the new boundary distorting the solution in the interior. Ideally, the boundary conditions should support a well-posed problem and

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Figure 3. Andy giving a lecture.

“absorb” any waves coming from the interior. An excellent mathematical tool for generating and analyzing such absorbing boundary conditions is pseudodifferential operators. Andy’s deep knowledge of microlocal analysis and his ability to rapidly grasp a new field was crucial. Pseudodifferential operators are however nonlocal and therefore not practical in computations. The next step is to approximate the nonlocal operators by local ones, which means differential operators. The obvious choice of using truncated Taylor expansions in this approximation turns out to produce ill-posed problems. By studying a wide class of local approximations in light of the recently developed normal mode analysis, it is possible to show that any order of approximations can be reached by using Pad´e approximations. This research was done during a few intensive weeks, and the first paper [EM77] came out in 1977. This remained one of Andy’s most-cited papers throughout his career. Andy made several significant contributions to numerical analysis, most often in connection with mathematical modeling of a particular physical process. In one important and purely numerical paper [CM80], Andy and Michael Crandall studied the challenging problem of numerical convergence in a nonlinear setting. This result together with a related paper by Kuznetsov and Volosin were game changers, which initiated a sequence of new results during the 1980s and 1990s. A difficulty is that the solution typically becomes discontinuous at a later time even if the initial value is smooth. A weaker notion of solution is needed for existence, and for uniqueness extra entropy conditions are needed and here the Kruzkov entropy is appropriate. Andy and Mike proved 𝐿1 convergence of monotone, conservation form schemes to the correct entropy 1650

solution of scalar nonlinear conservation laws in the multidimensional case. A key tool is a lemma on 𝐿1 contraction. If Andy’s work on absorbing boundary conditions [EM77] started his career in applied mathematics, another step was taken when he joined the faculty at Berkeley, 1979–1984. Using his talent to address open problems in analysis and applied mathematics, he focused on an application area, in this case reacting flows. He developed relevant mathematical models and analyzed their properties. The so-called, “Majda’s model” plays the same role for combustion as Burgers’ equation does for general compressible flow [Maj81]. This simple but ingenious model is phenomenological but explains the interplay between the advection, diffusion, and chemical reactions by just two quantities. It has been used extensively for developing and tuning numerical methods for the more general equations. Andy contributed to the area of reacting flows with a number of other results, but also continued with important works in applied analysis: the existence and stability of multidimensional shock fronts [Maj83, Maj84], and singular limits of quasilinear hyperbolic systems and in particular the zero Mach number limit for compressible flows [KM81], just to name a few.

2. Fluid Dynamics Andy Majda’s contributions to the modern development of mathematical fluid dynamics are immense, and it is not possible to provide a detailed account of his work on this subject in this short note. Instead, we will focus on two topics that have withstood the test of time, where the results of Majda and his collaborators are to date essentially the best available results: criteria characterizing putative finite time singularities in incompressible models, and the stability of multidimensional shock solutions to hyperbolic systems of conservation laws. The quest for finite-time singularities in ideal fluids. The fundamental macroscopic model describing the motion of inviscid flows (these neglect viscous effects) are the Euler equations, written down by L. Euler in 1757. These PDEs encode the conservation of mass, momentum, and energy. When the change in density with pressure is negligible (which is the case for most liquids) the fluid flow is modeled as being incompressible. In spite of being around for more than 250 years, some of the fundamental properties of the resulting PDE model, namely the incompressible Euler equations, remain to date open. Chief among these is the question of finite time singularities: Given an infinitely smooth initial distribution of velocity (and hence pressure), is it possible that in finite time the incompressible Euler dynamics creates nonsmooth solutions (meaning, with an unbounded velocity gradient)? Majda appreciated the importance of this question for our understanding of fluids (e.g.,

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the dynamical creation of small scales, cascades in inertial range turbulence), and his research program from the early 80s to the mid-1990s had a very large component investigating this problem. Majda’s most cited contribution to this subject (joint with Thomas Beale and Tosio Kato) is the so-called BealeKato-Majda criterion [BKM84], which states that the threedimensional incompressible Euler equations form a finite time singularity at time 𝑇 < ∞ if and only if 𝑇 ∫𝑇initial ‖𝜔(𝑡, ⋅)‖𝐿∞ 𝑑𝑡 = +∞, where 𝜔 = ∇ × 𝑢 is the fluid vorticity (the curl of the fluid velocity). That is, finite time singularities from smooth initial data can occur if and only if the magnitude of the vorticity blows up in a nonintegrable way. To date, this is essentially the best available blowup criterion for incompressible three-dimensional Euler (up to log log corrections), and is widely used as a “test” in numerical simulations of putative Euler singularities. The paper [BKM84] foreshadowed a common theme in Majda’s work on finite time singularities in ideal fluid models: the fundamental role played by vorticity dynamics and Lagrangian geometry. In hindsight, emphasizing the role of vorticity may not seem like that novel a perspective, especially since this was the leading point of view at the start of the 20th century; but one has to understand that mathematical fluid dynamics in the 70s and 80s was to a large extent influenced by a perspective through which fluid motion is an Eulerian dynamical system in a suitable Banach space, where the only important feature of the nonlinear term was that it obeyed certain cancellation properties, and satisfies certain bounds. The Lagrangian vorticity perspective championed by Andy Majda, Peter Constantin, and others was instrumental in the revival of mathematical fluid dynamics at the turn of the 21st century. Subsequently, Majda approached the problem of finite-time singularities for the incompressible threedimensional Euler equations through the analysis (both numerical and analytical) of a hierarchy of simpler toy models in which the potentially catastrophic effect of vortex stretching is in competition with the regularizing effect of advection. Perhaps the simplest such PDE model is the so-called Constantin-Lax-Majda model [CLM85]. Recall that the vorticity field 𝜔 in three-dimensional incompressible Euler solves is Lie-advected: 𝐷𝑡 𝜔 = 𝜔 ⋅ ∇𝑢, where 𝐷𝑡 = 𝜕𝑡 + 𝑢 ⋅ ∇ is the material derivative. The map 𝜔 ↦ ∇𝑢 may be represented as a matrix of Calderon-Zygmund ´ operators (iterated Riesz transforms) applied to the vorticity vector, and thus the rough bounds for 𝜔 ⋅ ∇𝑢 (say in 𝐿𝑝 spaces for 𝑝 ∈ (1, ∞)) are the same as the bounds for |𝜔|2 . This is the motivation behind the analogy with the Riccati ODE 𝑥̇ = 𝑥2 in Lagrangian coordinates. The ConstantinLax-Majda (CLM) model incorporates nonlocality in this analogy, by proposing the one-dimensional evolution NOVEMBER 2023

equation 𝜕𝑡 𝜔 = 𝜔𝐻𝜔, where 𝐻 is the Hilbert transform (the canonical one-dimensional Calderon-Zygmund ´ operator). In turns out that upon denoting 𝑧 = 𝐻𝜔 + 𝑖𝜔, the CLM evolution becomes the complex Riccati equation 𝑧 ̇ = 𝑧2 /2, from which finite time blowup can be directly deduced. This remarkably simple model has inspired a lot of deep subsequent works aimed at balancing the effect of nonlocal vorticity stretching and nonlocal advection. For instance, De Gregorio proposed a generalization of the CLM model in which the 𝜕𝑡 𝜔 term is replaced by the nonlocal advection operator (𝜕𝑡 + 𝑢𝜕𝑥 )𝜔, with 𝜕𝑥 𝑢 = 𝐻𝜔. In certain regimes, this model is also known to blow up in finite time, but remarkably, a number of blowup questions remain open. Note that for a few decades the mathematical fluid dynamics community has criticized such simple toy models of the three-dimensional incompressible vortex dynamics, because they cannibalize the Euler equations to a seemingly absurd degree, and so drawing conclusions about finite time singularities seems not to be justified. We want to emphasize however that very recently Tarek Elgindi has proven the finite time breakdown of classical solutions of the three-dimensional incompressible equations from minimally smooth initial data (𝐶 1,𝛼 for some 0 < 𝛼 ≪ 1). Remarkably, at the heart of Elgindi’s construction lies an asymptotic (as 𝛼 → 0+ ) simplification of the three-dimensional Euler equations in selfsimilar variables, which has an exact solution obtained by solving (a version of) the CLM model! Another toy model (which by now is considered to be “classical”) of the three-dimensional incompressible Euler equations coansidered by Majda (jointly with Peter Constantin and Esteban Tabak) is the so-called surface quasigeostropic (SQG) equation [CMT94]. The SQG equation has its origins in atmospheric dynamics (see the paper of Isaac Held and collaborators on this subject); at the same time, it has a very deep analytic and geometric connection with the three-dimensional Euler vorticity dynamics, while being an equation in two space dimensions (hence, making it more amenable to direct numerical simulations). The SQG equation may be written as the active scalar equation 𝐷𝑡 𝜃 = (𝜕𝑡 + 𝑢 ⋅ ∇)𝜃 = 0, where 𝑢 = ∇⟂ Δ−1/2 𝜃 is the nonlocal velocity. In particular, a quantity similar to vorticity is ∇⟂ 𝜔, is also Lie-advected 𝐷𝑡 (∇⟂ 𝜃) = (∇⟂ 𝜃) ⋅ ∇𝑢, and the map ∇⟂ 𝜃 ↦ ∇𝑢 is given by a matrix of Calderon-Zygmund ´ operators. The numerical simulations reported in [CMT94] indicated a strong and potentially singular front formation in the shape of a hyperbolic saddle for the active scalar 𝜃. Later numerical studies of Ohkitani and Yamada indicated that the direct numerical simulations were instead consistent with super-exponential growth for ‖∇⟂ 𝜃(⋅, 𝑡)‖𝐿∞ instead of finite time blowup, and this collapsing hyperbolic saddle scenario was ruled out

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rigorously as a path toward finite time blowup in the PhD thesis of Diego Cordoba. ´ The work of Constantin, Majda, and Tabak [CMT94] has sparked an immense amount (over 700 citations) of beautiful mathematical works dedicated to the analysis of the SQG equation. It has led to a proof of finite time blowup in the case of patches, and in the presence of a solid boundary. Nonetheless, the question of whether the SQG equations posed on the plane (or on the two-dimensional periodic box) is capable of evolving smooth initial data into a finite time singularity, remains to date open; this is considered to be one of “the” outstanding mathematical challenges of inviscid incompressible fluid dynamics. Coming back to the incompressible three-dimensional Euler equations themselves, Majda returned in the mid1990s to the subject of blowup criteria. Motivated by the coherent structures observed to be ubiquitous in the inertial scales of fully developed hydrodynamic turbulence, Majda, Peter Constantin, and Charles Fefferman [CFM96] considered a geometric vorticity-based blowup criterium for three-dimensional Euler. The so-called ConstantinFefferman-Majda (CFM) criterion states that certain regularity information on the direction of the vorticity vector 𝜔(𝑡,𝑥) gives information on the size of the vorticity vector. |𝜔(𝑡,𝑥)|

The mechanism is that of a hidden cancellation present in the vortex stretching term 𝜔 ⋅ ∇𝑢. The CFM criterion implies in particular that if a pair of potentially singular vortex tubes were to smoothly align during the stretching process, then the growth of vorticity is eventually subdued. This is yet another beautiful example of a leitmotif present in Majda’s work: the Lagrangian geometry of the vorticity in three-dimensional incompressible Euler leads via delicate mathematical analysis, to bounds on the magnitude of the vorticity vector itself, which plays the key role in determining whether or not finite time singularities occur. Multidimensional shocks. Systems of conservation laws are ubiquitous in the modeling of physical processes. In the case of scalar problems, these problems have already been quite well understood for a few decades, in part because several unreasonably nice things are available to the mathematical analyst: maximum principles, monotonicity formulas for the entropy, BV-contraction, etc. This was essentially done in the work of Kruzkov. The story is however very different in the case of systems and, in particular, systems in multiple space dimensions, which is the relevant case for continuum mechanics. For systems, very few rigorous results were known before Majda’s fundamental contributions. There were only special solutions to piecewise constant initial values, so-called Riemann problems, and Glimm’s existence proof for restricted problems in one dimension.

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The prototypical example of a multidimensional conservation law is the compressible Euler equation of gas dynamics, in two or three space dimensions. Given a smooth initial condition for density, velocity, and energy, these equations have, at least for a small amount of time, a unique smooth solution. It is, however, well known that a 𝐶 1 smooth continuation of this solution cannot be defined for all times, due to the emergence of singularities. The prototypical stable singularity is a multidimensional shock: a codimension-one oriented hypersurface, such that on either side of this surface the unknowns (density, momentum, internal energy) are 𝐶 1 smooth functions and such that across the shock surface the density, normal velocity, and energy are discontinuous. The natural requirement that the globally defined fields define a weak solution to the compressible Euler system means that the jumps in these quantities across the shock surface are coupled to the evolution of the shock surface as a free boundary, through the classical Rankine-Hugoniot jump conditions. This naturally leads to the study of discontinuous weak solutions to systems of conservation laws. Majda’s monumental works [Maj83] and [Maj84] gave the first rigorous mathematical results for the existence and stability of multidimensional shock fronts for the compressible Euler equations; more generally, for systems of genuinely nonlinear hyperbolic conservation laws. Starting from a “shock-front” data, i.e., the shock front is given at the initial time together with compatible density, momentum, and internal energy fields, which are smooth on either side of the initial shock, Majda’s results provide a rigorous short-time existence and structural stability of such shock fronts within the compressible Euler framework. Some of Majda’s novel contributions to the mathematical analysis of this mixed hyperbolic free-boundary value problem include the notion of “uniform stability” for shock-front solutions based on normal mode analysis, the associated stability analysis for the linearization of a curved shock front, the nonlinear stability as an extremal form of linear stability, and the construction of the discontinuous solution via a nonlinear iteration scheme. Majda’s monographs [Maj83, Maj84] have essentially closed the subject for the local-in-time propagation of multidimensional shock fronts arising from discontinuous initial data for more than a decade. To date, these are essentially the best-known results in this generality. These works have been extremely influential in the hyperbolic community; for instance, they paved the way for the work of Gu´es, M´etivier, Williams, and Zumbrun who proved the existence and stability of viscous multidimensional shocks. Modern research in this area is dedicated to connecting Majda’s picture of the fully developed multidimensional shock, to the local smooth evolution arising from

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smooth initial data. How do multidimensional shocks emerge from smooth data in the first place? What is their geometry? How do they propagate? Is this propagation unique? Do other singularities emerge at the same time as the shock? This is an extremely active research area.

3. Turbulent Diffusion Starting in 1990, Majda turned his attention to problems related to turbulent diffusion and combustion. Turbulent diffusion is associated with the problem of describing and understanding the transport of some physical entity, such as heat or particulate matter, which is immersed in a fluid flow. In most models, it is assumed that the fluid is undergoing some disordered or turbulent motion. If the transported quantity is passive, that is, it does not significantly influence the fluid motion, it is said to be passive, and its concentration density is termed a passive scalar field. Weak heat fluctuations in a fluid, dyes utilized in visualizing turbulent flow patterns, and chemical pollutants dispersing in the environment may all be reasonably modeled as passive scalar systems in which the immersed quantity is transported by ordinary molecular diffusion and passive advection by its fluid environment. Turbulent combustion is about the propagation of fronts in environments undergoing some disordered or turbulent motion. The general and typical problem of describing turbulent diffusion of a passive quantity is the study of the quenched and or annealing properties of scaled versions of solutions of diffusion partial differential equations and solutions of stochastic differential equations, both transported by a velocity field which is assumed to be either periodic or random–typically a Gaussian random field of prescribed spectrum. The problem is really challenging and there are several important analytic and probabilistic contributions, which even to date do not give the complete answer. In his joint program with Avellaneda which started in [AM90] and continued with several influential papers [AM92a,AM92b], Majda attacked a simplified version of the problem and obtained several exact results. This work represents another example of Majda’s approach to Applied Mathematics—study a difficult problem by considering a simplified but yet challenging version which can be analyzed mathematically and obtain rigorous and new results which show the possible behavior of the underlying application. Instead of looking at the most general problem, Avellaneda and Majda studied the asymptotic behavior of a simple model problem for a passive scalar with incompressible shear velocity fields admitting a statistical description and involving a continuous range of excited spatial and/or temporal scales, and developed a complete renormalization theory with full mathematical rigor. The

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analysis yielded explicit formulae for the anomalous time scaling in various regimes as well as the Green’s function for the large-scale, long-time, ensemble averages, and the renormalized higher order statistics. The simple form of the model problem was deceptive. Indeed the renormalization theory in [AM90] provided a remarkable range of different phenomena as parameters in the velocity statistics vary. These included the existence of several distinct anomalous scaling regimes as the spectral parameter varies as well as explicit regimes where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity. Turbulent diffusion combustion models are reaction diffusion equations perturbed by incompressible velocity fields. The aim is to understand the influence of the velocity field on the speed of evolving fronts resulting from the reaction. In a series of papers, Majda and Souganidis provided the first rigorous results for the effective speed of models with spatio-temporal periodic velocity fields, came up with explicit bounds for a model problem with a velocity field with incompressible shear velocity, and provided several important counterexamples to the general conjecture that the front velocity should be given by the so-called G equation.

4. Climate–Atmosphere–Ocean Science In the 1990s, Andy made the decision to focus his research efforts on problems from climate–atmosphere–ocean science. In approaching this new area, Andy brought his modus operandi of modern applied mathematics, summarized in Figure 2, which is a synergy between rigorous mathematical theory, efficient computational methods, and physical insight. Furthermore, he aimed his sights at the biggest and most challenging open problems. The tropics: Multiscale models, singular limits, moist convection, and dispersive waves. The tropics were seen as the biggest challenge, so they naturally grabbed Andy’s attention. In fact, Andy would frequently recall being told that the tropics are too difficult, which to him only served as more motivation. In jumping into the tropics, one of Andy’s first and biggest impacts was the derivation of multiscale models. With Rupert Klein, Andy used the systematic asymptotic procedure of applied mathematics to derive different governing equations for the different scales of interest, including multiscale interactions between scales [MK03]. Later, with Alexandre Dutrifoy, Andy formulated rigorous proofs of some of these singular asymptotic limits. One factor that drew Andy toward the tropics was that the tropics are home to a very interesting set of dispersive waves. They are called equatorial waves, since they are

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trapped near the equator and propagate around the globe with the equator acting as a waveguide, due to the vanishing of the Coriolis force at the equator. It was only in the 1960s that equatorial waves were discovered and their mathematical description was formulated (by Yanai, Matsuno, and others). Andy described his attraction to these waves by saying (perhaps overly simplistically) that he saw the dispersion curves for equatorial waves and knew he wanted to work on the tropics. Another factor that drew Andy toward the tropics was a major open problem: the convective parameterization problem. The word “convective” refers to clouds and storms, which are essentially manifestations of convection within the atmosphere. They must be parameterized— i.e., represented as an unresolved, subgrid-scale process— in global climate models, since climate models must use large grid spacing to remain computationally feasible. The grid spacing is in the range of 10 to 100 km, which is too large to resolve the fluid dynamics of clouds and storms. The convective parameterization problem has persisted since around 1960 in the earliest days of global climate modeling or general circulation modeling. These two factors—equatorial waves and the convective parameterization problem—come together in a phenomenon called convectively coupled equatorial waves (CCEWs). CCEWs are, as the name implies, equatorial waves that are coupled with clouds and storms: from an overly simplistic point of view, you can imagine that one phase of the wave is rainy, and the other phase of the wave is clear and sunny. The waves themselves have wavelengths of roughly 1,000 km or larger, and so in principle the waves could be resolved by a climate model with a grid spacing of 10 to 100 km. However, CCEWs are coupled with clouds and storms, which suffer from the convective parameterization problem, so climate models have struggled to adequately simulate CCEWs. Furthermore, many aspects of CCEWs were not well understood from a theoretical point of view when Andy began his research on the tropics in the 1990s and 2000s. One of Andy’s biggest contributions in atmospheric science is aimed at these two factors, and it is the multi-cloud model that he developed with Boualem Khouider (see Figure 4). It started as a nonlinear wave model to explain the mechanisms of CCEWs, and it grew into a sub-grid-scale model of clouds and convection for climate models. This work showed how the multicloud model could help to improve CCEWs and other tropical phenomena in global climate models (GCMs). One mysterious phenomenon that doesn’t fit in with equatorial wave theory is the Madden–Julian Oscillation (MJO). Despite many advances in understanding since its discovery in 1971, a generally accepted theory for the

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Figure 4. Schematic diagram of the physical processes in Andy Majda and Boualem Khouider’s multicloud model. It illustrates the great amount of detail in the physical processes that Andy considered in his modeling efforts. From [KM08].

mechanisms of the MJO has remained elusive. Many theories have been proposed but none has been generally accepted. In light of this, the MJO has been called the holy grail of tropical meteorology. Of course it grabbed Andy’s attention. Andy brought the ideas of multi-scale asymptotics to try to explain the workings of the MJO. With Joseph Biello, he showed that CCEWs can impact the larger-scale MJO and help to shape the MJO’s circulation structure. Later, with Sam Stechmann, he formulated an idea of how the MJO works as a multiscale phenomenon in a dynamical model, called the MJO Skeleton model [MS09]. It is a nonlinear PDE model, based on a nonlinear oscillation, and it isn’t a multi-scale model per se but is based on multi-scale ideas. Andy wanted to name it the skeleton model because it captured only the most basic and large-scale features of the MJO (i.e., its skeleton), and not the smaller-scale details that would be part of a true multiscale description. Computational methods: Data assimilation, ensemble forecasting, uncertainty quantification, and information theory. Another major theme of Andy’s weather and climate research was computational methods and probabilistic forecasting, and what we would now refer to as data science. In these areas, data assimilation was one of Andy’s

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favorite topics. Data assimilation is the process of creating the initial conditions for a forecast: a first guess at today’s weather is provided by yesterday’s forecast, and a better estimate is obtained by assimilating observational data into the model. Algorithms for data assimilation have undergone substantial advances in the decades since the advent of weather forecasting, and many of the most widely used algorithms are from the 1990s and 2000s with continuing advances today. Part of the improvement in weather forecast skill over the past decades is due to improvements in data assimilation algorithms. Andy often said that “data assimilation” is a particularly unexciting name for such a vibrant and interesting research topic, but that is the terminology that is in common use. Other names for data assimilation are state estimation and filtering, and Andy became absorbed in this field enough that he wrote a book with John Harlim titled Filtering Complex Turbulent Systems. Two of Andy’s results on data assimilation are described in the next paragraphs, as examples of his many novel ideas in these areas. A first example began when Andy was attending a workshop at the National Center for Atmospheric Research (NCAR) in Boulder, Colorado. As Andy told the story, he was sitting in the audience during a presentation on data assimilation, and was surprised to see the results of a numerical calculation. He noticed that the numerical time step Δ𝑡 was large and violated the Courant–Friedrichs– Lewy (CFL) stability condition, which should have caused the numerical solution to diverge toward an infinitely large value. However, the numerical solution didn’t diverge. Andy noticed this on the spot and was intrigued. He suspected that the assimilation of observational data was providing a stabilizing effect. He soon went on to write a paper with Marcus Grote on this topic, and described how the traditional CFL stability criterion is modified in the presence of data assimilation. As in his modus operandi of modern applied mathematics, he developed rigorous mathematical theory and also computational examples. A second example on data assimilation is Andy’s work on “catastrophic filter divergence.” Andy said that he coined the phrase himself to describe a type of divergence that might seem unexpected: a numerical solution with filtering/data assimilation can sometimes tend toward infinite values in finite time, even though the underlying stochastic dynamical system without filtering/data assimilation is dissipative and stable with the absorbing ball property. So in this case, the assimilation of observational data actually provides a destabilizing effect. Andy said he noticed this property for the first time while he was working on a project with John Harlim. They were developing computer codes for some data assimilation algorithms and their numerical simulations were

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diverging to infinite values. Since they were investigating examples that they thought were tame and well-behaved, they thought their computer code must have a bug. However, when they were unable to discover a bug, they began to suspect that the code was correct and searched for a possible explanation for the numerical divergence that they were seeing. In the end, Andy wrote several papers on this topic with collaborators including John Harlim, David Kelly, Georg Gottwald, and Xin Tong. Again following his modus operandi, Andy developed simple example models to illustrate the mechanisms of catastrophic filter divergence, investigated many numerical examples, and proved rigorous theorems about his ideas. Three additional themes of Andy’s work on computational methods were stochastic modeling/parameterization, uncertainty quantification, and information theory. He had many interesting results over the years, from some relatively early work on stochastic climate modeling with Ilya Timofeyev and Eric Vanden Eijnden [MTVE01], up to his last years with some of his last students, Nan Chen and Di Qi [MQ18]. Among his excellent results, he was particularly fond of his prediction, using empirical information theory in work with Bruce Turkington, of the location of the Great Red Spot of the planet Jupiter. Anyone who worked with Andy or met him at a conference probably saw that Andy approached his work with great passion. He took his work very seriously and wanted it to have an impact on the world, and he expected others to approach their work in the same way. With such passion always turned on, Andy could be blunt in his criticisms, which Andy himself refered to as “tough love.” A discussion with Andy might sometimes feel like the words of a tough sports coach. Perhaps this aspect of Andy’s personality was shaped by his own experiences with sports coaches in his youth. He would often recall his time as a football player in high school and his later attraction to tennis. He was a sports fan and sports were always part of his life. He approached math and science with the type of intensity that an athlete might bring to a sports event. Many people who worked with Andy will miss the energy and enthusiasm that he always carried with him. Andy was the founder and leader of two centers. The first was the Center for Atmosphere Ocean Science (CAOS) at New York University (NYU), a unit within the Courant Institute for Mathematical Sciences. From its founding in the 1990s to his last days there, CAOS grew to approximately 10 professors, and continues to foster a graduate program of approximately 20 current PhD students. The second center founded by Andy was the Center for Prototype Climate Modeling at NYU–Abu Dhabi. It provided a place where new ideas could be aided in the transition from an incipient research idea to a component of an

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operational global climate model. Among many successes of the center was the transition of Andy and Boualem Khouider’s multi-cloud model into a convective parameterization in a GCM. Andy had many colleagues and friends around the world, and perhaps two of his most substantial interactions were with colleagues in China and India. He received an honorary doctoral degree from Fudan University in Shanghai in 2008, and in a span of years he was a special visitor at Fudan University and cohosted several workshops there. Also, in India, he organized several workshops and formed collaborations with experts on the Indian summer monsoon. Another leadership role for Andy was when he led a team of researchers (professors, postdocs, and students) as principal investigator (PI) of a Multidisciplinary University Research Initiative (MURI). The theme of the initiative was “Physics-Constrained Stochastic-Statistical Models for Extended Range Environmental Prediction” and it was funded by the Office of Naval Research with program managers Reza Malek–Madani and Scott Harper. The five years (2012–2017) of close interactions between Andy and the team were some of the most productive of Andy’s career. He published approximately 20 papers per year in this period, and even 30 papers in 2015. He organized annual meetings for the team that were stimulating and enjoyable and were the source of many fond memories and some of the stories retold here.

5. Friends and Collaborators Throughout his career, Andy worked with many people including many students. Part of his legacy is the mark he left not only on mathematics and science but also on the people who make up the mathematical and scientific communities. Below is a selection of memories of Andy from many of his friends, collaborators, and students. Based on Andy’s website and the Math Genealogy Project, Andy had 29 PhD students: Rafail Abramov, Robert Almgren, Miguel Artola, Andrea Bertozzi, Anne Bourlioux, Noah Brenowitz, Jonathan Callet, Dongho Chae, Nan Chen, Ryszard Dziuzynski, Pedro Embid, Jonathan Goodman, Roy Goodman, Margaret Holen, David Horntrop, Peter Kramer, Ding-Gwo Long, Richard McLaughlin, Robert Palais, Robert Pego, Di Qi, Victor Roytburd, Daniel Ruprecht, Steven Schochet, Sang-Yeun Shim, Sam Stechmann, David Stuart, Enrique Thomann, and Qiu Yang. Included in the memories below are contributions from several of his PhD students, and many friends and colleagues.

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Stan Osher I met Andy in 1974, almost half a century ago. The reason these years were as good as they were for me was largely because of this meeting. I was at SUNY Stony Brook, spending a lot of time at Courant and somehow scored a 13th floor office there. I was told about this brilliant young postdoc, also interested in initial-boundary value problems. We met and wrote around seven papers together. But, more importantly, we became friends. Neither of us came from academic backgrounds, but rather from the streets of lower-class ethnic urban neighborhoods. I remember fondly a discussion we had with Mike Crandall where we tried to figure out whose background was more lower-class. Although Andy is seven years younger than I, I learned a lot from him. What I remember most is his belief that the best applied math research should also impact the larger scientific community and society in general. I tried to recruit Andy for Stony Brook, but he wisely chose to go to UCLA. He then recruited me to UCLA Math, a department trying, with ambivalence, to build up in applied math. This turned out well and I am truly grateful. Eventually, Andy moved on from UCLA and we lost touch, but reconnected around twenty years ago. We attended and spoke at each other’s 60th birthday conferences and enjoyed each other’s company in the remaining years. It was always a pleasure to hang out with Andy. He was a terrific applied mathematician, of course, but also an interesting person with a capacity for great compassion. His passing is a great loss to the community and a personal loss for me.

Peter Constantin I met Andy in 1983–1984, introduced by Sergiu Klainerman. We started talking about blow-up in a simple model of Euler equations we were calling “the baby vorticity” model. This turned into our first paper, joint with Peter Lax. Andy liked to think in terms of a hierarchy of models. Next on his blow-up list was the contour dynamics equation for two-dimensional vortex patch boundaries, because, just as the baby vorticity model, it had a nonlinear structure resembling the quadratic nonlocal nonlinearity of vortex stretching in three-dimensional Euler equations. Vortex patch boundaries were proved later to remain Stan Osher is a professor of mathematics and computer science, electrical engineering, and chemical and biomolecular engineering at the University of California, Los Angeles. His email address is [email protected]. Peter Constantin is the John von Neumann Professor of Mathematics and the director of PACM at Princeton University. His email address is const@math .princeton.edu.

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smooth for all time, albeit at the price of super-exponential growth of norms measuring their smoothness. Interested in incompressible turbulence, Andy was closely following the numerical and physics-theoretical approaches of the time. Numerical computations were showing certain Beltrami-like aspects of turbulent flows. This led us to our second joint paper, on the Beltrami spectrum. We continued to talk often, I would visit him in Princeton and he would visit me in Chicago, combining these trips with visits to his mother who lived close by, in East Chicago, across the Indiana border. I was thinking about a class of hydrodynamic models related to Euler equations which I was calling “active scalars” to contrast them with the “passive scalars” which were popular in the physical field-theoretical approaches to turbulence in those days. Andy told me he was talking with Isaac Held in Princeton about a quasi-geostrophic model, which was the same as one of the active scalars. We joined forces, and, together with Esteban Tabak we wrote a paper on the SQG model, with numerical predictions of blow-up. The scenario we were putting forward turned out differently, and numerical studies by Ohkitani-Yamada, and later Schorgoffer and Nie, Jiahong Wu and coauthors, and rigorous results of Diego Cordoba, showed that this scenario cannot sustain blow-up. In fact, an interesting geometric depletion of nonlinearity takes over the dynamics. We wrote a paper, joint with Charlie Fefferman on the subject of depletion of nonlinearity in three-dimensional Euler equations, due to the local alignment of vorticity direction. Andy continued to be interested in incompressible fluids for a while. Our last joint paper was a 1997 paper with Ciprian Foias and Igor Kukavica on the density of the set of ancient solutions of two-dimensional Navier-Stokes equations. Around that time Andy decided to get involved in climate science. He immersed himself in the technical literature and then started to work in the area. Although we have not collaborated since then, I continued to see him from time to time. Last time I saw him was in Victoria in 2019, at a celebration in his honor. Although gravely ill, he was in good spirits, and was challenging his friends to roast him more vigorously, without much success. Andy had a strong influence on many applied mathematical areas: multidimensional shock fronts in hyperbolic PDEs, detonation models in combustion, singular limits in fluid mechanics, incompressible formation of singularities, climate science, statistical modeling, and more. The mathematical sciences community lost in Andy Majda a major figure.

Victor Roytburd Andy Majda was a great mathematician and an amazing man. I had the honor of being the first PhD student Andy graduated. I met him (and became his student) more or less incidentally. Before coming to the US as a Jewish refugee in 1979, I had been a graduate student (aspirant) in Russia for three years (without getting a degree) and worked in industry for nine years on some optimization issues totally unrelated to my graduate research. After some unsuccessful and feeble attempts to find a job, friends of mine gave me wonderful advice to try getting into graduate school, no matter that I came in June and the application process to all graduate schools was long over. I met Joe Keller through some Jewish contacts of my mother-in-law who lived in Palo Alto at the time. Joe very generously listened to me and told me of a star-quality mathematician doing a very important fantastic work in applied mathematics, Andy Majda, and suggested that I talk to him. Andy was a professor at Berkeley but usually spent summers at Stanford, and Keller connected me with Andy. Andy came to our first meeting quite sweaty after a tennis game with his tennis gear in tow (which in my eyes was a serious argument in his favor as a thesis advisor). After looking through my reprints, Andy said that he was not interested in the linear stuff and offered that I work under his guidance on combustion problems, which were one of his interests at the time. Needless to say, I agreed then and there and one of those combustion problems led to my dissertation. Among other achievements, Andy is known for his early groundbreaking work on multidimensional shock waves. Once I asked him why he did not continue working in this or other areas of PDEs. His answer was that while there were several mathematicians who could do work of high quality (comparable to his) in analysis, he saw his principal and rather unique strength in extracting underlying mathematical structures from messy and unwieldy physical phenomena and analyzing those by all methods available, be it rigorous analysis, asymptotics, or numerical computation. From pure analysis he resolutely moved to more applied problems. Andy was second to none, a real artist, in designing simple conceptual models that were amenable to analysis while preserving salient features of underlying phenomena. One of his first models of this kind was the Majda’s model of detonation. By the way, he used to say that scientists are divided into artists and art Victor Roytburd is a professor of mathematical sciences at the Rensselaer Polytechnic Institute. His email address is [email protected].

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critics; I don’t know what he meant by art critics but he definitely saw himself as the former, and an artist he was. As an advisor, Andy was extremely generous and forgiving. He did not intervene much in what I was doing; however, he provided some crucial advice. Also, he made some attempts to teach me how to write mathematics. My Russian papers were very brief announcements of the Doklady type. Andy taught me to forget about this mathematical tendency to economize paper; “important ideas and notions deserve to be repeated.” It was an amazing learning experience to write joint papers with him. Right on the early stages of the work he developed a detailed plan of the future paper that was headed with the title. He knew which scientific questions he wanted to resolve and apparently had no doubt that they would be resolved. Andy was very passionate and brutally honest about science. I remember sitting next to him at a colloquium lecture by Vladimir Arnold on singularities theory that, according to Arnold, was developed mostly by him and his collaborators. Among other examples, he mentioned that shock waves dynamics could be given an elegant interpretation through his results. Andy was literally sitting on his hands not to jump up, telling me that it was complete nonsense (he used a stronger expression though). Finally, he asked the speaker how the scalar theory that Arnold was talking about could have anything to do with the shock waves which are inherently multidimensional. Arnold tried to talk his way out of this unpleasant situation but Andy did not let him off the hook. As a result, I believe the department chair either disinvited Andy from the colloquium party at somebody’s house or elicited from him a promise not to get into scientific discussions with Arnold. Andy secured a sabbatical position for me at Princeton from 1988 to 1989, and I worked with him and his very talented student Anne Bourlioux on detonations. Once, when visiting him at home, Andy stunned me by saying that he was preparing to move in the new direction, atmospheric science. He showed me a rather substantial pile of books and reprints he was ploughing through. Here again he was resolutely planning a turn in a new direction that he considered as an outstanding scientific direction for applied mathematics. Andy was a multifaceted man. I think he played football in high school, and he was an excellent tennis player, although he never agreed to play matches with me; (“Victor, I am so much better than you that it does not make any sense for us to play games”). He surprised me once by admitting that he took ballet lessons while a postdoc at NYU. I asked him why in the world he would take ballet lessons. He answered, “Victor, you don’t understand; that is where the girls are.” He was a cinephile and we shared a love of movies by Bob Altman (even to some of those that

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were not universally accepted as the good ones). He was a loyal and supportive friend to me and others. I remember him regularly visiting Ron DiPerna during his terminal illness in Princeton. Andy Majda was a real mensch. He left us too early; he will be missed.

Robert Pego I was one of Andy’s first PhD students, thanks to a tip from Jerry Bona. It was a period of exciting progress in Andy’s work on fluid mechanics, conservation laws, and numerics, with major works emerging on radiation boundary conditions, monotone difference schemes, vortex blob methods, multidimensional shock fronts, and instabilities of combustion waves. A lively circle of activity included visitors such as Mike Crandall, Ron DiPerna, Tai-Ping Liu, Tom Beale, Ruben Rosales, and Barbara Keyfitz. Andy could switch topics with incredible swiftness and seemed to have all his knowledge instantly on tap. As a teacher, he gave clear and inspiring lectures, and as a mentor he was conscientious. He had played football in his youth and had an expansive and sometimes blunt sportsman’s personality. He was famously not shy about expressing sharp opinions, but was extremely supportive of his students. He would explain to you enthusiastically what was interesting about the little lemma you proved and how it fit into the big picture. After my PhD studies, I felt for a while that math was in black-and-white, while it had been in technicolor with Andy. In the 1980s, Andy’s work diversified rapidly into fields like turbulence and stochastic modeling. At the end of that decade he made a major shift to focus on the mathematics of atmospheric flows and climate. Andy always articulated an inspiring vision of applied mathematics as part of the scientific enterprise in service of humanity. He esteemed the development of good models and better computational methods alongside rigorous analysis of fundamental ideas. Although my work has been analytical for the most part, on more that one occasion I took heart from this heritage to address a problem by a formal approach that could benefit a broad scientific audience. It behooves us all to pick up the torch that Andy has been forced to lay down.

Robert Pego is a professor of mathematical sciences at Carnegie Mellon University. His email address is [email protected].

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Pierre-Louis Lions On top of numerous seminars and lectures at Collège de France, Andrew Majda was invited twice to deliver a research course at Collège de France and thus twice received the Medal of Collège de France. Few mathematicians have received it once and Andrew Majda is the only mathematician who received it twice! This should not be a surprise in view of Andrew Majda’s depth and breadth. His lectures in Paris always attracted large audiences from many different scientific communities: applied mathematics, fluid mechanics, combustion theory, turbulence theory, meteorology, and climatology. . . . He also had real friends in Paris and I am proud to count myself among them. These facts are a good illustration of the unique nature of Andrew Majda’s work—in French, we would say “oeuvre”—and of his singular contributions to Mathematics and more generally to Science.

Charles Fefferman When I knew Andy best, his interests were concentrated on classical problems of fluid mechanics. He had a vision of how that field should progress. He preached that neither numerical simulation, physical intuition, nor mathematical rigor alone could crack the tough, important problems. Rather, it was essential to combine them. In those days, Andy was one of very few people able to work that way: Andy could think like a fluid and also like a mathematician, an exceedingly rare combination of gifts. It was a big mistake to try to put over any BS in Andy’s presence. I repeatedly watched in a mixture of delight and awe as his sharp questions punctured pompous public pronouncements from eminent mathematicians and physicists in a matter of seconds. I wonder what Andy would say now about my preceding paragraph.

Dave McLaughlin Andy was convinced that “modern applied mathematics” has a great deal to contribute to both the basic and the applied sciences—a conviction that Andy so often expressed and reiterated. To Andy, modern applied mathematics Pierre-Louis Lions is the professor of partial differential equations and their ap´ plications at the Collège de France and Ecole Polytechnique. His email address is [email protected]. Charles Fefferman is the Herbert E. Jones, Jr. ’43 University Professor of Mathematics at Princeton University. His email address is [email protected] .edu. Dave McLaughlin is a Silver Professor of Mathematics and Neural Science at the Courant Institute of Mathematical Sciences, New York University. His email address is [email protected].

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meant the union of today’s computational science with both the philosophy and methodology of mathematics. He was convinced that the revolutionary advances that have occurred in computing power and data acquisition would be terribly under-utilized without the precision, perspectives and methodology of mathematics. As summarized in this article, Andy’s work over the years has resulted in confirmation after confirmation of his conviction, with perhaps the best examples provided by the many results in theoretical atmosphere–ocean science that he obtained during the last 25 years of his career.

Russ Caflisch Andy Majda was a powerhouse as a mathematician working on PDEs, fluid dynamics, atmospheric science, and scientific computing, but he was also much more. He had a large network of former students and postdocs, collaborators, and others who looked to him for advice and inspiration. At the Courant Institute he was the founder and longtime leader of the Center for Atmosphere/Ocean Science (CAOS). He also had great influence on the scientific direction of Courant and on faculty hiring through his wideranging knowledge and passion for math and science.

Eric Vanden Eijnden I met Andy 25 years ago when I came to Courant fresh from my PhD. I was very much looking forward to talking to him about his work with Marco Avellaneda on turbulent diffusion, which was the topic of my thesis. The meeting did not go very well. True to form, Andy threw me out of his office after about a minute, declaring that I didn’t know anything about anything. He was right of course, as Andy was often right. This first meeting was not our last, however: Andy invited me back and we ended up working together for many years afterward, first on turbulent diffusion (with Peter Kramer) then on stochastic parameterization (with Ilya Timofeyev, on what became known as the MTV framework, no pun intended). Working with Andy was intense, but always rewarding: even though he was tough, always telling it straight like it is, he was also very caring, both about the science and the people he was working with. Just as he had done for many others, Andy helped me a lot at the beginning of my career, with advice Russ Caflisch is the director of the Courant Institute of Mathematical Sciences, and a professor of mathematics at New York University. His email address is [email protected]. Eric Vanden Eijnden is a professor of mathematics at the Courant Institute of Mathematical Sciences, New York University. His email address is eve2@cims .nyu.edu.

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about what to do and what to learn to be able to do it— Andy had very good taste in problems and the ability to make a dent in them. Andy was an extraordinary character in many ways. He was forward looking, always searching for new challenges with vision—even though he was well-established at a young age, he never rested on his laurels and kept going after new and timely topics throughout his career. Andy was also one of the most driven people I ever met. This was true all his life, but it became even more apparent after he had his stroke. He never gave up and the courage with which he faced his health challenges was truly inspirational. His attitude reminds me of a beautiful poem by Dylan Thomas whose first verse is: “Do not go gentle into that good night, Old age should burn and rave at close of day; Rage, rage against the dying of the light.” That drive to push the limits is how I remember Andy—I miss him very much.

Weinan E Andy was a master of using simple models to explain complex physical phenomenon. We have seen this again and again in his work on gas dynamics, combustion, singularity formation in fluid equations, and atmospheric flows. In this regard, he was the “Landau” in applied mathematics. In somewhat of an opposite direction, Andy also defined a certain style of applied mathematics: getting to the bottom of things using a combination of rigorous analysis, asymptotic analysis, and numerical analysis. In it, we see the power of applied mathematics shown at its fullest. This style has greatly impacted a generation of applied mathematicians like myself. It is not just useful for fluid dynamics, but for many other problems across a wide range of fields. In fact, right now we are witnessing applications to the understanding of neural network models.

Andrea Bertozzi I was introduced to Andy Majda when he first arrived at Princeton. I was starting my third year as an undergraduate and I was told to take his graduate course in Partial Differential Equations. This was a good transition from the Weinan E is a professor at Peking University. His email address is weinan @math.princeton.edu.

Andrea Bertozzi is the Distinguished Professor of Mathematics and Mechanical and Aerospace Engineering at the University of California, Los Angeles, and the Betsy Wood Knapp Chair for Innovation and Creativity. Her email address is [email protected].

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rigorous pure math curriculum given to Princeton Math Majors and a research career in applied mathematics. He gave me a delightful problem to work on for my undergraduate thesis—something that became my first publication— in dynamical systems. Andy had incredible intuition, not only for good problems to work on, but about areas in which one would find future good problems. He steered me toward nonlinear PDE in the late 1980s during a time when desktop computing was just beginning to have an impact in science and when there were many open problems on the boundary between science, mathematics, and computing. His critical stance on other people’s work gave me great perspective and helped to form my own taste in research problems. He had a great capacity to see well beyond “where we are now” into the realm of where we should be a decade or two down the road. This foresight has had an immense impact on both science and mathematics and also on the careers of many younger applied mathematicians.

Rupert Klein According to Andy, I was his first postdoc. Our first encounter dates back to the Conference on Hyperbolic Systems in Bordeaux, France, in the summer of 1988. Forman Williams, an eminent combustion scientist and a good friend of both Andy and my PhD advisor Norbert Peters, had quizzed me earlier about what I was up to for my postdoc time. When I mentioned “combustion” and “turbulence,” he asked whether I was interested in working with Andy Majda or Steve Orszag. “Sure,” I said, “but I don’t know either of them.” He promised to ask them, and he did. The next thing I knew is that Andy had said I could come, that he would provide an office, but that I should bring my own salary. A grant proposal with the German Science foundation was successful, and everything was arranged without Andy and I ever having met or even talked to each other. Then came the Bordeaux conference, which I attended for the chance of meeting my prospective host. Early during the week I heard all sorts of intimidating rumors about Andy, and these made me procrastinate a bit on approaching him. So, by Wednesday he had sensed me sneaking around in his vicinity, and over a coffee break—obviously quite annoyed—he stormed right at me booming “WHO ARE YOU??!!” Timidly I uttered my name, and in a split second Andy changed gears, opened his arms, beamed at me “you are Rupert Klein. . . ” and then excitedly told the bystanders the whole story of the postdoc arrangement. Rupert Klein is a professor at the Freie Universität Berlin. His email address is [email protected].

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Throughout the rest of the week we had tremendous fun discussing the weakly nonlinear asymptotics of acoustic waves, Andy and Miguel Artola’s “kink-modes on a vortex sheet,” the acoustic-chemistry-interactions I had been working on for my diploma, and the like. Later in the year, I moved to Princeton, looking forward to working on my (self-funded) project in combustion theory and to learn all the pertinent math background from my host. But, alas, in the course of our first meeting in Andy’s office up on the ninth floor of Fine Hall, he told me he was not working on combustion anymore, but that I would be welcome to discuss these issues with his PhD students Anne Bourlioux and Rob Almgren. Well, I did, and my exchange with the two of them was enjoyable and very interesting, for all involved, as I hoped. But there was no scientific exchange with Andy for the time being. That is, until a few months into my stay, when Andy approached me over coffee and asked what I was doing with Lu Ting at the Courant Institute every second Thursday; and would I not be willing to give an Applied Math seminar talk on this one of these days. Andy had ventured into vortex dynamics and turbulence theory in the meantime and he was interested in finding a reduced analytically tractable model that would describe the self-stretching of vorticity. He had suspected for a while that Lu’s pioneering matched asymptotic analyses of slender vortex filaments could provide a solid point of departure for the derivation of such a reduced model, but had thus far not found the time to go through these intricate derivations himself. Lu and I were indeed working on lecture notes that summarized that work, and so my seminar presentation was right on the money. Andy asked several pointed questions afterward with an eye to the self-stretching issue, and then he and I took off for our first adventure of close cooperation. Two years earlier, my PhD advisor Norbert had not only suspected already that Lu’s work should be of high interest for turbulence theory, but he also thought that Hidenori Hasimoto’s paper on “A soliton on a vortex filament,” which showed that the local induction approximation for slender vortices was equivalent to the focusing cubic nonlinear Schrödinger equation, could probably play an intriguing role in the context of Lu’s work. So, he had me give lectures on both of these theories in his Turbulence Seminar at RWTH Aachen, and thus I was familiar with both. Hence, when I had gotten stuck with the geometric complexities of our asymptotic filament model and Andy suggested that I see whether Hasimoto’s theory could help untangle the mess, I had the tools already and the door to the total of five papers on the self-stretching of vortex filaments with Andy was wide open. I will never forget the enthusiasm and excitement Andy and I shared in the course of these developments, nor the

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incredible efficiency of communication that was possible with him. Yes, I was mathematically well-trained as far as mechanical engineers are concerned, but I was a “dyedin-the-wool” fluid mechanicist. Andy could seamlessly shift from, e.g., the abstract math perspective on nonlinear Schrödinger to intuitive discussions of self-induced vortex motion and back, however, and this way we always found the right level of language very quickly and were able to move forward in big strides. This, in hindsight, is one of the most remarkable experiences in my life as a scientist. I have had this amazing and immediate feeling of mutual scientific understanding with no more than three colleagues over the past 30-odd years. After my postdoc period I went back to Aachen and Andy and I lost touch for a while. Yet, in the late 1990s, when Andy had already moved on to the applied math of atmosphere–ocean science, I joined the PotsdamInstitute for Climate Impact Research in Potsdam, Germany. Shortly thereafter “it happened again,” as Andy would later say: We met in Breckenridge for the 13th Atmospheric and Oceanic Fluid Dynamics Conference. He told me about flow regimes in the tropics and how he envisioned the construction of an entire scale-dependent model hierarchy, and two years later our next paper on “Systematic multi-scale Models for the Tropics” came out. This paper, he often jokingly told meteorologists in the audience of his various science lectures, was his revenge on the insiders of atmosphere–ocean science: Their field, he said, kept potential intruders from other disciplines from intruding by the impenetrable jungle of acronyms they had grown around it. So, here we went with MEWTG, SPEWTG, IPESD, QLELWE, and so on in the 2003 paper. Later in 2006, we incorporated moist process models in our multiscale asymptotics framework, and studied the interaction of internal waves with arrays of deep convective towers. Then in 2009, we ventured into advanced time series analyses for atmospheric flow applications together with Christian Franzke and Illia Horenko. Andy loved interdisciplinary advanced applied mathematics research, and he helped foster related workshops and conferences where ever there was a promising opportunity. Thus, in 2002, 2006, and 2010, he and I organized Oberwolfach workshops on atmosphere–ocean science together with Oliver Bühler and Bjorn Stevens, and today these have developed into a very productive multileaved workshop series at the institute. Our last major joint enterprise—again with Bjorn Stevens on board—was the stimulating long program on “Model and Data Hierarchies for Simulating and Understanding Climate” at the Institute for Pure and Applied Mathematics (IPAM), in Los Angeles in 2010. The spin-offs of this long program I think still reverberate quite positively in the community.

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Andy has been a great supporter and a fabulous cooperating partner throughout my career, and this leaves me deeply grateful. As partnerships tend to have it, we have had our challenging moments, but these never rattled the foundations of our great mutual respect or the shared enthusiasm about our joint and mutual scientific achievements.

Leslie M. Smith Andy was my close colleague, mentor, and friend for 30 years. His influence on pure and applied mathematics was extraordinary, as was everything about Andy. On the applied side, he made pioneering contributions to the theory of turbulence, combustion, and climate–atmosphere– ocean science. Andy’s work never stopped evolving, and it was next to impossible to keep pace with him. One of his most important legacies was mentoring junior mathematicians and his peers, always ready to share his vision and insight. I was not Andy’s student or postdoc and could not help but be surprised by his generosity, but there it was, genuine and with no strings attached. Conversations with Andy changed my career at least twice, the first time as a beginning postdoc and later at midcareer. For my first postdoc at the Stanford Center for Turbulence Research, I was given the task of penetrating a new theory of turbulence based on renormalization that had just been introduced into the fluid dynamics community by Victor Yakhot and Steve Orszag. Two years later, after summing many loop diagrams, I went to work with Victor and Steve at Princeton, where Andy was separately working on a mathematically rigorous renormalization theory for advection-diffusion of a passive scalar. Thus, I came to know Andy in the middle of a heated rivalry. On request to meet with Andy, his assistant was instructed to tell me “Professor Majda says that you should read his papers.” That I did, including the beautiful papers by Avellaneda and Majda on renormalization. After a bit of time and persistence, Andy and I became friends as well as colleagues. I admired him as both the mathematician and the person, insistent upon rigor, honest and straightforward to a fault, rough on the outside but big-hearted on the inside. My friendship with Andy developed over decades and became one of the most important relationships in my professional life. Much later, during a shuttle trip from Banff to Calgary, I mentioned a desire to move beyond my current research projects, and Andy proceeded to give me ideas for tackling moist atmospheric dynamics that would Leslie M. Smith is a professor in the department of mathematics and the department of engineering physics at the University of Wisconsin-Madison. Her email address is [email protected].

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be a foundation of my research for the next few years. I have a special memory of meeting Andy in Beijing when he received the Lagrange Prize at ICIAM 2014, and we had dinner with his spouse Gerta Keller, Boualem Khouider, Rupert Klein, and Sam Stechmann. The last time I saw Andy was in Victoria, at one of his 70th birthday parties, where Gerta gave a public lecture on her theory of the dinosaur extinction, and we shared another celebratory dinner. I was lucky to know Andy, and am extremely grateful for his brilliance, leadership, kindness, and companionship.

Richard McLaughlin Andy Majda was an applied math pioneer. He made a strong impact in so many different areas of science including compressible flow, combustion, turbulent transport, vortex dynamics, and climate modeling. In each of these fields, he brought powerful mathematics to bear upon real world phenomena, often utilizing mathematically exact solutions which captured many features of the system under study. He taught us to dive deep into the science whilst using the best mathematics possible to uncover new behavior and make new predictions about complex systems. His scientific impact and influence will be felt for a long time.

Peter Kramer The profound impact of Andy’s mentoring while I was his PhD student would presumably echo largely what others are saying in more colorful ways so I’ll share instead my encounter with Andy while I was an undergraduate physics major at Princeton. Not knowing what the subject really meant, I had never considered taking an “applied mathematics” class until Andy offered an undergraduate “Mathematical Fluid Mechanics” class my senior year. Beyond the usual introductory mathematical formalism and analysis of various fluid equations, Andy infused into this class his famous triangle of mathematics interacting with computing and the scientific discipline for insight and demonstrated this principle with concepts distilled from recent research by himself and his collaborators. His class was revelatory in illuminating the power of applied and computational mathematics toward understanding a richer spectrum of systems in the world than I was seeing in my physics classes. The following semester Andy directed a smaller undergraduate course in which we studied Richard McLaughlin is a professor of mathematics at the University of North Carolina. His email address is [email protected]. Peter Kramer is a professor of mathematics at the Rensselaer Polytechnic Institute. His email address is [email protected].

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several key research papers in mathematical fluid mechanics. These classes with Andy, on their own, persuaded me to pursue graduate studies in applied mathematics rather than physics. Indeed, I had been having difficulty identifying an area of physics whose research prospects resonated with my interests, and Andy’s uniquely inspirational undergraduate classes steered me to a career that I believe has been much more satisfying than it would otherwise have been. These, I believe, were the last undergraduate classes Andy ever taught (1992–1993), and I feel so privileged to have been introduced to Andy and his scientific world view by them.

dynamics in the past two decades. His work has changed the way many people think about tropical atmospheric dynamics. Because of Professor Majda’s contributions, the field of tropical atmospheric dynamics is now in a new era in which nonlinear scale interactions are no longer a speculation, a concept, or an observational perception. His mathematical models and computational strategies and algorithms for the multiscale interactions for atmosphere and ocean dynamics have paved new paths and solid grounds for CAOS research.

Boualem Khouider

I was very fortunate to be Andy’s PhD student as well as his postdoc. Andy taught me everything about research, offered many precious opportunities to improve my abilities, and provided enormous help to my career. I was extremely touched by how hard Andy worked to guide me in research and how generous he was in spending time helping me. In addition to being my research advisor, Andy was my close friend. He also often made me feel that he was a close family member of mine. I sincerely cherish all the time spent with Andy. Andy told me that the most important thing for someone who wants to do great scientific work is is that “he/she loves science.” Andy himself was a person who really loves science. His spirit will continue to influence all of us. I keep in mind all that I learned from Andy and regard Andy as a model for my life and career. The last time I saw Andy was October 10, 2019, at his home. Andy brought me to the front door when I left. He had a slow pace, but a big smile. That was the scene that will stay in my memory forever.

The passing of Professor Andy Majda on March 12th, 2021, sadly marks the end of a lifetime full of new mathematical and scientific discoveries and of breaking barriers between highly complex mathematics and very important applications including but not limited to climate, atmosphere and ocean sciences (CAOS). Majda’s ingenuity, endless vision, contagious energy, and generosity in sharing and broadcasting his ideas and nonconservative approach to scientific research will be forever remembered. There is no doubt that his legacy will outlive his academic children and grandchildren. Professor Majda has made several groundbreaking advances in the theoretical description and understanding of many important atmosphere and ocean phenomena. His contributions to the theories of the Madden-Julian Oscillation (MJO), convectively coupled equatorial waves, and El Nino, are just a few examples. By the time he decided to engage in climate change science, Andy Majda was one of the world’s most famous applied mathematicians of his generation, known for his groundbreaking work in numerical analysis, shock waves, turbulence theory and combustion. Andy’s “intrusion” to the CAOS community was received with a mixture of skepticism and awe. While a few people were saying that the problems he was tackling were too hard for a newcomer, many have admired his original and unconventional approach to these problems. With his impeccable mathematical skills, his extensive experience in applying mathematics to describe and understand observed physical phenomena, his sharp vision to quickly grasp the most fundamental features from observations, and his tireless efforts in training young scientists, Professor Majda has made, unquestionably, some of the most exciting contributions in the area of atmospheric Boualem Khouider is a professor of mathematics and statistics at the University of Victoria. His email address is [email protected].

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Nan Chen

Di Qi Andy Majda was my PhD advisor and later Postdoctoral mentor. He was a great mathematician, sharp thinker, enthusiastic teacher, and most of all a true friend with an ardent heart. No doubt Andy has left an indelible mark on me in my scientific pursuit as well as in my perception of life. During the years working with him, I was always amazed by the functioning of Andy’s magical mind to quickly digest and employ new ideas with skill. He was always eager to spread his knowledge and share his thoughts with the people around him. It was well-known that Andy always showed up earliest in the morning in his office. He was able to continually discuss science on vastly different Nan Chen is an assistant professor of mathematics at the University of Wisconsin-Madison. His email address is [email protected]. Di Qi is an assistant professor of mathematics at Purdue University. His email address is [email protected].

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research topics tirelessly from morning till late afternoon. It is still a wonder to me how Andy could smoothly shift subjects and stay sharp in his many ongoing projects all at the same time. Even when his health began to decline, Andy maintained his vitality to continue his everyday scientific routine. He never stopped the weekly scientific meetings with all of us. He forced himself to overcome great pain and showed little sign of his suffering. Besides the numerous scientific achievements, it is his shear strong-minded determination to keep on going that leaves the deepest impression for me. Andy remains a persistent inspiration and gives me the courage to carry on. I was lucky to be guided by Andy, with his insight, energy, and intellectual power at the early stage of my career. I will forever miss the many exhilarating discussions around the long couch in his office, with warm sunlight pouring in through the wide window panes, where so many brilliant ideas emerged.

Reza Malek-Madani I knew Andy Majda for over 35 years, first meeting him in 1975 as a graduate student at Brown when he gave a talk in the PDE Seminar. It was clear from that meeting that Andy not only was a great researcher, he was already a gifted teacher. His lectures throughout his career were organized as if meant to tell a story. We are so fortunate that many of his talks have been recorded and are available on YouTube and on other platforms. I was of course aware of the remarkable contributions he was making during the 80s and 90s, but that was mostly as a distant observer. In the year 2000, I became the program officer at ONR, running the applied math program, and it was at that time that I got to interact with Andy regularly. Andy had just started his new research area in atmospheric and oceanic sciences, an area of critical importance to the Navy. In the early 2000s, each year Andy organized a Friday–Saturday workshop in early December and brought together applied mathematicians and oceanographers to discuss the many topics that our colleagues have already described here: Predictability for the atmosphere and ocean (2003), Large-Scale turbulence in the atmosphere and ocean (2004), Vortices and Waves in Geophysical Flows (2006), to mention just the first few. The brainstorming in these workshops were always stimulating, and on occasion heated and contentious—in those years, on the way home late on Saturday, my train would stop in Philadelphia. Many of my students who had just attended Reza Malek-Madani is a professor of mathematics at the United States Naval Academy. His email address is [email protected].

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the Army-Navy game would get on board, telling stories about the battle they had just witnessed on the field, and I would wonder which of us had seen a more competitive event. The net result of watching Andy during those formative years, and learning from him how to learn, was to give me the tools and the courage to approach Scott Harper, my counterpart at ONR in charge of GFD research funding, to propose that we could come up with an interdisciplinary approach to our programming design to take advantage of what was being developed at Courant, at NCAR, at JPL, and elsewhere. Scott, a Princeton graduate, quickly agreed that we were observing something special. For the next two decades I would visit Andy regularly, and I was grateful for his enormous patience while I tried to keep up with him. On the Fridays of those workshops, Andy would take me aside at lunchtime, we would get a sandwich and then walk the streets of New York while he told me what was on his mind. My job for the next several months was to digest his thoughts and fill in the gaps in what I understood. I slowly began to understand what he meant by his modus operandi, which others have already alluded to. That concept impacted my career in two ways: Andy’s framework for what modern applied mathematics should be ended up defining the underpinning for my program; and my attempt to understand Andy’s approach became the focus of a course I taught annually at the Naval Academy for over 15 years. What touched me the most about Andy was the courage he showed when he switched his research focus so dramatically to atmospheric sciences. ONR and the Navy benefitted enormously from this decision. Our community is so much richer because of Andy’s brilliance, and the generous and unselfish manner in which he treated all of us.

Gerta Keller The first time I met Andy Majda was in the summer in Palo Alto, California. From the very beginning, I called Andy by his last name “Majda,” which is an endearing name for the Swiss. As the summer progressed, Majda and I became close friends. He often worked at the Stanford Coffee House and I met him for caf´e lattes. At sunset, we ran across the Bay Area mud flats among thousands of birds and talked about all sorts of interests from science to nature, but I never was interested in Majda’s favorite sport topics from football to tennis. And so began our lifelong friendship and love.

Gerta Keller (also known as Gerta Majda-Keller) is a professor of paleontology and geology in the Geosciences Department of Princeton. Her email address is [email protected].

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is my second favorite Alps trip and has fantastic desserts. And then you can walk to Piz Bernina.” And so, Majda planned his last trip to his favorite mountains and the best desserts in the Val Rosegg valley. It will happen in the summer of 2023. References

Figure 5. Andy Majda and Gerta Keller in the Swiss Alps.

Majda and his younger twin brothers grew up in the middle of East Chicago’s oil refineries, where they played among the slag heaps and roasted potatoes. Their father grew up as a Jesuit and learned many languages in Poland. When he left for the USA, he immediately enlisted in the US army as a radio technician during World War II. After the war, he worked in East Chicago’s oil refineries. Andy Majda was a precocious youngster who quickly taught himself everything there was to learn in the encyclopedia, which was the only book in their home. He had a photographic memory and never forgot anything. Even during the first years of his encyclopedic memory, he was called “the little professor.” His twin brothers followed the little professor’s example. In a short time, Andy Majda became an extraordinary mathematics professor with full credentials by age 28. He followed the twins: George, also a mathematician, and John an oncologist. They were a remarkable trio of academics. Majda fell in love with the Swiss Alps. To introduce Majda to the beauty of the Swiss Alps, we organized a major hiking and climbing trip over three days. It was Majda’s first trip into the high mountains. We steadily hiked upward toward the glacier, known as the Forno Glacier, and then carefully picked our way between ice, water, and rocks. After leaving the Forno glacier, we began a very hard steep switchback climb to the top of the mountain that ended the first day adventure at the Forno Hut, a Swiss Alp club house. Near the end of Majda’s life, I asked him: What was your favorite Alps trip? He smiled happily and said “Forno Hutte.” We both laughed at our memory. “I will take you there” I said. He looked at me and replied, “G, I can’t walk anymore.” “But Majda, I will take you there,” I repeated. He looked at me and smiled, realizing what I proposed and he was happy. On second thoughts he said, “G, you can’t do it anymore, you’re not strong enough.” “I will do it,” I replied. “G, it’s better to take me to Val Rosegg, which NOVEMBER 2023

[AM90] Marco Avellaneda and Andrew J. Majda, Mathematical models with exact renormalization for turbulent transport, Comm. Math. Phys. 131 (1990), no. 2, 381–429. MR1065678 [AM92a] Marco Avellaneda and Andrew J. Majda, Approximate and exact renormalization theories for a model for turbulent transport, Phys. Fluids A 4 (1992), no. 1, 41–57, DOI 10.1063/1.858499. MR1140130 [AM92b] Marco Avellaneda and Andrew Majda, Mathematical models with exact renormalization for turbulent transport. II. Fractal interfaces, non-Gaussian statistics and the sweeping effect, Comm. Math. Phys. 146 (1992), no. 1, 139–204. MR1163672 [BKM84] Thomas J. Beale, Tosio Kato, and Andrew J. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984), no. 1, 61–66. MR763762 [CFM96] Peter Constantin, Charles Fefferman, and Andrew J. Majda, Geometric constraints on potentially singular solutions for the 3-D Euler equations, Comm. Partial Differential Equations 21 (1996), no. 3-4, 559–571, DOI 10.1080/03605309608821197. MR1387460 [CLM85] P. Constantin, P. D. Lax, and A. Majda, A simple onedimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38 (1985), no. 6, 715–724, DOI 10.1002/cpa.3160380605. MR812343 [CM80] Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21, DOI 10.2307/2006218. MR551288 [CMT94] Peter Constantin, Andrew J. Majda, and Esteban Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), no. 6, 1495– 1533. MR1304437 [DM87] Ronald J. DiPerna and Andrew J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689. MR877643 [DM88] Ronald J. DiPerna and Andrew Majda, Reduced Hausdorff dimension and concentration-cancellation for twodimensional incompressible flow, J. Amer. Math. Soc. 1 (1988), no. 1, 59–95, DOI 10.2307/1990967. MR924702 [EM77] Björn Engquist and Andrew Majda, Absorbing boundary conditions for numerical simulation of waves, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1765–1766, DOI 10.1073/pnas.74.5.1765. MR471386 [KM08] Boualem Khouider and Andrew J. Majda, Multicloud models for organized tropical convection: Enhanced congestus heating, Journal of the Atmospheric Sciences 65 (2008), no. 3, 895–914.

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[KM81] Sergiu Klainerman and Andrew Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), no. 4, 481–524, DOI 10.1002/cpa.3160340405. MR615627 [Maj00] Andrew J. Majda, Real world turbulence and modern applied mathematics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 137–151. MR1754773 [Maj81] Andrew Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math. 41 (1981), no. 1, 70–93, DOI 10.1137/0141006. MR622874 [Maj83] Andrew Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983), no. 275, iv+95, DOI 10.1090/memo/0275. MR683422 [Maj84] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984, DOI 10.1007/978-1-4612-1116-7. MR748308 [MK03] Andrew J. Majda and Rupert Klein, Systematic multiscale models for the tropics, Journal of the Atmospheric Sciences 60 (2003), no. 2, 393–408. [MQ18] Andrew J. Majda and Di Qi, Strategies for reducedorder models for predicting the statistical responses and uncertainty quantification in complex turbulent dynamical systems, SIAM Rev. 60 (2018), no. 3, 491–549, DOI 10.1137/16M1104664. MR3841156 [MS09] Andrew J. Majda and Samuel N Stechmann, The skeleton of tropical intraseasonal oscillations, Proceedings of the National Academy of Sciences 106 (2009), no. 21, 8417– 8422. [MTVE01] Andrew J. Majda, Ilya Timofeyev, and Eric Vanden Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math. 54 (2001), no. 8, 891–974, DOI 10.1002/cpa.1014. MR1829529

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Panagiotis Souganidis

Samuel N. Stechmann

Vlad Vicol Credits

Figures 1, 3, and 5 are courtesy of Gerta Keller. Figure 4 is courtesy of ©American Meteorological Society. Used with permission. Photo of Panagiotis Souganidis is courtesy of Panagiotis Souganidis. Photo of Samuel N. Stechmann is courtesy of Samuel N. Stechmann. Photo of Vlad Vicol is courtesy of Vlad Vicol.

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In Memory of Steve Zelditch Coordinated by Bernard Shiffman and Jared Wunsch

Figure 1.

Steve Zelditch, our beloved and admired colleague and a major figure in spectral theory, semiclassical analysis, quantum chaos, and Kähler geometry, died on September 11, 2022, at the age of 68. Steve is survived by his wife, Ursula Porod and their two sons, Benjamin and Phillip. He died during a hurriedly organized Zoom conference with a star-studded speaker lineup and enormous attendance, celebrating his achievements and their impact on a huge range of analysis and geometry. Steve, who had an unquenchable thirst for mathematics, was present at this meeting and discussing as much math as he could, all through the first three days of talks. He died on the night before the fourth and final day, when the grief-stricken attendees had to carry on without him. The outpouring of admiration, sadness, and appreciation of Steve’s mathematical and human dimensions has been overwhelming, and a small sample is provided below. Bernard Shiffman is an emeritus professor of mathematics at Johns Hopkins University. His email address is [email protected]. Jared Wunsch is a professor of mathematics at Northwestern University. His email address is [email protected]. Communicated by Notices Associate Editor Daniela De Silva. For permission to reprint this article, please contact: [email protected].

Steve grew up in Palo Alto. He was an undergraduate at Harvard, where his initial ambition was to become a novelist; he got distracted by mathematics along the way. (His love for literature stayed with him, however, and he was incredibly well read and opinionated on literature of all kinds.) Steve received his PhD from Berkeley in 1981 under the direction of Alan Weinstein. He was subsequently a Ritt Assistant Professor at Columbia, then joined the faculty at Johns Hopkins in 1995. In 2010, Steve moved to Northwestern, where he was the Wayne and Elizabeth Jones Professor of Mathematics. Steve was an ICM speaker in 2002, won the Stefan Bergman Prize in 2013, and was a Fellow of the AMS. You can read about many aspects of Steve’s research in the tributes that follow, but for those who didn’t know the breadth of his work, it is perhaps instructive to point to one central theme and four main threads within that theme. Steve loved asymptotics. Any problem phrased in terms of asymptotic expansions lit his enthusiasm. He had a remarkable capacity for seeing common features in seemingly disparate asymptotic questions, and in particular, for finding ways that “semiclassical” asymptotics, expressing the relationship of quantum mechanics and classical mechanics as Planck’s constant is allowed to tend to zero, could be employed in surprising new areas. Steve’s first great success was the story of quantum ergodicity, describing how in a quantum system whose underlying classical dynamics are chaotic, the energy eigenfunctions must be correspondingly scrambled up in both position and momentum. This became a major area of mathematics, with Zelditch as its unquestionable leader. Steve was the first to systematically use semiclassical tools in Kähler geometry, where the “Tian–Yau–Zelditch” expansion dictates how pulling back the Fubini–Study metric under the Kodaira embeddings via powers of an ample line bundle can approximate any Kähler metric. The TYZ expansion has become a key tool in Kähler geometry. In addition, Steve developed a new area of mathematics,

DOI: https://doi.org/10.1090/noti2807

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directly. Some of this work has turned out to have a rich relationship with problems in the theory of string vacua. We have just scratched the surface here: MathSciNet currently lists 184 publications, and Steve had many active collaborations at the time of his death. He was, moreover, interested in everything, in mathematics and in life. We miss him.

Nalini Anantharaman

Figure 2. Steve Zelditch with his sons, Phillip and Ben, in 2002.

Figure 3. Jared Wunsch and Steve Zelditch at the New Chair Investiture Ceremony, 2012.

“stochastic Kähler geometry” to which he further applied asymptotics. Mark Kac famously asked “Can one hear the shape of a drum?” By work of Gordon–Webb–Wolpert, we know the answer to be “no.” But one can hear a lot about the drum, and Steve’s positive inverse-spectral results (on analytic domains with symmetry, and on nearly circular ellipses) are the best known. Finally, somewhat more loosely, we remark on a large body of Steve’s work involving asymptotics of randomized objects, sometimes studied for their own sake, and sometimes as a proxy for deterministic objects (like individual Laplace eigenfunctions) that are too elusive to cope with 1668

I feel shy about taking up the pen to write about my friend and collaborator Steve Zelditch, in a language that is not my mother tongue. Steve enjoyed words and literature, his conversation was full of savor and he liked to play with the American language—you could guess when he was the referee of one of your papers. He was eager to help you improve your style, both in mathematics and in English. Steve is famous, among other things, for a large body of articles concerning “quantum ergodicity.” After Alexander Shnirelman, in 1974, stated a theorem relating classical ergodicity of a Hamiltonian flow to the equidistribution of eigenfunctions of the associated Schrödinger operator, Steve Zelditch developed a pseudodifferential calculus on hyperbolic surfaces that allowed him to give the first full proof of the theorem. Interestingly, he told me that his work aroused no interest in the US at the time, but received quick recognition in France. Steve developed the subject in all possible directions, he showed how rich a subject this is, and he is largely responsible for the popularity of the subject nowadays: quantum ergodicity for Laplacian eigenfunctions on Riemannian manifolds without and with boundary, for eigenfunctions of Dirac operators, quantum ergodicity for abstract ℂ∗ dynamical systems, for restrictions of Laplacian eigenfunctions to hypersurfaces, relations between quantum ergodicity and counting of nodal domains. Steve made fundamental contributions to several other areas, such as zeroes of random polynomials, random Kähler geometry, and inverse spectral problems. It strikes me that I never heard Steve criticize a colleague or a mathematical result, based on anything other than scientific grounds. He helped me a lot when I was preparing my Bourbaki talk about random nodal domains and was struggling to compare the contributions of various teams: instead of describing the various contributions in terms of competition, he tried to explain to me the vision and merits of each author. He liked talking and I liked listening, which was both fun and tiring, as he could become enthusiastic about all sorts of unexpected things. Because of, or Nalini Anantharaman is a professor of mathematics at the University of Strasbourg. Her email address is [email protected].

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maybe “thanks to” him, I bought CDs of Green Day, System of a Down, Arcade Fire—which I never really listened to afterwards. . . . When you lose a friend who lives on the other side of the Atlantic and whom you used to see about once a year, it’s easy to imagine he’s still there. These days, when I go through moments of great intellectual enthusiasm, I think of him and how intense his intellectual life was, I think “this is a moment that Steve would prize”—and I suddenly recall that he is lost forever.

Michael R. Douglas My relationship with Steve Zelditch began with mathematical physics, and quickly grew into friendship. In early 2002, I was studying what would soon be dubbed “the string landscape,” the set of solutions of superstring and M theory which might describe our universe considered not one by one (as was usual in physics) but as a totality. The original example is the set of three-dimensional Calabi–Yau (Ricci-flat Kähler) manifolds, candidates for the “hidden” dimensions of superstring theory. Over the years many more solution sets were proposed, and the goal of describing them was extended to defining a probability distribution over solutions, called the “measure factor” in quantum cosmology. A basic problem of the string landscape is to find and study natural random distributions over algebraic geometric objects: varieties, vector bundles, sections, their zeroes, and so on. With this in mind, I started poring over the mathematical literature, and sometime in 2002 I ran into a paper of Bleher–Shiffman–Zelditch [BSZ00], which studied zeroes of random sections of a holomorphic line bundle 𝐿. Since these spaces of sections are finite dimensional and linear, the normal distribution is well defined, and one can ask for the distribution of simultaneous zeroes of 𝑛 = dim 𝑀 sections. The basic result of [BSZ00] was that, considering a sequence of bundles 𝐿, 𝐿2 , … , 𝐿𝑁 , as 𝑁 → ∞ the limiting normalized distribution is the 𝑛-th power of the curvature of 𝐿. Now one of the physics problems I was looking at was to find critical points of a “random flux superpotential,” a holomorphic section drawn from a finite-dimensional linear space. These are simultaneous zeroes of the components of the covariant derivative, very similar to the zeroes studied in [BSZ00]. Even better, the techniques (such as the Kac–Rice formula) were familiar from random matrix theory.

Michael R. Douglas is a research scientist at the Center of Mathematical Sciences and Applications at Harvard University. His email address is mdouglas @cmsa.fas.harvard.edu.

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I was delighted to learn that Zelditch would be at ICM 2002, and we arranged a meeting there. He explained his work, and I asked him whether they had considered doing the same for critical points of a random section. Indeed they had considered it, but they had thought that nobody would be interested. So that is where our collaboration (with Bernie Shiffman as well) began. This led to [DSZ04, DSZ06b, DSZ06a], which solved my problem, and many subsequent works. It also led to many interactions with geometric probabilists at workshops Steve invited me to. I should also mention Steve’s long collaboration with my student Semyon Klevtsov which further deepened his impact on physics. My interactions with Steve made a great impression on me, going far beyond these specific works. As every mathematician who has worked with physicists knows, even when you are talking about the same mathematical objects, and even when the language barriers have been overcome, there are great differences in how you think. You ask different questions, and you can have very different opinions about when they have been answered. Steve had very broad interests and was flexible in what he would consider, but at the end of the day uncompromising. And in these difficult days for fundamental physics, lacking much experimental guidance, rigorous standards are all the more valuable to keep us on track. Steve had a great love of life which made him a pleasure to be around, and I have wonderful memories of times together, in particular of a wine tour in central California we took with Bernie. I will greatly miss him.

Boris Hanin The first time I met Steve we were both new to Northwestern. I was a first year PhD student and he had just moved from Johns Hopkins. We got to talking over lunch (faculty would sometimes eat lunch in the common room of Lunt Hall) about a curious relationship between zeros and critical points of high degree polynomials. I had no idea that Steve was in the middle of writing an influential series of articles, mainly with Bernie Shiffman, studying zeros and critical points of random polynomials and holomorphic sections. He immediately reframed the result I mentioned into a question that could be approached using Bergman kernel expansions and suggested that after a few weeks of reading his papers I could find an answer. Boris Hanin is an assistant professor of operations research and financial engineering at Princeton University. His email address is bhanin@princeton .edu.

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In this way, Steve was a magician. His mind would key into unexpected mathematical facts and, seemingly out of thin air, he’d produce a connection to something he knew. This was often followed by a question and a plan of attack. After around a year of work, I excitedly came to Steve’s office to explain my new theorem on zeros and critical points. His answer: “I don’t believe the result.” Not knowing that such a response was even possible, I replied “but I have a proof.” Steve went on to assure me both that my proof was certainly wrong and that this wasn’t the issue. Results—especially unexpected ones—demand conceptual explanation. In this way, Steve was a purist. He insisted on understanding mathematics in a manner so deep that the technical and the intuitive merged. Finding an explanation that Steve found satisfying took me several months and taught me what it’s really like to understand my own work. Far from being frustrated by me, when I finally came to him with a simple heuristic derivation, Steve was overjoyed and arranged for me to speak about it at a conference on random geometry in Montreal the following summer. I had never attended a conference before, and I still remember Steve asking virtually every speaker questions, with follow-ups in the breaks and even during dinners. That image of Steve sitting in the front row, engaging with the content of the summer schools, workshops, and conferences we both attended over the years is how I’ll most remember him. In this way, Steve was a mentor. He taught me to see opportunities for growth as a mathematician, but didn’t prescribe how to use them. He also encouraged me to learn new fields, even when I became fascinated by neural networks as a postdoc and probably should have been writing more articles on spectral asymptotics. Over the past few years it was my great pleasure to continue working with Steve, and we submitted the revisions for our final joint paper a few days before he passed away. Though he was quite unwell by that point, he was still intent on pursuing mathematics, both insisting that we include certain oscillatory integral estimates in our revision and, in the same breath, asking me to send him a recent paper on quantum computing that I had gotten excited about. I miss him dearly.

Andrew Hassell I first met Steve when I was a graduate student in the early 90s, at various spectral theory/microlocal analysis Andrew Hassell is a professor of mathematical sciences at Australian National University. His email address is [email protected].

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conferences in the US. Steve stood out in such gatherings: he would ask many questions both during and after talks, and because of his remarkable breadth of knowledge, he could ask searching questions on seemingly any topic that came up. He was as comfortable with representation theory, geometric quantization, complex geometry or random matrices as he was with more “core” topics such as Fourier integral operators or eigenvalue counting functions. Steve continued his lively (and loud—he had a penetrating voice) questions and discussions during breaks and over meals. His questioning style was intense, passionate and sometimes verging on aggressive! He was generous with ideas, throwing out numerous questions to groups of participants arising from the talks or from his own research. A junior mathematician could do very well for him or herself by listening to Steve! It was certainly a good way to acquire research questions, and if things developed well, it sometimes led to a joint publication with Steve. Over the course of my career, at least eight research articles were influenced by or were the result of answering questions posed by Steve. I’ll describe some of these in the remainder of this note. Isospectral problems. Steve was interested in isospectral problems (popularized by Kac’s famous article [Kac66]) throughout his career, see for example [Zel92, Zel00, Zel09, HZ22]. When I was a postdoc, knowing I was interested in scattering theory, Steve asked what we could say about a class of isophasal domains — the scattering theoretic analogue of isospectrality. Could we show that such a class is compact, similar to the famous result of Osgood–Phillips–Sarnak? We worked on this when Steve visited Brisbane and Canberra in 1997, and showed this in our first joint paper [HZ99]. Quantum ergodicity. Steve’s seminal result on quantum ergodicity [Zel87] was obtained early in his career. Yet it took some time for its importance to be appreciated. In fact, MathSciNet lists no citations of this paper until 1997 (it now has 249, at the time of writing). Steve was fascinated by the equidistribution of eigenfunctions and researched aspects of this question throughout his career. This includes questions on the rate of quantum ergodicity, 𝐶 ∗ -algebraic aspects of quantum ergodicity, ergodicity of billiards, quantum ergodicity of boundary values and restrictions to hypersurfaces, quantum mixing and quantum variance. I worked with Steve on quantum ergodicity of boundary values. The paper arose from Steve sending me what he called an “embryo,” that is, an unfinished manuscript that contained his attempt (usually a very significant attempt) to prove the result, with a detailed strategy and much preparatory work. I was fortunate to receive several embryos from Steve over the years. We investigated the microlocal distribution of boundary values of eigenfunctions

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of the Laplacian on bounded Euclidean domains, for various different self-adjoint boundary conditions [HZ04]. In 2008, I found a simple way to prove the widely conjectured statement that stadium billiards are not quantum unique ergodic (QUE) (for almost every aspect ratio of the central rectangle). This work was not in collaboration with Steve but was informed by an earlier paper of his [Zel04b] together with several conversations we’d had over the years. In 2008, during an MSRI program, he asked whether the method, which involves studying the spectral flow as one varies the aspect ratio, could be adapted to show non-QUE for a system with classically KAM dynamics. Almost a decade later, I gave this problem to my graduate student Sean Gomes. Sean was able to show not just non-QUE but non-QE for 1-parameter perturbations of completely integrable systems, again for almost every value of the perturbation parameter. Shortly afterward, when Sean was a postdoc at Northwestern, we showed that a stronger statement could be made in the case of two-dimensional KAM systems. Semiclassical asymptotics of scattering matrices. In the late 1990s, Steve asked me what one could say about the semiclassical asymptotics of the spectrum of a scattering matrix, say for the Schrödinger operator ℎ2 Δ+𝑉(𝑥)−𝐸, where 𝑉 is a 𝐶𝑐∞ potential and 𝐸 is a positive energy level. It was motivated by equidistribution results that he obtained for quantized contact transformations [Zel97]. Fifteen years later, I started working on this problem, initially with Datchev, Gell-Redman, and Humphries in the centrally symmetric case. When Steve saw the result he realized that this could be combined with his ideas in his early paper on quantized contact transformations. Steve, Jesse Gell-Redman and I showed that the spectrum can be divided into two parts, one of which lies very close to 1 on the unit circle, and the other is equidistributed [GRHZ15]. Steve was a great supporter of early career researchers. He loved discussing mathematics, and he gave people in his audience equal respect whether they were legendary mathematicians or lowly PhD students. He always believed that he could learn from whoever he was talking to, and was never happier than when suggesting research problems to his audience. Personally, I felt very encouraged by Steve in my first few years post-PhD and my mathematical life was greatly enriched by interacting with Steve. I miss him severely.

Hamid Hezari I feel privileged to call myself a former PhD student and collaborator of Steve Zelditch. In fact, I became his student in a fascinatingly lucky way. What later became the key to connect me to Steve was an Iranian math magazine, given to me on the first day of my undergraduate education, which contained a translation into Farsi of Kac’s famous 1966 paper, “Can one hear the shape of a drum?” The catchy title caught my attention. I tried to read it, but understood close to nothing of the mathematical content and methods except that the author raised the question of whether one can find the shape of a (not necessarily circular) drum from its frequencies of vibrations, and showed that one can actually hear the shape of a perfectly round drum (a disk). Later in 2004, when I was admitted to the PhD program of Johns Hopkins University, Steve was in charge of the graduate analysis course I was taking. My intention was to pursue number theory. Steve’s generosity was striking, both in terms of his time and his mathematical ideas. At that time, he was focused on the inverse spectral problem for analytic domains, and after one of his lectures, he openly shared with me the challenges he was facing. Steve’s contagious enthusiasm quickly got me interested in the inverse problem. In 2007, after a series of three long and technical papers, Steve proved that generic analytic plane domains with one axial symmetry are distinguishable from each other by their sound frequencies. This theorem still stands as one of the strongest results in the subject. A natural problem was to extend this result to higher dimensions. In 2008, in my first joint work with Steve, we proved an analogous inverse result for generic analytic domains in ℝ𝑛 , 𝑛 ≥ 3, under the condition that they are symmetric with respect to all coordinate axes. Removing the symmetry assumptions, even one of them, still remains as a big challenge. Steve’s next mission was to investigate the inverse problem for smooth plane domains. One of Steve’s main approaches in doing mathematics was to always work out a simple, and at the same time important, example first. For our case, ellipses were a natural choice because of their unique billiard dynamical properties that seem to characterize them amongst other planar smooth domains. The famous Birkhoff conjecture states that ellipses are the only completely integrable billiard tables. One then asks whether ellipses are unique from the quantum mechanical point of view; i.e., are the eigenvalues of the Laplacian associated to an ellipse (with Dirichlet or Neumann boundary conditions) unique amongst all smooth domains? This is Hamid Hezari is an associate professor of mathematics at the University of California, Irvine. His email address is [email protected].

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Figure 4. From left to right: Yanir Rubinstein, Steve Zelditch, Hamid Hezari. Baltimore, Spring 2009.

a difficult problem and remains open in this generality. In 2011, we proved a partial result about the spectral rigidity of ellipses within the class of smooth domains with two axial symmetries. A big development was subsequently announced in 2014 by Avila, de Simoi, and Kaloshin, who proved a version of Birkhoff’s conjecture for nearly circular ellipses. I remember that Steve got very excited about this result, knowing that there was something valuable for us to use. Indeed, in 2019, we managed to show that one can hear the shape of a nearly circular ellipse among all smooth domains. This is a strong result, but raises the question: what about ellipses of arbitrary eccentricity? In fact, this was a problem we investigated until the last few months of Steve’s life. Without a doubt, Steve was the most influential person in my life. He taught me how to do, read, write, and speak mathematics, and even how to fully live life. While I did not learn to his standards, what I could absorb helped me enormously with my career and personal life, for which I am forever thankful. He is greatly missed by his entire mathematical community.

Semyon Klevtsov I met Steve in January of 2009 at a conference on random geometry in Quebec that he co-organized. I was finishing my thesis under Mike Douglas, providing another derivation of the celebrated Tian–Yau–Zelditch–Catlin expansion of the Bergman kernel, using a quantum mechanical path integral parametrix. I was looking forward to talking to Steve about holomorphic sections, balanced metrics, and Kähler geometry. We started talking right from the moment we met. Steve was easily approachable, very friendly, and always eager to discuss math. I immediately felt “on the same wavelength” with him after just a first Semyon Klevtsov is a professor of mathematical physics at the University of Strasbourg. His email address is [email protected].

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few minutes of the conversation. During the conference dinner Steve sketched an idea of random Bergman metrics. (It must have been done on a napkin as my memory tells me it was definitely during the dinner.) I liked the idea a lot and we started working together. Our collaboration, which also included Frank Ferrari for a while, and our regular meetings ran until his last days and it was the most exciting intellectual adventure in my scientific life. I moved to Europe after finishing my PhD and Steve and I would work by Skype and then meet between semesters or at conferences. As transatlantic collaborations go, the first days of in-person meetings consistently began with the jet-lagged person trying not to doze off in front of the blackboard. At some point into our collaboration I realized that holomorphic sections can be used to describe quantum Hall states. I then branched out parttime to develop this subject with Steve’s continued support and influence. We even organized a conference together on geometric aspects of the quantum Hall effect in 2015. One of Steve’s deep and original contributions to modern mathematics is that he was one of the pioneers of “marrying” probability and geometry, often via the ideas from quantum field theory and quantum gravity. In fact, as he told me a few times, he decided early on that he would work on math related to quantum theory. In his grad school days he even took Richard FeynFigure 5. Steve Zelditch, man’s quantum mechanics Antwerp, 2014. class, although apparently that experience turned out be somewhat disappointing. Steve definitely was a topnotch expert in all things quantum. I would guess that his ideas about random geometry stem from his earlier work in spectral theory and quantum chaos. Later on, with Bernie Shiffman they launched a very successful area of random holomorphic sections—one of the first random geometry models. My later work with Steve on random Kähler metrics continued this line of thought. Together with several of Steve’s friends and colleagues, we long planned to organize a conference in his honor. Covid interfered and we finally got to do this only when we learned about his illness. It is quite extraordinary that about 500 people signed up to participate despite very short notice. This and all the outpouring of emotions following his passing away on the last day of the conference

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are testaments to his influence and lasting impact on so many people in very diverse areas of mathematics. I am truly blessed to have known Steve, worked with him, and enjoyed his friendship and mentorship. He was and is a role model, not only scientifically, but also as a human being. He will be dearly missed.

William P. Minicozzi II

it has been studied for hundreds of years. Steve made a number of important contributions to this problem and to related questions in complex geometry. A personal favorite is the paper [SZ11] inspired by a conjecture of Yau. Steve Zelditch was truly one of a kind. He brought energy and life to the department and the community. He will be sorely missed.

Duong H. Phong

I was fortunate to overlap with Steve Zelditch for about fifteen years at Johns Hopkins University. Steve was a great colleague and a remarkable mathematician—technically powerful, broadly knowledgable and always curious. He had virtually limitless energy and enthusiasm, he was great to talk math with, and he was enormously fun to be around (his protracted discussions causing me to miss my bus to the train station too many times). Steve had very broad interests, mathematically and more generally, but he had a particular interest in eigenfunctions. Fourier analysis describes the spectral theory of the circle of radius one: the eigenvalues are square integers 𝑘2 , and the eigenfunctions are sin 𝑘𝑥 and cos 𝑘𝑥. For compact manifolds, the spectral theory of elliptic operators gives a complete basis of eigenfunctions with eigenvalues going to infinity. There are some universal features, but the eigenfunctions behave very differently depending on the geometry of the manifold 𝑀. One of the themes in Steve’s work is the mysterious analogy between classical and quantum mechanics. In the quantum perspective, for each 𝑥 ∈ 𝑀, the value 𝑢2 (𝑥) is the probability density of the quantum particle being at 𝑥. This theme appears early in his influential 1987 paper on hyperbolic surfaces [Zel87], where he showed that the (quantum) eigenfunctions become uniformly distributed just like the (classical) geodesic flow. This perspective leads to natural questions. For example, on which spaces do the eigenfunctions “concentrate” the most? One way to measure concentration is to look at the ratios of various 𝐿𝑝 norms. If we normalize the 𝐿2 norms to be one, then how large can each 𝐿𝑝 norm be and on which spaces is this achieved? Steve and Chris Sogge proved beautiful results in this direction with geometry and dynamics playing key roles; see, e.g., their results on 𝐿∞ norms in [SZ02]. Their ideas generated a lot of activity and this continues to be an important area of research. Instead of looking at the places most likely to find the quantum particle, what can we say about the places least likely to find it? These are the points where the eigenfunction vanishes; this zero set is known as the nodal set and

I am heartbroken to write these lines in memory of Steve Zelditch, who passed away so suddenly on September 11, 2022. I can only share the grief of Steve’s entire family, and especially his wonderful wife Ursula, who extended the warm hospitality of their home to me so many times. In this context, I can’t even count the number of times when Steve confided to me how happy he was, and how lucky he was to have met Ursula. I am probably among those mathematicians and colleagues who have known Steve the longest, since 1981 when he joined Columbia University as a Ritt Assistant Professor. Mathematics and life are long and hard journeys, and it was a privilege for me to travel much of it in his company, often side by side, and always, I believe, in communal spirit. He graduated with a thesis on Schrödinger equations and microlocal analysis, and I witnessed first hand his growing interest in dynamical systems, and the emergence of his foundational paper on geometric quantum chaos. While we were not in as close contact after his departure from Columbia for Johns Hopkins in 1985, I followed his regular great works as well as I could after this early period, including the asymptotic expansion of the Bergman kernel, the many amazing applications he found for which he won the Bergman Prize, practically the creation of a whole new field of random complex geometry, and the first positive advance for decades on M. Kac’s famous question on whether one can hear the shape of a drum. Steve was the undoubted master of semiclassical analysis, transfigured with insights from other fields. From this vantage point, he would cast a new and unexpected light into a wide area of mathematics, including complex geometry, probability, dynamical systems, mathematical physics, and reveal phenomena that even seasoned experts in these areas would not suspect had existed. Steve excelled in every intellectual enterprise which he set his mind to. One example is the speed with which he built the Northwestern Mathematics department into the powerhouse in complex geometry which it is today. This can probably be traced in large part to the enormous

William P. Minicozzi II is the Singer Professor of Mathematics at Massachusetts Institute of Technology. His email address is [email protected].

Duong H. Phong is a professor of mathematics at Columbia University. His email address is [email protected].

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influence which he had on all his friends and those around him. While very much aware of other people’s opinions, he always had to form his own, which he would base on careful thought and study. He was always fair, and never overlooked the positive aspect of things. Even though referee reports of mathematics papers are anonymous, I suspect that anyone who got a report from Steve would instantly recognize it as such: it was always detailed, positive, informed, and raised unexpected interesting questions. And those who have served on the same National Science Foundation panels with him can vouch for his lucidity and eloquence. I myself learned a lot from him, from topics that I did not know at all before to subjects where I had had some familiarity, such as semiclassical analysis. Since Steve’s opinions were so well thought out to begin with, it was not easy to get him to change them. So it is with a bit of childish pride that I can report one instance when he came to me and said, “You know, after all these years, I have now come to your view that Richard Gere is a very bad actor.” It is terribly sad for me to think that Steve and I won’t be discussing mathematics, literature, and movies again, or be taking rides in his car listening to Armenian duduk music, or simply be arguing, as close friends are prone to, from the most mundane topics to the ones that we care most about. But his work will live on in mathematics, and his memory will be with me always.

Yanir A. Rubinstein I realize that in this type of memorial, exaggerations frequently happen. But to stay true to Steve’s legacy, I will do my best to say things as straight as I can. Steve Zelditch was my postdoc advisor for the academic year 2008–2009. Surprisingly, there was something of a mutual first in that relationship. I was Steve’s first NSF postdoc mentee. In 2007, when I asked Steve if he would agree to sponsor my application, he told me that if successful, I would be his first such mentee. I was astonished. To me, it was already shocking that Steve was not, say, at MIT or Stanford. When I first asked Steve about this, he shrugged it off and gave me the “one day you’ll understand, boy” reaction. On one occasion I pressed him hard on the issue as I felt it was unjust—at the time I was young and idealistic and believed that belonging to a top university was decided purely on the basis of the level of one’s mathematical originality and production. But it is not. And Steve explained that to me thoroughly. Yanir A. Rubinstein is a professor of mathematics at the University of Maryland. His email address is [email protected].

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Steve was honest down to his very last bone. Early on, he chose very hard problems that did not always have elegant solutions. Unwittingly, sometimes his results and choices created rivals. Also, it did not help that it took quite some decades for some of Steve’s most foundational work to become mainstream and appreciated for its pioneering value. Finally, it did not help that Steve was obsessive about mathematics. He could talk mathematics for hours on end, oftentimes interrupting his interlocutor repeatedly. Luckily for me, I came to know of Steve through his work on Kähler geometry, and the Kähler community accepted Steve with open arms almost immediately following his much-cited 1998 paper on the asymptotic expansion of the Szego˝ kernel. In a community with some rivalries and big egos, Steve was a soothing presence, universally appreciated and admired in the community. Steve didn’t take sides and often served as an ambassador making crucial connections. Steve loved talking to me about Kähler geometry, in which I was supposed to be an expert. In retrospect, I understood that his making me the purported expert actually somewhat contributed to my becoming one—it pushed me to deliver the answers he wanted and uphold that image. When he spoke about microlocal analysis, his true passion, I felt he really lit up. On the other hand, in Kähler geometry he put on the student gown, which he loved as well. Either way, regardless of the discussion topic, Steve always seemed to have unbelievable levels of energy—like nothing I have ever seen then or since. During my postdoc, I was in my 20’s while he was in his 50’s. Yet, that would be impossible for an outside observer to tell judging from his energy. It was the same in our collaborations. He would work on a problem by collecting books on a topic and putting dozens of papers in a folder, many of which he had read quite thoroughly. I imagined that his bullet reading skill came from his early life as an English major, but maybe it was just his genius. He then wrote several notes in another folder about different aspects of the theory, either summarizing results from the first folder, or trying to work out ideas on his own. He had many such projects at any given moment. It was breathtaking for me over the years to see him venture into completely new fields, from Kähler geometry to probability and mathematical physics and biology, certainly an inspiration for me, as I have also been a happy mathematical nomad throughout my career. In our last conversation, shortly before his passing, I expressed my deep admiration and gratitude to him. He listened but then insisted on emphasizing to me that “these relationships are very much mutual.” It was one of the most moving things a mentor has ever told me. He also told me, “you have truly surpassed yourself, Yanir.” I share

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this with the readers not to compliment myself but rather to try to communicate the magnitude of greatness and generosity of Steve, who was not thinking of himself, even as he was dying, and tragically since he was in possession of his full intellectual forces and in the midst of one of the most creative periods of his career. Let me share another recollection from about two months prior. Steve first mentored me in grant writing. As a small tradition, I would call Steve whenever I got an NSF grant. When I called him in June 2022, I felt something was off. He didn’t sound quite like himself. I asked him what was the matter. He shared that he was undergoing some tests and had some health issues. I asked him if there was anything I could do for him, which he dismissed. When it turned out that the disease was terminal, he wrote to remind me that I had asked what I could do, saying that there actually was something: take care of one of his famous folders, and see to it becoming published papers. In the following two months up until his death we spoke a few times about that folder. He fervently cared about mathematics and wanted to make sure those ideas got worked out. For him, each one of his folders had a life of its own, much beyond whether his name appeared on it. Three more things are unforgettable to me about Steve. First, his idealism; second, his relentless support of young mathematicians; third, his keen dislike of “declaring victory.” He sought to tackle hard problems that often did not have a nice and beautiful solution, but required many long, ingenious computations. It was not easy to write a paper with Steve because of his extremely high standards. Others will undoubtedly talk about Steve’s sense of humor, which deserves its own separate essay. For my part I will end with the following anecdote from our last conversation. I told Steve that he had achieved more than most mathematicians achieve in two lifetimes. He replied, “Well, I wish I had a third.” I already miss Steve dearly and doing mathematics will never be the same for me.

Bernard Shiffman I’ve known Steve Zelditch since he came to Johns Hopkins in 1985 as an Assistant Professor. When I called Steve to offer him a job at Hopkins (as I was on the hiring committee), I knew that he was an outstanding hire, but didn’t expect that I would ever work with him, since Steve hadn’t done Kähler geometry and I knew very little about microlocal analysis. I didn’t know then how Steve would latch onto and learn about almost any subject he knew nothing about and rapidly become an expert. Steve jumped headfirst into complex geometry with his 1998 seminal paper on what is now called the NOVEMBER 2023

Figure 6. Steve Zelditch and Bernie Shiffman, Santa Barbara, 2005.

Tian–Yau–Zelditch asymptotic expansion of the Bergman kernel for powers of a positive line bundle on compact Kähler manifolds. He was able to see connections between different areas of mathematics that others wouldn’t notice—our collaboration began when he heard a talk I gave on complex dynamics and saw a connection to quantum ergodicity. Then over the next 24 years, Steve developed a new area of mathematics, “stochastic Kähler geometry,” with the help of numerous collaborators including myself. Stochastic Kähler geometry involves the asymptotics of probabilistic invariants such as distribution and correlation functions of zeros and of critical points of random holomorphic sections of line bundles. Recently, together with Ferrari and Klevtsov, Steve began the study of random “Bergman metrics.” Steve was inspirational to his students and postdocs, as we know, and also to his colleagues and collaborators. He was generous with his ideas, in fact too generous, as the number of ideas he would come up with in one afternoon could take up many years. Steve was also a very gracious host and was generous with his time. Working with Steve was not only inspiring, but also fun. As all his colleagues know, Steve could talk entertainingly for hours—when you got in a conversation with Steve, on the phone or in person, you could expect the discussion to last two or more hours. His loquaciousness wasn’t only with mathematics. Steve would discourse at length on myriad topics, from stamp collecting to literature to whatever. Steve didn’t do anything halfway, from setting up an aquarium in his house, which at one time was the only furniture in his living room, to being a wine connoisseur, to following politics—in 2000 Steve convinced me to go with him to a rally in Washington for Ralph Nader, who was then the Green Party candidate for president. When working on a paper with Steve, he wouldn’t want to stop after obtaining the desired result, but would keep

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pushing the result to more settings and generalizations, until I had to insist we stop and submit the paper. I last spoke on Zoom with Steve on August 3, 2022, after his cancer had progressed beyond treatment. He began the conversation by saying that he couldn’t talk long and that I would have to do most of the talking—he then talked for an entire hour. I miss Steve. His passing leaves a large void.

Chris Sogge I first met Steve Zelditch at a microlocal analysis conference in Irsee, Germany, in the summer of 1990. He made a big impression on me. Even though, at the time, we were at the beginning of our careers and had much different backgrounds, we really hit it off and started a professional relationship and friendship that was one of my most important ones and would grow over the next three decades. A few years later, in 1996, I moved from UCLA to Johns Hopkins University. I was happy at UCLA and my career was going well, but Steve was always a master of persuasion. The move to Baltimore was great for me and my family, and I especially loved the 14 years that we overlapped until Steve left for Northwestern University. There are so many fond memories. Our families quickly became very close and we spent much time together, either at each other’s house or at our children’s sporting events. Our two youngest children are the same age as the Zelditch children, and they became very good friends. The family dinners would always end the same way. Steve would snag me away from everyone else and attempt to spend hours either talking about mathematics or his latest obsession. The worst was during his stamp collecting phase. Steve’s long soliloquies about Grauert tubes, Austrian stamps, Georgian wines, 1930s Shanghai music,. . . , would be interrupted (usually to my relief) by family members wanting to go home. When our children competed against each other in a sporting event, such as soccer, Steve would always be working out a calculation on a pad of paper sitting in his fold-up Home Depot chair. It was always remarkable how he was able to look up at exactly the right time to cheer on one of his sons louder than any of the rest of us parents. Steve really was great at multitasking. I really grew as a mathematician through my collaborations and many discussions with Steve over the years, attempting to become a practitioner of what Steve liked to call “Global Harmonic Analysis.” During the time he was at Johns Hopkins he tended to bring out the best in us, and Chris Sogge is the J. J. Sylvester Professor of Mathematics at Johns Hopkins University. His email address is [email protected].

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Figure 7. Steve Zelditch and Chris Sogge in 2004.

he perked up the department in so many ways. He was a very impactful and successful department chair (1999– 2002), who, among many other things, was instrumental in starting our very important and thriving J. J. Sylvester Assistant Professor (postdoc) program. Steve was somebody who rarely lost arguments, and I am sure that this was the case in his negotiations with the JHU administration. Also, even though he was so loquacious, Steve could be a great listener. This was especially true during seminar talks when his multiple interruptions would inevitably force the speaker to really tell us what he or she was attempting to say. Steve’s encyclopedic knowledge could be intimidating, but his charm and sense of humor would always result in a smile after a couple of well-directed questions. Steve really was a force of nature and, without a doubt, the most interesting person I have had the pleasure of knowing throughout my career. Two days before Steve sadly passed away I was honored to speak in the amazing online conference, “Global Harmonic Analysis,” which was quickly but skillfully and lovingly put together by several young mathematicians whom Steve had impacted. I ended my talk with a quote from our friend and former colleague, Bill Minicozzi: “Steve is a unicorn. Unique on the planet.”

Joel Spruck In 1990, while I was at the University of Massachusetts at Amherst, I was contacted by the hiring committee of the mathematics department at JHU and asked if I was interested in a senior position. I learned that Steve Zelditch was the only “hard analyst” in a department dominated by homotopy theory, algebraic geometry, and number theory. I Joel Spruck is the J. J. Sylvester Professor Emeritus of Mathematics at Johns Hopkins University. His email address is [email protected].

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had never met Steve as we moved in different circles. I visited the department soon thereafter and I gave a talk on my recent joint work with Craig Evans on the level set mean curvature flow. Steve must have liked my talk because I quickly received an offer. I joined the department in the 1992–1993 academic year. Steve quickly became my favorite person to talk to and have lunch with because he was so magnetic. I came to learn over the ensuing years that Steve was perfectly comfortable with the abstract high-powered algebraic side of the department because he knew and understood so much mathematics. I don’t think Steve fully realized that other math people were not as broad as he was. This could sometimes be frightening to students when he was on their oral exam committees. Steve was, apart from being a brilliant mathematician, a wonderful and delightful person who was devoted to his wife Ursula and sons Benjamin and Phillip. He was endlessly curious and talkative and made you smile inside and out. Steve was also what in Yiddish is called a mensch, roughly translated as an honorable person, someone full of integrity. We will all miss him dearly.

Alexander Strohmaier When I was a young postdoc I became interested in the intriguing relation between eigenvalues of the Laplacian and the geodesic flow. After working out some consequences of spectral properties of the geodesic flow on the clustering of eigenvalues, I was going to give a talk about this at a conference in Montreal in June 2004. Talking to the other participants I quickly learned that what I had done was contained in the work of Steve Zelditch. It was at this conference that I first met Steve. He was mathematically firm whilst very kind on a personal level. He told me that these things happen all the time and I should not be discouraged. I would like to describe here the simple and beautiful correspondence between the spectral measures of the Laplacian and the spectral measure of the geodesic vector field on the unit tangent bundle. The spectrum of the Laplacian −Δ on a closed Riemannian manifold (𝑀, 𝑔) has been of interest to mathematicians for a long time. Let 𝜆𝑗 be the positive roots of the eigenvalues and 𝜙𝑗 the corresponding eigenfunctions. The formula 𝜔𝑗 (𝐴) = ⟨𝐴𝜙𝑗 , 𝜙𝑗 ⟩ defines a state 𝜔𝑗 on the ∗-algebra of pseudodifferential operators of order zero and therefore on its norm completion 𝒜, which is a 𝐶 ∗ -algebra. Any weak-∗-limit point descends to a state on 𝒜/𝒦 ≅ 𝐶(𝑆 ∗ 𝑀). A theorem by Helton Alexander Strohmaier is a professor of mathematics at the University of Leeds. His email address is [email protected].

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from 1977 links the clustering properties of eigenvalues to the geodesic flow. Namely, the existence of a single nonclosed geodesic implies that the set {𝜆𝑗 − 𝜆𝑘 ∣ 𝑘, 𝑗 ∈ ℕ} is dense in ℝ. This theorem can be made more precise and follows from a trace formula that originates from the work of Helton and Zelditch, which I would like to explain here. Let 𝑍 be the geodesic vector field on 𝑆 ∗ 𝑀. Then i𝑍 generates a unitary group on 𝐿2 (𝑆∗ 𝑀) and therefore defines a self-adjoint operator. Next, one can define the Riesz means of the spectral measures 𝜇𝐴,𝑗 associated with the Gelfand–Naimark–Segal representation of the states 𝜔𝑗 . It has been noted by Zelditch in [Zel96], that the sequence 𝜇𝐴,𝑘 (𝑓) converges to the measure ⟨d𝐸𝜆̃ 𝑎, 𝑎⟩, where 𝑎 ∈ 𝐶(𝑆 ∗ 𝑀) is the principal symbol of 𝐴, and d𝐸𝜆̃ is the spectral measure of the generator of the geodesic flow. This shows the following trace-formula 𝑘

1 ∑ ⟨d𝐸 𝐴𝜙 , 𝐴𝜙𝑗 ⟩ → ⟨d𝐸𝜆̃ 𝑎, 𝑎⟩ 𝑘 𝑗=1 𝜆−𝜆𝑗 𝑗 with convergence in the weak-∗-sense as 𝑘 → ∞, where d𝐸𝜆 is the operator-valued spectral measure of the root √−Δ of the Laplacian. This immediately implies that any point in the spectrum of i𝑍 must be a cluster point of {𝜆𝑗 − 𝜆𝑘 }. This gave rise to a finer analysis of the interplay between the geodesic flow and quantum ergodicity, much of which owes to Zelditch.

Figure 8. Steve Zelditch and Alexander Strohmaier, seen looking at their reflection.

Steve Zelditch and I have recently collaborated on several projects related to the generalization of spectral theory of the Laplacian to the more general relativistic situation of stationary spacetimes. The Gutzwiller–Duistermaat– Guillemin trace formula was shown in this context in [SZ21]. Without going into details, the framework set up in [Zel96] is very general and it is therefore likely that

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Helton’s observation holds for stationary spacetimes with compact Cauchy surfaces. This would imply spectral clustering if there exists a single nonperiodic null geodesic in the space-time. Steve has had a profound influence on the field with many different results, ranging from nodal sets and restriction theorems to complex properties of eigenfunctions. He was able to describe in a single sentence the essence of a paper. It is this type of interaction that is so important amongst mathematicians. I owe him a lot and he will very much be missed.

Jacob Sturm I met Steve in 1981 at Columbia. He was just starting his postdoc; I just finished mine and had recently moved to Johns Hopkins. But I missed NYC terribly, and although I had just met Steve and he barely knew me, we somehow hit it off and he suggested that I could stay with him on weekends in his two-bedroom apartment on 113th Street between Amsterdam and Broadway, so that’s what I did. I have very fond memories of those days. At that time, I was a number theorist (a student of Goro Shimura) and Steve was interested in geodesic flows on compact Riemann surfaces, so we didn’t have much in common except for the upper half plane. Nevertheless, we had a lot to talk about, both mathematically and about “life.” We went to parties, movies, bars, restaurants, hosted dinners, etc., and just enjoyed the exuberance that living in NYC inspires. Little did I know at the time that 20 years later, when I switched fields to complex differential geometry, that his work would have a profound influence on my own. The Tian–Yau–Zelditch theorem was first recognized as a very powerful tool in Kähler geometry by Simon Donaldson, who used it in several papers to prove some marvelous theorems. After Donaldson’s work, many other researchers in the area took notice of Steve’s work and applied it to great advantage. TYZ says that Kähler metrics, which are rather transcendental sorts of objects, can be approximated in a very precise sense by Bergman metrics, which are “algebraic.” Phong and I realized how one could use TYZ to show that geodesics in the space of Kähler metrics can be approximated by Bergman geodesics and how geodesic rays could be approximated by test configurations. We wrote several papers about this topic, and Steve was very interested. He invited me to Hopkins a couple of times, and then he and Jian Song worked out in beautiful detail (obtaining much more precise results) the geodesic theorems for toric varieties. I think they wrote three papers on Jacob Sturm is Distingushed Professor of Mathematics at Rutgers UniversityNewark. His email address is [email protected].

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this subject. So, it ended up that we influenced each other, which made me very happy. Many people have said that Steve was not just a brilliant mathematician—he was just plain brilliant, and I couldn’t agree more. I remember a dinner I once had with Steve and John Morgan and Phong at the Lion’s Head (a famous Greenwich Village bar/restaurant, now defunct, that was a favorite hangout for journalists and writers like Jimmy Breslin, Norman Mailer, etc). Steve, John, and I had (separately) seen a recently released movie and were discussing it over drinks. John and I were saying the sorts of things that people often say: “the plot was formulistic,” “the acting was great,” “I didn’t like the ending,” but Steve’s take was at a completely different level. He spoke for an uninterrupted 15 minutes or so, comparing it to other films by the same director, pointing out subtle symbolism, the role played by the history of the setting, the influence of Greek mythology, . . . . It was as if he had written a detailed film review and was reading it aloud! All of this delivered without pretension, in fact seeming unaware of his own brilliance. I think we were all a bit awestruck. Steve was a lot of fun to be around: he had a great sense of humor and could talk about virtually anything. At the end of a long math day at a conference or during a visit, the group would often go out for drinks, usually wine, usually pinot noir. Steve was fond of saying “Pinot noir isn’t just a wine. It’s a way of life.” Once, during a dinner at Pasha, a Turkish restaurant in NYC with a limited selection, I told him that I enjoyed the Kendall Jackson pinot noir that we had ordered. He told me that there was a lot to be experienced, and that KJ was just scratching the surface. He went on to say something like “Toric varieties are probably the Kendall Jackson of Kähler geometry. But if one doesn’t start scratching the surface, how does one get deeper into things?” Quintessential Steve. Steve often talked about his family—one instance I recall was a workshop at Park City in July of 2013 where Steve was giving a minicourse on eigenfunctions of the Laplacian. We spent a lot of time hanging out together, and Steve seemed a bit homesick. The fact that he could connect with Ursula at many levels (including mathematics) meant a great deal to him. He was very proud of Benny who was a top student, an award-winning guitarist, and highly motivated. And Philly was Steve’s great buddy with whom, despite the fact that Philly was only 14 years old at the time, he was able to have long, stimulating intellectual discussions. I was planning to call Steve after the September 2022 conference in his honor, not realizing how far his illness had progressed. Now I regret that I didn’t call him earlier when I first learned he was sick. I miss him a great deal.

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John A. Toth I first met Steve Zelditch while I was a graduate student in the early 1990s. Steve organized a workshop at Johns Hopkins on Birkhoff normal forms and the computation of the associated spectral invariants. I distinctly recall Steve’s infectious enthusiasm for the field and his unusual generosity in sharing his ideas on a wide variety of topics, and in inviting me to visit. Our collaboration, which began in the late 1990s and lasted more than two decades, was a continuous source of inspiration for me, and his emails, sometimes multiple in a single day or night, reflected his enthusiasm that continued unabated through the years. Our joint projects all dealt with the semiclassical asymptotics of eigenfunctions of self-adjoint elliptic operators on compact manifolds in various settings. Starting in around 1997, we began working on the asymptotic properties of joint eigenfunctions of quantum completely integrable (QCI) systems. On a compact 𝑛-manifold, these systems are characterized by the existence of a family of 𝑛 selfadjoint pseudodifferential operators 𝑃𝑗 ; 𝑗 = 1, … , 𝑛 with the property that [𝑃𝑖 , 𝑃𝑗 ] = 0; 𝑖 ≠ 𝑗. Given the associated principal symbols 𝑝𝑗 = 𝜎(𝑃𝑗 ) one forms the associated classical moment map 𝒫 ∶= (𝑝1 , … , 𝑝𝑛 ) ∶ 𝑇 ∗ 𝑀 → ℬ ⊂ ℝ𝑛 . Given a regular value 𝑏 ∈ ℬ𝑟𝑒𝑔 , by the Liouville–Arnold theorem, the invariant sets 𝒫 −1 (𝑏) are a finite union of Lagragian tori. However, given a singular value 𝑏 ∈ ℬ𝑠𝑖𝑛𝑔 , these tori can degenerate in a rather complicated fashion. The associated semiclassical blow-up properties of the joint eigenfunctions of the 𝑃𝑗 ’s are closely linked to the properties of the singular leaves of the moment map via the quantum Birkhoff normal form associated with the commuting operators. In the period between 1997 and 2006, Steve and I wrote several papers on the concentration properties of QCI eigenfunctions and their link to the singular leaves of the moment map. Around 2008, in a discussion with Steve at a conference in Austria, he raised the question of whether the celebrated QE theorem of Shnirelman, Zelditch, and Colin de Verdière extends to generic hypersurfaces 𝐻 𝑛−1 ⊂ 𝑀 𝑛 . A few years earlier, Hassell and Zelditch [HZ04] and, independently, Burq had answered this question in the affirmative for Neumann (or Dirichlet) eigenfunctions in the special case where 𝐻 was the boundary of a piecewise-smooth domain with ergodic billiards. Steve and I began working on the general question in 2009 and published a series of papers proving that the QE theorem was indeed true for a full density of restrictions of QE Laplace eigenfunctions under a generic microlocal John A. Toth is a professor of mathematics at McGill University. His email address is [email protected].

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asymmetry property on the hypersurface. Specifically, given a Riemannian manifold (𝑀, 𝑔) with ergodic geodesic flow 𝐺 𝑡 ∶ 𝑆 ∗ 𝑀 → 𝑆 ∗ 𝑀 there is a density one subsequence of QE Laplace eigenfunctions {𝜙𝜆𝑗 }, 𝑘 ∈ ℕ such that given 𝑘

any zeroth-order pseudodifferential operator 𝐴 ∈ Ψ0 (𝐻) and provided 𝐻 is microlocally asymmetric with respect to the geodesic flow, the Dirichlet data 𝜙𝜆𝐻𝑗 ∶= 𝜙𝜆𝑗 |𝐻 sat𝑘

𝑘

isfies lim𝑘→∞ ⟨𝐴𝜙𝜆𝐻𝑗 , 𝜙𝜆𝐻𝑗 ⟩ = ∫𝐻 𝜎(𝐴)𝑑𝜇𝐻 , where 𝑑𝜇𝐻 de𝑘

𝑘

∗ notes the restriction of Liouville measure to 𝑆𝐻 𝑀. This is the quantum ergodic restriction (QER) theorem that we proved in the papers [TZ12,TZ13] both on manifolds with or without boundary. The corresponding result for general Schrödinger operators was proved by Dyatlov and Zworski. Steve and I together with Hans Christianson also proved a companion QER theorem for Cauchy data (𝜙𝑗 |𝐻 , 𝜕𝜈 𝜙𝑗 |𝐻 ) in [CTZ13]. Over the last decade, most of our joint work dealt with applications of eigenfunction restriction results (including QER) to upper bounds on the Hausdorff measures of intersections of eigenfunction nodal sets with general hypersurfaces in the real-analytic setting. We proved sharp upper bounds on the measure of such nodal intersections first for piecewise-analytic bounded planar domains and then in the general analytic setting in arbitrary dimension in [TZ21]. Our work together was just one of many different collaborations that Steve fueled with his enormous energy and his wide and deep mathematical interests. He was an extraordinary mathematician and a force of nature as a person. I will miss him immensely.

Ben Weinkove I met Steve when I was a graduate student in the early 2000s, giving a talk at Johns Hopkins. Steve had recently brought powerful new techniques into Kähler geometry, in his proof of the Tian–Yau–Zelditch expansion. Later I learned that this was typical of Steve’s style. His knowledge of many disparate areas of mathematics gave him a large tool box which he exploited in whatever problem sparked his interest. I was surprised and flattered by Steve’s interest in my work. Again, this was classic Steve. It didn’t matter if a graduate student or a Fields Medalist was giving a talk, Steve wanted to understand it, and would keep asking questions until he did. I admired this attitude which represented to me the best of mathematical culture: interest driven by genuine curiosity, not credentials. Ben Weinkove is a professor of mathematics at Northwestern University. His email address is [email protected].

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About a decade later I became Steve’s colleague at Northwestern as part of a group in geometric analysis, including Valentino Tosatti and Aaron Naber, built with his help. Steve had a huge presence in our department. In seminars, the question, “Does anyone have any questions for the speaker?” often became, “Steve: do you have any more questions?” Steve had a strong sense of responsibility toward the discipline and the department. He loved to teach graduate functional analysis so much he even offered to teach it for free. In hiring matters, Steve offered well informed opinions on candidates in almost every field. Not content with merely reading letters, Steve scrutinized the papers of candidates, and often had specific questions to follow up with them. Steve’s devotion to mathematics continued even as his health was failing. He still met with students, wrote reference letters and took the time to write preliminary exam problems. Steve’s untimely passing was a terrible blow to us all and to me personally. He had been a large part of my mathematical life from the beginning of my career. He was an inspiration to me. Thank you, Steve—it was an honor to have known you.

MathSciNet, one of them having appeared in the Annals of Mathematics.

Richard A. Wentworth Steve Zelditch was an exceptional individual; intensely smart, gregarious, energetic, funny, and with an insatiable appetite for intellectual engagement. There was almost no part of mathematics, or science more generally, that he didn’t find interesting. His scholarship and vast knowledge were exemplary, and his enthusiasm was infectious. His presence left such an indelible impression on all who knew him that his disappearance is difficult to comprehend.

Alan Weinstein Steve Zelditch was one of my early PhD students, and it was clear almost immediately that he was exceptional. Many students later, I still found him one of the best. Shortly after Steve finished his PhD, his thesis inspired me to write a paper on a symbol calculus for Schrödinger operators on ℝ𝑛 . The analysis in the paper was essentially that of the thesis, to which I added geometric interpretation. In the introduction, I wrote, “I would like to thank Steven Zelditch for many stimulating discussions concerning [his thesis], and frequent reassurance that integrating by parts would work whenever I needed it to.” After that paper, since I figured that my kind of microlocal analysis was in very good hands (an assumption which turned out to be absolutely correct), I took a long break from the subject to pursue other interests. Of course, Steve went on to deepen and broaden his work to encompass many aspects of the theory of eigenvalues and eigenfunctions of differential operators. To the personal sadness of his passing to those close to him is added the loss of someone still in his scientific prime. In recent years, in his mid-to-late 60s, he continued to produce excellent work, with many papers on Alan Weinstein is an emeritus professor of mathematics at the University of California, Berkeley. His email address is [email protected].

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Figure 9. Steve Zelditch in 2012 with Jacob Sturm (right) and Richard Wentworth (left).

Steve approached mathematics as a scientist, in a way that I always found unique and inspiring. He would often say that what he looked for in a person’s work was whether they were “discovering new phenomena.” His lectures were replete with references to quantum mechanics and pictures of Chladni plates. While having a clear perspective from his own expertise in microlocal analysis, he was fearless in incorporating whatever new methods might be needed for the problem at hand. I once heard someone describe his research as resembling a “big truck rumbling down the street.” On any topic, Steve was a formidable debater and a master of dispassionate discourse. He frustrated his opponents by taking apart the logic of their arguments and exposing inconsistencies, all in a calm yet persistent way. He was also a remarkable judge of character and human nature, and with this came an understanding of and a compassion for people. Similarly, despite his intense focus, what I think everyone remembers about Steve was his terrific sense of humor. His talks were invariably a blend of Richard A. Wentworth is a professor of mathematics at the University of Maryland. His email address is [email protected].

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scholarly prowess and self-deprecating jokes, always welltimed and always well-received. I likely first met Steve at an AMS meeting in Ann Arbor in the early 1990s, where he, Lizhen Ji, and I discussed spectral problems on surfaces with cone singularities. While Steve and I never formally collaborated, he gave me crucial input on several projects related to determinants of elliptic operators, and he vastly broadened my knowledge of what was happening in the mathematical world. He was always generous in sharing his wisdom and experience. I remember and use to this day many examples of his sound advice on how to approach all aspects of our profession. He was a great colleague. In talking to people who didn’t know him I was always in the habit of describing Zelditch as “the most remarkable person I have ever met.” This somewhat Gurdjieffian formulation, one that I hope Steve would have appreciated, was meant only half in jest. With his far-too-early passing, indeed, it seems to me truer than ever.

Shing-Tung Yau I have known Steve for more than thirty years. He was at Johns Hopkins and I was very impressed by his deep insight in geometry and in modern analysis. Both he and I are fond of eigenvalues and eigenfunctions of the Laplacian. I was pleasantly surprised to find out that he graduated from Berkeley and was a student of Alan Weinstein, whom I knew well. About 40 years ago, I proposed a program to construct Kähler–Einstein metrics on Fano manifolds. I was convinced that their existence is related to stability of the manifold in the sense of Mumford’s geometric invariant theory. The first step was to show that the Kähler–Einstein metric can be recovered from the Fubini–Study metrics obtained from embeddings into projective space. I assigned this problem to Gang Tian for his thesis. I suggested using the ideas of peak sections in my work with Siu on holomorphic isometric embeddings. Tian was able to do so, and the higher regularity was accomplished by my other student Ruan. The Fubini–Study metric is actually the Bergman metric for the projective embedding, and it turned out that Steve was able to look at this problem from the point of view of the asymptotic expansion of the Bergman kernel. Several mathematicians followed his insight and made important contributions to my original conjecture on the existence of Kähler–Einstein metrics on Fano manifolds. The idea contributed in a key manner to Shing-Tung Yau is a professor of mathematics at Tsinghua University and an emeritus professor of mathematics at Harvard University. His email address is [email protected].

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the solution of my conjecture provided by Donaldson et al. There were of course many other important contributions made by Steve in geometry and in analysis. A notable contribution was his beautiful work on spectral rigidity for a large number of domains in Euclidean space. Steve was so direct on revealing his insight to other people that sometimes I thought he was arrogant—but each time he proved to be correct. I admired him. I tried several times to nominate him to be elected to the National Academy of Sciences. Although we did not have much chance to communicate, I believe that Steve liked me, because he insisted on listening to my talk in honor of his birthday even while he was dying. I tried my best. But I was giving the talk in Beijing through Zoom. I could not tell his response to my talk. He passed away a short time after my talk. I lost a good friend. But I am glad that I gave my talk in his honor right before he passed away. All of us will remember his deep contribution to mathematics and his friendship.

Maciej Zworski Steve Zelditch spent part of the academic year 1987–1988 at MIT and that is where we met for the first time. I was a third-year grad student working with Richard Melrose while Steve, who was already at Johns Hopkins, was visiting as an NSF postdoctoral fellow. He very quickly became a strong and irresistible presence in my mathematical life, which had perhaps been all too comfortable till then. He talked about everything and asked questions about everything. He was particularly aggressive in trying to find out from me if the Lax–Phillips semigroup was a Fourier integral operator. I was lost and he felt somewhat uncharacteristically guilty, apologizing that if “one comes from the Guillemin-style school, one would ask that kind of question about your father and mother.” Not long afterward he invited me to Johns Hopkins and while sitting in Baltimore harbor he was talking about the Langlands program. I understood nothing but when I heard the words meromorphic continuation I mumbled if “it isn’t something like Lax–Phillips automorphic scattering.” Steve turned to me and bellowed: “You see this skyscraper, you see this water hydrant—that is how the two compare!.” This type of passion mixed with humor (half self-deprecating, half wicked) has been a sometimes endearing, sometimes infuriating, force for good in my mathematical life and that of many others. Maciej Zworski is a professor of mathematics at the University of California, Berkeley. His email address is [email protected].

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Figure 10. Lizhen Ji, Steve Zelditch, and Maciej Zworski.

A few years later, when thanks to Steve’s good offices I was also at Johns Hopkins, he introduced me to one of his favorite subjects, and one which he pioneered in the West: quantum ergodicity. Ten years before, Steve had discovered an announcement by Shnirelman stating that for a compact Riemannian manifold 𝑀 with an ergodic geodesic flow, almost all eigenfuctions equidistribute. That means that if 0 = 𝜆0 < 𝜆1 ≤ 𝜆2 ≤ ⋯ is the complete list of eigenvalues with eigenfunctions, 𝑢𝑗 , −Δ𝑢𝑗 = 𝜆𝑗 𝑢𝑗 , ∫𝑀 |𝑢𝑗 |2 = 1, then there exists a density one subsequence 𝑢𝑗𝑘 such that ∫ 𝜑|𝑢𝑗𝑘 |2 → |𝑀|−1 ∫ 𝜑, 𝑘 → ∞, 𝑀

(1)

𝑀

for all 𝜑 ∈ 𝐶 ∞ (𝑀). In fact, this equidistribution is valid in a stronger position and momentum sense. Steve Zelditch provided a proof in the constant negative curvature case, including the finite volume noncompact case. (When in addition, a surface is arithmetic, a celebrated work of Lindenstrauss later showed that the sequence of 𝑢𝑗 ’s can be chosen so that there is no need for a subsequence—the case of so-called unique quantum ergodicity). Colin de Verdière then gave a proof for closed manifolds. He recalls how Steve, without any prior arrangements, drove up to Institut Fourier in Grenoble, found him in his office, explained his work on quantum ergodicity and drove off. While at Johns Hopkins, we generalized this work to the case of compact manifolds with piecewise smooth boundaries. Many advances, a lot of which are by Steve and his collaborators, have been made since and it is impossible to survey them here. I conclude with a very recent one: if in (1) we take 𝜑 ≥ 0 to be equal to 1 on a nonempty open set Ω, then there exists a constant 𝑐(Ω) > 0 such that ∫Ω |𝑢𝑗𝑘 |2 > 𝑐(Ω). Dyatlov, Jin, and Nonnenmacher (building on earlier work of Anantharaman, Bourgain–Dyatlov, 1682

Figure 11. Phillip Zelditch, Ben Zelditch, and Ursula Porod at the Northwestern University memorial for Steve, October 2022.

and Dyatlov–Jin) showed that for negatively curved surfaces ∫Ω |𝑢𝑗 |2 > 𝑐(Ω) where 𝑢𝑗 is any sequence of eigenfunctions. We do not know if all 𝑢𝑗 ’s are equidistributed but at least there cannot be any holes in their supports, uniformly as 𝜆𝑗 → ∞. References

[BSZ00] Pavel Bleher, Bernard Shiffman, and Steve Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000), no. 2, 351–395, DOI 10.1007/s002220000092. MR1794066 [CTZ13] Hans Christianson, John A. Toth, and Steve Zelditch, Quantum ergodic restriction for Cauchy data: interior que and restricted que, Math. Res. Lett. 20 (2013), no. 3, 465–475, DOI 10.4310/MRL.2013.v20.n3.a5. MR3162840 [DSZ04] Michael R. Douglas, Bernard Shiffman, and Steve Zelditch, Critical points and supersymmetric vacua. I, Comm. Math. Phys. 252 (2004), no. 1-3, 325–358, DOI 10.1007/s00220-004-1228-y. MR2104882 [DSZ06a] Michael R. Douglas, Bernard Shiffman, and Steve Zelditch, Critical points and supersymmetric vacua. II. Asymptotics and extremal metrics, J. Differential Geom. 72 (2006), no. 3, 381–427. MR2219939 [DSZ06b] Michael R. Douglas, Bernard Shiffman, and Steve Zelditch, Critical points and supersymmetric vacua. III. String/M models, Comm. Math. Phys. 265 (2006), no. 3, 617–671, DOI 10.1007/s00220-006-0003-7. MR2231684 [GRHZ15] Jesse Gell-Redman, Andrew Hassell, and Steve Zelditch, Equidistribution of phase shifts in semiclassical potential scattering, J. Lond. Math. Soc. (2) 91 (2015), no. 1, 159–179. MR3335243 [HZ99] Andrew Hassell and Steve Zelditch, Determinants of Laplacians in exterior domains, Internat. Math. Res. Notices 18 (1999), 971–1004. MR1722360

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[HZ04] Andrew Hassell and Steve Zelditch, Quantum ergodicity of boundary values of eigenfunctions, Comm. Math. Phys. 248 (2004), no. 1, 119–168. MR2104608 [HZ10] Hamid Hezari and Steve Zelditch, Inverse spectral problem for analytic (ℤ/2ℤ)𝑛 -symmetric domains in ℝ𝑛 , Geom. Funct. Anal. 20 (2010), no. 1, 160–191. MR2647138 [HZ22] Hamid Hezari and Steve Zelditch, One can hear the shape of ellipses of small eccentricity, Ann. of Math. (2) 196 (2022), no. 3, 1083–1134. MR4502596 [Kac66] Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, part II, 1–23. MR201237 [SZ02] Christopher D. Sogge and Steve Zelditch, Riemannian manifolds with maximal eigenfunction growth, Duke Math. J. 114 (2002), no. 3, 387–437, DOI 10.1215/S0012-7094-0211431-8. MR1924569 [SZ11] Christopher D. Sogge and Steve Zelditch, Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett. 18 (2011), no. 1, 25–37, DOI 10.4310/MRL.2011.v18.n1.a3. MR2770580 [SZ21] Alexander Strohmaier and Steve Zelditch, A Gutzwiller trace formula for stationary space-times, Adv. Math. 376 (2021), Paper No. 107434, 53. MR4178908 [TZ12] John A. Toth and Steve Zelditch, Quantum ergodic restriction theorems. I: Interior hypersurfaces in domains wth ergodic billiards, Ann. Henri Poincar´e 13 (2012), no. 4, 599– 670, DOI 10.1007/s00023-011-0154-8. MR2913617 [TZ13] John A. Toth and Steve Zelditch, Quantum ergodic restriction theorems: manifolds without boundary, Geom. Funct. Anal. 23 (2013), no. 2, 715–775, DOI 10.1007/s00039-0130220-0. MR3053760 [TZ21] John A. Toth and Steve Zelditch, Nodal intersections and geometric control, J. Differential Geom. 117 (2021), no. 2, 345–393, DOI 10.4310/jdg/1612975018. MR4214343 [Zel87] Steven Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), no. 4, 919–941. MR916129

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[Zel92] Steven Zelditch, Isospectrality in the FIO category, J. Differential Geom. 35 (1992), no. 3, 689–710. MR1163455 [Zel96] Steven Zelditch, Quantum ergodicity of 𝐶 ∗ dynamical systems, Comm. Math. Phys. 177 (1996), no. 2, 507–528. MR1384146 [Zel97] Steven Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 1, 305–363. MR1437187 [Zel99] Steve Zelditch, Spectral determination of analytic biaxisymmetric plane domains, Math. Res. Lett. 6 (1999), no. 34, 457–464. MR1713144 [Zel00] Steve Zelditch, Spectral determination of analytic biaxisymmetric plane domains, Geom. Funct. Anal. 10 (2000), no. 3, 628–677. MR1779616 [Zel04a] Steve Zelditch, Inverse spectral problem for analytic domains. I. Balian-Bloch trace formula, Comm. Math. Phys. 248 (2004), no. 2, 357–407. MR2073139 [Zel04b] Steve Zelditch, Note on quantum unique ergodicity, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1869–1872. MR2051153 [Zel09] Steve Zelditch, Inverse spectral problem for analytic domains. II. ℤ2 -symmetric domains, Ann. of Math. (2) 170 (2009), no. 1, 205–269. MR2521115 Credits

Figures 1 and 3 are courtesy of Northwestern University. Figures 2, 6, and 10 are courtesy of Ursula Porod. Figure 4 is courtesy of Hamid Hezari. Figure 5 is courtesy of Semyon Klevtsov. Figure 7 is courtesy of Will Kirk/Johns Hopkins University. Figure 8 is courtesy of Alexander Strohmaier. Figure 9 is courtesy of Paul Feehan. Figure 11 is courtesy of Ben Weinkove.

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When Small Changes Lead to Big Impact: Hysteresis in Mathematics Teaching Chris Rasmussen and Estrella Johnson The purpose of this article is to highlight some insights gleaned from the emerging body of research that focuses on undergraduate mathematics teaching. Unlike the domain of research that examines student reasoning and what it means to understand particular content, it is only within the last dozen or so years that the field has closely examined instructional practices at the undergraduate level. Indeed, Speer and colleagues in their 2010 comprehensive review referred to collegiate mathematics teaching as “an unexamined practice” (Speer et al., 2010, p. 99). The situation today, however, is quite different and reflects a growing interest in undergraduate mathematics teaching. For comprehensive reviews of this literature we point readers to Rasmussen and Wawro (2017) and Melhuish et al. (2022). Our goal in this report is to present these insights in a way that is accessible to a wide range of readers (often mathematics education research reports are steeped in terminology and theory because they are aimed at an educational research audience). In pulling out these researchbased insights, our aim is to contextualize, motivate, and provide some accessible instructional recommendations and resources that are relevant for a wide range of teaching approaches. Chris Rasmussen is a professor at San Diego State University. His email address is [email protected]. Estrella Johnson is a professor at Virginia Tech. Her email address is strej@vt .edu. Communicated by Notices Associate Editor William McCallum. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2805

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We structure our discussion of research on undergraduate mathematics teaching by first highlighting three articles that examine the benefits and limitations of lecture followed by a brief review of the value of instruction that actively engages students. We then shift to an overview of research-based instructional strategies that reflect easy entry points for actively engaging students in class. We conclude this brief review with a collection of resources where readers can learn more.

There Are Good Reasons to Lecture Lectures have a long history in undergraduate mathematics instruction. Advantages of lectures include efficient delivery of a large amount of material, opportunities to motivate the topic, inspire students, and provide the necessary background for study outside of class, and the chance for instructors to model mathematical ways of thinking and doing mathematics. Indeed, mathematicians conversant with the culture and practice of mathematics are wellpositioned to motivate, inspire, and be models for how mathematics is done (e.g., Byers, 2007). Research that illuminates the practice of lecture, therefore, can highlight its value and nature, as well as its limitations. For example, Weber and Fukawa-Connelly (2023) investigated what mathematicians learn from attending other mathematician’s lectures. These researchers interviewed 13 mathematicians who participated in a two-week workshop on inner model theory, a topic in set theory. Central findings include an appreciation of the big picture of the domain and how experts in this area think about the topic, the highlighting of important results and techniques, and the feeling of being well-equipped to embark on their own subsequent study of the material. These findings illustrate

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how lectures differ from textbooks or research papers that are necessarily concerned with rigor and correctness, and often mask the intuition and big-picture ideas that lectures can convey. Instead, findings from this study illustrate the potential for undergraduate mathematics lectures to model doing mathematics, provide a road map of the domain, and motivate students and hence facilitate independent study. We conjecture that because of the distance in mathematical maturity between where undergraduate students are and where mathematicians are, specific pedagogical actions are needed so that students can benefit from lectures in the same ways that mathematicians and graduate students do. Another relevant and illuminating study that examined the practice of lecturing is the international study by Artemeva and Fox (2011). Based on lecture recordings and interviews with over four dozen lecturers from Australia, Canada, Israel, Spain, Sweden, the United Kingdom, and the United States, these researchers identified a collection of common practices, which they refer to as “chalk talk.” Chalk talk practices include verbalizing everything written on the board along with metacommentary about what was written, using rhetorical questions to signal transitions, the use of pointing to highlight key ideas, verbal references to the text, assignments, and notes, and checks for understanding. It is precisely these chalk talk practices that enable lecturers to convey the practice of how experts do and think about mathematics. Some differences across the cultural-historical educational contexts include the acceptability of making mistakes, the willingness of students to ask questions, and instructor use of notes during the lecture. Studies like those by Weber and Fukawa-Connelly and Artemeva and Fox are valuable because they highlight qualities of good lectures and how they might contribute to student learning. In addition to some of the positive aspects of lectureoriented instruction, the research literature has also revealed nuanced ways that account for why students do not always take away with them what the instructor intends. For example, Lew et al. (2016) studied what advanced undergraduate students learned from a real analysis lecture and compared this to what the professor thought to be the central points of the lecture. These researchers conducted a case study of one professor, Dr. A, who had an excellent reputation as a real analysis instructor with over 30 years of experience. As a case in point, they report on one 11-minute proof that a sequence {xn } with the property that |xn – xn+1 | < rn for some 0 < r < 1 is convergent. They first interviewed Dr. A and asked him about the goals of his lecture and why he presented this proof to students. They then showed him a video recording of the lecture and asked him to stop the recording at every point he thought

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he was trying to convey mathematical content and to specify the content. They then conducted interviews with three pairs of students with the following four “passes” through the lecture: Pass 1: Students recalled what they learned from the proof by reviewing their notes. Pass 2: Students watched the lecture again in its entirety, took notes, and were asked what they learned and what they thought the instructor was trying to convey. Pass 3: Students were shown short specific clips of the video and asked what they thought the professor was trying to convey. Pass 4: Students were asked whether particular content highlighted by Dr. A in his interview could be gleaned from the proof they just watched. Content conveyed by professor

Pair #1

Pair #2

Pair #3

To show sequence is convergent without a limit candidate, show it is Cauchy

Pass 3 Pass 3 Never

Triangle inequality is important for proofs in real analysis

Pass 2 Pass 3 Pass 3

Geometric series in one’s “toolbox” for working Never Never Never with bounds and keeping quantities small How to set up proofs to show a sequence is Cauchy

Pass 4 Pass 2 Pass 4

Cauchy sequences can be thought of as “bunching up”

Pass 3 Pass 3 Pass 3

Figure 1. When students recognized the intended content conveyed by Dr. A.

As shown in Figure 1, even strong mathematics students with an experienced and accomplished instructor miss many of the intended points made by the instructor. In fact, none of the students picked up on Dr. A’s intended content during the first pass, which is typically the only chance they get to glean insights from a lecture. A primary reason for this was that the instructor conveyed most of the important points orally, but the students’ focus was what was written on the board. This finding resonates with the description of chalk talk where metacommentary is a prominent feature of lecture. The authors conclude with two important implications. First, as educators we can help students become better note takers, attending to what is verbalized as well to what is written on the board. Second, as lecturers, we can be more attuned to the possibility that important ideas which we convey via metacommentary may not be remembered by students. This suggests the need for heightened awareness of what is and what is not written on the board. Lecture has a long history in undergraduate mathematics teaching and is not without merit. Research,

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however, is increasingly spotlighting how alternative forms of instruction that actively engages students can improve student success, especially in introductory mathematics courses required of STEM-intended students. Informed by this growing body of literature, professional societies are calling on the field to look toward other forms of instruction. For example, the AMATYC, AMS, ASA, MAA, and SIAM came together to produce the Common Vision document, which states the following: We see a general call to move away from the use of traditional lecture as the sole instructional delivery method in undergraduate mathematics courses … Even within the traditional lecture setting, we should seek to more actively engage students than we have in the past. (Saxe & Brady, 2015, p. 19) In the next section we provide a few highlights from the literature on active learning as well as easy to use strategies that can, as Saxe and Brady urge, be used in more traditional lecture settings.

inquire into student thinking and teachers foster equity in their design and facilitation choices). Importantly, this definition does not define inquiry in terms of prescriptive behaviors, but rather lays out four aspirational goals that can be achieved in different ways that fit the style of each instructor. For an overview of how inquiry-based mathematics education has been taken up in many European countries we refer readers to the Platinum Project1 (GomezChaco et al., 2021). While inquiry-based learning is a rather large-scale shift from traditional lecture courses, with the classes often devoting more than half of each class session to small-group work, student presentation of problems at the board, and whole-class discussion, there are also ways to improve student learning outcomes with much more incremental changes. In the next section we highlight several of these easy-to-implement strategies.

Engaging Students During Class Time is Good

Trying to characterize instruction as strictly “lectureoriented” or “active learning” is often an unnecessary division as most of our teaching is not one or the other. For instance, when we look at “lecture-oriented” classes there are lots of examples where instructors frequently ask questions to increase student engagement (e.g., Artemeva & Fox, 2011) and a recent study identified a sizable subgroup of inquiry-based learning instructors whose reported instruction looked almost exactly like the reported instruction of people who had never heard of inquiry-based learning (Vishnubhotla et al., 2022). In fact, there are many student engagement techniques that could be incorporated into a predominantly lecture-based class with only minor alterations. In fact, going back to the Freeman et al. (2014) meta-analysis, they defined “active learning” classrooms as ones that included “vaguely defined ‘cooperative group activities in class,’ in-class worksheets, [or] clickers” (p. 8414)—with as little as 10% of the class time devoted to active learning. This suggests that minor active learning modifications to lecture may still go a long way in terms of improving student engagement and learning. Here we discuss some of these lecture-compatible, student engagement techniques that can easily be fit into an existing class. Exit Cards: Exit cards (aka one-minute papers) are a quick way to gather information at the end of class on how students are thinking about the class material. Exit cards can be asked quite generally, (e.g., what is one thing you learned today and one question) or they can be specific to one of the main ideas from class (e.g., draw a diagram

Across undergraduate Science Technology Engineering and Mathematics (STEM) courses, the educational research literature fairly consistently finds that teaching practices that encourage student engagement during in-class time are beneficial to student learning. Most comprehensively, Freeman et al. (2014) conducted a meta-analysis of 225 research articles that compared outcome variables between classes that incorporated some sort of “active learning” techniques to classes that did not. In their meta-analysis, they found that student exam scores in classes that incorporated active learning techniques were 6% higher and failure rates were 12% lower, on average, than in classes that did not incorporate active learning techniques. Research studies conducted specifically in undergraduate mathematics classes have found increasing student engagement in class, through the use of nonlecture teaching techniques, has a positive impact on learning outcomes, persistence, and success. For instance, across multiple studies Laursen and colleagues have found higher student reports of confidence and learning gains in inquiry-based learning courses than in more traditional lecture-based courses (e.g., Kogan & Laursen, 2014). A retrospective account of two different threads of inquiry as well as a definition of inquiry-based mathematics education is detailed in the overview paper by Laursen and Rasmussen (2019). Laursen and Rasmussen (2019) define inquiry in terms of four aspirational pillars. Two of these pillars emphasize what students do (students engage deeply with coherent and meaningful mathematical tasks and students collaboratively process mathematical ideas) and two emphasize what teachers do (teachers 1686

It Doesn’t Have to Be All or Nothing, Here Are Some Ideas About How to Get Started

1 https://munispace.muni.cz/library/catalog/view/2132/5995 /3467-1/0#preview

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to illustrate Rolle’s theorem). Given their versatile nature, they can be used to support students in reflecting on their own learning or as a source of information for the instructor about how their students are thinking about the material and what could use more attention in either the homework or the next class session.

reflections. Given the real-time data that is collected through the use of clickers and other means, they not only serve as a way to help students stay engaged with the material in real time, they can also serve as a real-time formative assessment, which differs from exams in that these assessments provide information that can be used by teachers to make better-founded decisions about the next steps in instruction.

Think-Pair-Share: Whereas exit cards, at most, require just a few minutes at the end of class, the think-pair-share method builds in more structured and intentional pauses during class time. For this method, you pose a quick question or task to the students and ask them to first reflect on this privately (think), they then turn to a neighbor to discuss (pair), before sharing within a small group or to the whole class (share). This is a highly versatile technique, as there are numerous opportunities during any class in which students could use just a few minutes to consider the material being presented. This includes: completing a computation, thinking about how they would approach a problem, generating a (counter)example, or considering the validity of a conjecture. As discussed by Braun et al. (2018), “giving students time to think about and discuss mathematics mid-lecture encourages their active participation in the class. . . and serves as an effective comprehension check in which students are able to refocus their attention during a lecture” (p. 125). Individual Response Systems (Clickers): There are certainly times when we want to provide students time to think about a question or task, but the “pair” and “share” aspects of the previous method do not match the teacher’s goals or instructional environment (e.g., high-enrollment lecture hall). Using individual response systems (or “clickers”) still provide students time to reflect and engage in class—and still provide the instructor with timely information about the extent to which the students are following along with the material. While some instructors prefer to use some technology-delivered individual response system (e.g., clickers), classroom voting can also be done with a simple show of hands, with colored index cards (red = a, blue = b, etc.), or by having students hold fingers in front of their chests to indicate the option number for which they are voting. The polling questions used can range from quick homework/reading checks at the beginning of class, to checks for understanding or assessments of prior knowledge, to end-of-class NOVEMBER 2023

Guided Notes: We know that, in addition to what they write on the board, most math instructors also provide a lot of commentary and insights that they do not write down—and that are virtually never written down by students (Artemeva & Fox, 2011; Lew et al., 2016). As an alternative to having the students write down for themselves every bit of the lecture, guided notes provide students with “a print out of the notes which contain gaps and they can fill the gaps in as the lecture proceeds” (Iannone & Miller, 2019, p.6). The idea behind the use of guided notes is that it frees up time and mental capacity for students to focus on the larger ideas of the lecture and the rich verbal commentary that they may miss otherwise. As explained by one of the students in a study on the use of guided notes “I think I am probably more engaged in the ones [lecturers] that do give us gappy notes because you can kind of stop writing and actually listen to what the lecturer is saying. Rather than just having to copy down everything for an hour.” (Iannone & Miller, 2019, p.14) A fairly recent census survey of all mathematics departments that offer a graduate degree found both a readiness and willingness to implement more active learning strategies, but also a need for guidance on how to more effectively implement active learning (Rasmussen et al., 2019). In the following section we point readers to a number of resources on ways to actively engage students in class and how to begin shifting department culture so that active learning, while perhaps not yet the norm, is at least not an outlier.

Where to Learn More An excellent comprehensive resource for actively engaging students both inside and outside the classroom is the MAA Instructional Practices (IP) Guide2 (Abell et al., 2018). The IP guide is a “how to” guide, full of concrete examples. The edited volume contains three foundational types of practices: classroom practices, assessment 2

https://www.maa.org/programs-and-communities/curriculum% 20resources/instructional-practices-guide

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practices, and course design practices. The book is intended for all mathematics instructors, from those with years of experience to those just starting out in the profession. The classroom practices section contains a wealth of research-based instructional strategies that range from quick and easy to implement to more elaborate strategies that require greater preparation. The assessment practices section contains guiding principles for both summative and formative assessments. Summative assessments include things like quizzes and exams where the purpose is to evaluate student proficiency. Formative assessments, on the other hand, are intended to provide evidence about student progress to help instructors determine appropriate and needed next steps. In the previous section we gave the example of “exit cards” as one example of a formative assessment technique. The IP guide provides other examples as well as effective methods for designing summative assessments. Lastly, the course design practices section discusses and provides engaging vignettes focused on the plans and choices instructors make prior to teaching and what they do after teaching to modify and revise for the future. For those instructors that are interested in using classroom voting to actively engage students, one can now find many tried and true good questions for a variety of content areas. One good source of information is the edited volume, Teaching Mathematics with Classroom Voting: With and Without Clickers3 by Cline and Zullo (2011). Another excellent source of inspiration for good classroom voting questions can be found at this website.4 Our goal here is not to provide a comprehensive list of resources, but rather to offer interested instructors a starting point for locating resources. Another useful resource is the Academy of Inquiry Based Learning,5 which provides a variety of professional development and learning community opportunities for faculty to learn more about inquiry-based mathematics education. A good place to start at their website is the IBL Blog Articles, which include articles on getting started, learning about what IBL is, nuts-and-bolts topics, classroom organization and management, productive failure and growth mindsets, and more. In addition to classroom-level resources, we also want to point readers to department-level resources that can help shift department culture to be more supportive of engaged student learning. While transforming practice can and often does happen one instructor at a time, research has pointed to the power and sustainability of change 3 https://www.maa.org/press/ebooks/teaching-mathematics-with -classroom-voting-with-and-without-clickers 4 http://mathquest.carroll.edu/resources.html 5 http://www.inquirybasedlearning.org/

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when the department, rather than the individual, is the focus of change. The recently edited book, Transformational Change Efforts: Student Engagement in Mathematics through an Institutional Network for Active Learning6 (Smith et al., 2021a), reports on a national study of departmental change and change mechanisms that allow math departments to incorporate and sustain active learning in Precalculus to Calculus 2. Another excellent resource is a triple special issue of PRIMUS7 that features 26 mathematics departments in the process of transforming their introductory mathematics courses typically required for all STEM majors (Smith et al., 2021b). We hope that these stories of on-the-ground change efforts will inspire readers to develop their own change efforts, ones that fit their departmental and institutional context. Thinking about making a change can be overwhelming, whether it be at the individual instructor level or the department level. For those interested in making use of some different active learning strategies, whether this is in a primarily lecture class or not, we hope the research insights and resources provided here, while limited, make your life easier. For those that are already familiar with some of the instructional approaches highlighted in this article, we hope that the numerous references are useful for you to dive into and expand your repertoire. For those interested in better understanding the process of departmental change, especially change that is aimed at infusing more active learning into the introductory mathematics courses, we hope that the department change references provide you with inspiration and lessons learned. References

[1] M. L. Abell, L. Braddy, D. Ensley, L. Ludwig, and H. Soto, Instructional Practices Guide, The Mathematical Association of America, Inc., 2018. [2] N. Artemeva and J. Fox, The writing’s on the board: The global and the local in teaching undergraduate mathematics through chalk talk, Written Communication 28 (2011), no. 4, 345–379, DOI 10.1177/0741088311419630. [3] B. Braun, P. Bremser, A. M. Duval, E. Lockwood, and D. White, What does active learning mean for mathematicians?, Notices Amer. Math. Soc. 64 (2018), no. 2, 124–129. [4] W. Byers, How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics, Princeton University Press, 2007. [5] K. S. Cline and H. Zullo (eds.), Teaching mathematics with classroom voting: With and without clickers, no. 79, Mathematical Association of America, 2011.

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[6] S. Freeman, S. L. Eddy, M. McDonough, M. K. Smith, N. Okoroafor, H. Jordt, and M. P. Wenderoth, Active learning increases student performance in science, engineering, and mathematics, Proceedings of the National Academy of Sciences 111 (2014), no. 23, 8410–8415. [7] I. M. Gomez-Chac ´ on, ´ R. Hochmuth, B. Jaworski, J. Rebenda, J. Ruge, and S. Thomas, Inquiry in university mathematics teaching and learning: The PLATINUM project, Masaryk University Press, Czech Republic, 2021, https://platinum.uia.no/download/. [8] P. Iannone and D. Miller, Guided notes for university mathematics and their impact on students’ note-taking behaviour, Educational Studies in Mathematics 101 (2019), 387–404. [9] M. Kogan and S. L. Laursen, Assessing long-term effects of inquiry-based learning: A case study from college mathematics, Innovative Higher Education 39 (2014), no. 3, 183–199. [10] S. Laursen and C. Rasmussen, I on the prize: Inquiry approaches in undergraduate mathematics education, International Journal of Research in Undergraduate Mathematics Education 5 (2019), no. 1, 129–146. [11] K. Lew, T. Fukawa-Connelly, P. Meija-Ramos, and K. Weber, Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey, Journal for Research in Mathematics Education 47 (2016), no. 2, 162–198. [12] K. Melhuish, T. Fukawa-Connelly, P. C. Dawkins, C. Woods, and K. Weber, Collegiate mathematics teaching in proof-based courses: What we now know and what we have yet to learn, The Journal of Mathematical Behavior 67 (2022), 100986. [13] C. Rasmussen, N. Apkarian, H. Ellis, E. Johnson, S. Larsen, D. Bressoud, and the Progress through Calculus team, Characteristics of Precalculus through Calculus 2 programs: Insights from a national census survey, Journal for Research in Mathematics Education 50 (2019), no. 1, 24–37. [14] C. Rasmussen and M. Wawro, Post-calculus research in undergraduate mathematics education, in J. Cai (ed.), Compendium for research in mathematics education, National Council of Teachers of Mathematics, 551–581, 2017.

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[15] K. Saxe and L. Braddy, A common vision for undergraduate mathematical sciences programs in 2025, Mathematical Association of America, 2015. [16] W. Smith, C. Rasmussen, and R. Tubbs (Guest Editors), Infusing active learning in Precalculus and Calculus, PRIMUS 31 (2021b), no. 3-5, 239–657. [17] W. M. Smith, M. Voigt, A. Ström, D. C. Webb, and W. G. Martin (eds.), Transformational change efforts: Student engagement in mathematics through an institutional network for active learning, vol. 138, American Mathematical Society, 2021a. [18] N. M. Speer, J. P. Smith III, and A. Horvath, Collegiate mathematics teaching: An unexamined practice, Journal of Mathematical Behavior 29 (2010), no. 2, 99–114. [19] M. Vishnubhotla, A. Chowdhury, N. Apkarian, E. Johnson, M. Dancy, C. Henderson, . . . and M. Stains, “I use IBL in this course” may say more about an instructor’s beliefs than about their teaching, International Journal of Research in Undergraduate Mathematics Education (2022), 1–20. [20] K. Weber and T. Fukawa-Connelly, What mathematicians learn from attending other mathematicians’ lectures, Educational Studies in Mathematics 112 (2023), no. 1, 123–139.

Chris Rasmussen

Estrella Johnson

Credits

Photo of Chris Rasmussen is courtesy of Chris Rasmussen. Photo of Estrella Johnson is courtesy of Estrella Johnson.

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BOOK REVIEW

Concepts at the Heart of Mathematics— Through the Centuries Reviewed by Katalin Bimb´o The Story of Proof Logic and the History of Mathematics

By John Stillwell. Princeton University Press, 2022, 456 pp. The preliminary title of this book was How Mathematics Works—as we learn from the preface. The author views the current title as less ambitious. It is really hard to give a concise and descriptive title to this very unique book, and the title of this review does not describe the content precisely either. Proofs are essential to mathematics. However, what was accepted as a proof changed through the history of mathematics. M. Kline in [5] mentions examples when famous mathematicians (e.g., J. J. Sylvester) proved “theorems” from false assumptions. For the sake of this book, Stillwell considers a proof to be what is presented in a paper or a book on mathematics, which is labelled as “proof” and is a correct demonstration. For example, Newton’s method of inversion of a power series was a heuristic argument at the time with no rigorous justification. I would guess Stillwell’s view on Katalin Bimb´o is a professor of philosophy at the University of Alberta. Her email address is [email protected]. Communicated by Notices Book Review Editor Emily Olson. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2793

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proofs would appeal to many readers, because it strikes a balance between formalized proofs (that logicians would consider to be rigorous proofs) and hand-waving arguments (that might be sufficient to convince a layperson). Before going into some details about the content, it might be useful to clarify what this book is not about. The research area of proof theory deals with formalized proofs that may be couched in one or another proof system in one or another logic. A sample problem from this area would be whether the cut rule is admissible in S. C. Kleene’s sequent calculus formulation of first-order intuitionistic logic. The book is not about proof theory, although there is a mention of a formal system reminiscent of K. Schütte’s sequent calculus for first-order classical logic. Stillwell’s book is not a history of mathematics; historical details are complementary to the mathematical content. It should be noted that anybody who would like to follow a strand of history touched upon in the text will find a slew of references in the bibliography. The title of the book refers to logic too, and a branch of mathematical logic deals with formalized mathematical theories. D. Hilbert called investigations of certain properties of formalized theories (especially, of consistency and decidability) metamathematics. Stillwell gives a synopsis of Gödel’s incompleteness theorems, however, the study of formalized mathematical theories using the tools of symbolic logic (or the study of systems of formal logic) is not the main topic of this book. To outline what the book is about, we can say that it describes the development of the core ideas of mathematics and their connections. Stillwell pays special attention to when a new proof method appears, and whether

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Book Review it emerges due to conceptual pressure in a theory, or it brings about new developments in a theory. The thread of the story runs through examples, though not all bits and pieces of proofs are labelled and categorized. It starts with proofs that are needed to deal with infinity and ends with proofs that secure the foundations of mathematics and reveal the comparative strength of famous theorems. What is in this book? A very simple view of mathematics is to say that mathematics is a study of numbers and shapes. Of course, in the 21st century, even a high-school student could give examples of mathematical statements or formulas that are neither about numbers nor about shapes such as the general form of the derivative of a polynomial function. But it is reasonable to look back at the work of ancient Greek thinkers and see that they elevated mathematics from practical know-how to a discipline by creating a theory of geometry and by proving theorems about numbers. Pythagoras’s theorem is a natural starting point for this book; it allows for the introduction of rational and irrational numbers, Pythagorean triples, and Euclid’s algorithm to find the greatest common divisor (gcd). Stillwell locates the motivation behind the invention of (deductive) proofs by the Greeks in the “fear of infinity” (or put less dramatically, in the need to justify statements about infinity). The ten self-evident statements of Euclid’s Elements are often translated as “postulates” and “common notions.” These axioms deal with different aspects of the subject, and they are listed, which is helpful for readers who wish to follow the quite informal proofs of Euclid’s Propositions 4 and 5. Seeing the axioms makes it easier to understand why mathematicians tried to deduce Postulate 5 (P5) from the rest of the axioms. A presentation of Euclid’s proof of infinitely many prime numbers is a good excuse to introduce induction as a proof technique and to mention concepts such as perfect numbers, prime factorization, Mersenne primes, and geometric series. The next chapter jumps to Hilbert’s axiomatization of geometry from 1899. Stillwell gives in parallel the geometric axioms and an axiomatization of complete ordered fields, thereby, describing two categorical formalizations of the reals. Hilbert’s axiomatization of geometry goes beyond Euclid’s geometry not only by filling a couple of gaps (e.g., the one found by M. Pasch), but by expanding the range of points, for instance, in a plane, from the set of constructible points to all points ⟨𝑥, 𝑦⟩ (𝑥, 𝑦 ∈ ℝ). Perspectival drawings and paintings led to the study of projections and to theorems such as those of Pappus and Desargues— well before Hilbert’s axiom system. Stillwell not only states the latter two theorems, but he also lists an axiomatization of projective planes. Archimedes gave surprisingly accurate approximations (in terms of fractions) for the value of 𝜋 in the 3rd

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century BCE. But algebra took off in earnest in the 9th century CE, then, it was propelled by the quest to solve cubic, quartic, and quintic equations. Notation for polynomials was nonexistent, and imaginary numbers were not even talked about at the time of H. Cardano, in the 16th century. Stillwell quotes Newton to illustrate a change within a 100 years or so, namely, that calculation with variables had become an accepted method to obtain results. He goes on to state and prove the factor theorem. R. Dedekind’s dimension theorem applied to a numerical formulation of the ancient Greek puzzle called “doubling the cube” gives an impossibility proof. The interplay between algebra and geometry is further illustrated by projective geometry. And another impossibility proof shows that there are no octonion projective spaces, which follows from the nonassociativity of the multiplication operation on octonions. Algebraic geometry is a continuation of the development of the algebraic methods in the 16th and 17th centuries, including the introduction of coordinate systems. Stillwell starts with the conic sections and their equations, and he quickly introduces tangents, singularities, curves given by polynomials, and nonalgebraic curves. He illustrates that claims occasionally have a peculiar fate: Newton stated what is called B´ezout’s theorem, which was proved much later by permitting complex coordinates, counting multiplicities of intersections, and placing the curves in a projective space. Thales’s theorem is also proved, but the proof is in the real vector space, which was introduced by H. Grassmann in the 19th century. The content of Chapter 6 will likely be familiar to most readers, as it discusses the origins of calculus. But some might not know that the divergence of the harmonic series was proved by N. Oresme in the middle of the 14th century, or that several mathematicians gave infinite series and infinite products to approximate the value of 𝜋, which was proved irrational in 1761 by J. H. Lambert. Stillwell manages to explain many concepts and their connections from the binomial coefficient through the calculation of slope, area, and volume to infinitesimals. The latter require careful handling, and it was proved by A. Robinson (in the 1960s) that infinitesimals can be added consistently to the reals. Chapter 7 begins with Euclid’s gcd algorithm, modular arithmetic, and the Pythagorean triples, and then the text moves on to rational points on a circle and parametric equations. Fermat’s little theorem and Fermat’s last theorem for the fourth power are proved. The latter can be translated into a proof about polynomials, which in turn leads to the parameterization of curves, elliptic integrals, and elliptic curves. Continuing the theme of divisibility into complex numbers and algebraic integers, the notion of primes is extended. E. Kummer’s determination to achieve unique prime factorization led to the notion of ideals. The latter are sets of numbers, and they may be

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Book Review seen to be related to R. Dedekind’s cuts (of ℚ) that define real numbers. Ideals (i.e., cotheories) became very useful objects in lattice theory [1]. The fundamental theorem of algebra and its proof have a thought-provoking saga. Stillwell devotes a short chapter to this theorem alone; he gives several versions of the theorem, explains that early proof attempts (e.g., by C. Gauss) contained gaps and that filling the gaps produced a definition of real numbers. Having Dedekind’s cuts at hand, we get quick proofs of the least upper bound theorem and the intermediate value theorem. The term “non-Euclidean geometry” usually refers to geometries that emerged once attempts to deduce P5 from Euclid’s other postulates failed. Stillwell has already made clear that projective geometry, at least in drawing manuals and scattered theorems, preceded the work of J. Bolyai and N. Lobachevskiˇı. Chapter 9 is devoted to non-Euclidean geometries starting with a treatment of the geometry of the sphere. This is an example where bits and pieces of a theory (non-Euclidean geometry) were worked out (due to the practical needs of astronomy and sailing) before a whole axiomatic system was formulated. The sphere has a constant positive curvature, while the Euclidean plane has curvature zero. In order to obtain an object (the pseudosphere) with constant negative curvature, transcendental curves, namely, the catenary and the tractrix, are introduced.

will reappear later in the book as theorems provable in certain subsystems of second-order Peano arithmetic. The next three chapters belong together: first, basic set theory is presented, then the axiomatic approach is revisited via an axiomatization of Zermelo–Fraenkel set theory (ZF). Lastly, the axiom of choice (AC) is dealt with at some length. AC is often used without mention in mathematics, and here some of the set-theoretic equivalents such as Zorn’s lemma, the well-ordering principle, and some of its uses such as the Bolzano–Weierstraß theorem, the existence of non-Lebesgue measurable sets, and the Hausdorff–Banach–Tarski “paradox” are described. Finally, G. Cantor’s continuum hypothesis is mentioned, which is independent of ZFC (= ZF + AC) as was proved by K. Gödel and by P. Cohen. The last two chapters quickly introduce predicate logic and computability, and the book is concluded with the incompleteness of arithmetic and set theory. Predicate logic proofs are certain trees here; a better-known proof system that uses trees is the method of analytic tableaux for which R. M. Smullyan’s [6] is a classic source. Computation has many models from recursive functions and combinatory logic to register machines and D. Scott’s model. Stillwell presents a version of Turing machines. A detailed proof of an incompleteness theorem is quite lengthy, as for example, G. Boolos and R. Jeffrey’s [2] illustrates. Stillwell sketches the proof together with Hilbert’s formalism and Brouwer’s intuitionism, which characterized acceptable mathematical proofs differently. Then he turns to reverse mathematics, the goal of which is to delineate sensible fragments of second-order arithmetic and to place theorems from analysis into these fragments according to their proof strength.

Figure 1. Trefoil and knot 10123 (with 10 crossings).

The next chapter is on topology, which may bring very different images to one’s mind. Knots (from knot theory, see [4]) have an accessible side to them such as the colorful diagrams in Fig. 1. But they belong to a modern field of mathematics that illustrates how various areas are tied together (pun intended!). Stillwell includes the Reidemeister moves and some knot invariants, and then moves onto graphs. Chapter 11 turns back to the completeness of the real line and how this facilitated the development of the notions of limit and continuity. Stillwell defines continuity and uniform continuity, convergence and uniform convergence, and gives proofs for several theorems such as the Heine–Borel, the Riemann integrability of continuous functions, and the extreme value theorems. Some of these

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X Figure 2. A partial incompletable and a proper complete 4-coloring of a map.

In sum, this book contains multiple proofs, none of which comes from a formalized theory (in the sense of Hilbert). However, Stillwell underscores some components in proofs with names for them. The most prominent steps are the method of exhaustion (i.e., reasoning by cases) and (weak mathematical) induction. A method that is often used, but is not labeled here is reductio (i.e., proof by contradiction). Some steps that are specific to mathematical reasoning include renaming of variables (which is

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Book Review something else than the renaming of variables in 𝜆calculus) and translating between areas (e.g., rephrasing Fermat’s last theorem in terms of functions). Furthermore, some proofs seem to use analogical reasoning (e.g., the usual proof of Goodstein’s theorem). And the prolific use of diagrams suggests that pictures are vital to some proofs. A cautionary tale is A. Kempe’s “proof” of the four-color theorem (4CT), where both the “proof” and the counterexamples were diagrammatic. Figure 2 shows one of the smallest counterexamples to Kempe’s color-swapping algorithm derived from Soifer’s graph. Arguably, the story of proofs starts around the time of Euclid. His axioms are self-evident, hence the deductive method provides secure foundations for geometry. As mathematics moved forward, occasionally, like a sputtering engine producing smoke rather than torque (or more prosaically, furnishing false claims supported by faulty justifications), not only new concepts were required but Figure 3. Kurt Gödel. the language of mathematics and the proof techniques had to be clarified. The thrust to formalize mathematical theories wilted in the mid-20th century, partly because of Gödel’s theorems in [3]. However, it seems that there have been new developments in the overarching story of proofs in the last 50 years or so. The proof of the 4CT is merely one of the examples that point to the use of computers. Stillwell allots a short section to the use of computers in checking proofs. Perhaps, an exploration of proof assistants, theorem provers, and proof checkers, which might lead to a new wave of rigorization in mathematics, should be the topic of an entire book. A book for many readers. The book appears to have been produced with remarkable care. Although it is not teeming with references to online sources, there are some url’s mentioned for valuable resources such as an online version of O. Byrne’s colorful pictorial rendering of the first six books of Euclid’s Elements and a knot atlas. This book will be a worthwhile read for anybody with some tertiary education in mathematics who would like to see how some core ideas developed and what was the driving force behind their evolution. A reader who does not have the time or patience to follow every detail in a proof can skip over it without losing an appreciation of the concepts involved. For instructors, this book could be a handy source to spice up a course with a bit of history or an

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outlook on the connections between disparate fields. Undoubtedly, anyone whose research area is touched upon in the book might become displeased, because it does not cover any of the topics at the level of detail that a research monograph or a research paper would. (I can attest to such feelings while reading the last chapters.) I imagine that experts will be able to overcome such perceptions and they will enjoy and profit from a more encompassing view of a significant portion of mathematics. References

[1] Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR0227053 [2] George S. Boolos and Richard C. Jeffrey, Computability and logic, 3rd ed., Cambridge University Press, Cambridge, 1989. MR1025336 [3] Kurt Gödel. Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I. In Solomon Feferman, editor, Collected Works, volume I, pages 144– 195. Oxford University Press, New York, NY, 1986. MR1549910 (MR0831941). [4] Louis H. Kauffman, Formal knot theory, Mathematical Notes, vol. 30, Princeton University Press, Princeton, NJ, 1983. MR712133 [5] Morris Kline, Mathematics: The loss of certainty, Oxford University Press, New York, 1980. MR584068 [6] Raymond M. Smullyan, First-order logic, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 43, SpringerVerlag New York, Inc., New York, 1968. MR0243994

Katalin Bimbo´ Credits

Book cover is courtesy of Princeton University Press. Figures 1 and 2 and author photo are courtesy of Katalin Bimbo. ´ Figure 3 is by Marcel Natkin and courtesy of St´ephane and Laurent Natkin.

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BOOKSHELF New and Noteworthy Titles on our Bookshelf November 2023

book also contains colorful figures of many symmetries and dynamics to aid in one’s understanding.

The Price We Pay for Symmetry

By Scott Crass. Chapman and Hall/CRC, 2022, 190 pp. The fundamental theorem of algebra states that a polynomial of degree 𝑛 with coefficients in ℂ will have 𝑛 roots in ℂ, counting multiplicity. However, it is often the case that knowing these roots exist gives no insight into how to locate them. The book makes the crucial observation that the roots are an unordered set and one can exploit the symmetry that arises from this fact. For instance, the solution set of the polynomial 𝑥2 − 14𝑥 + 40 could be thought of as either of the ordered pairs (4, 10) or (10, 4). Thus the line 𝑦 = 𝑥 provides symmetry in this small case. Higher dimensional cases have symmetry as well. In Polynomials, Dynamics, and Choice, the author exploits symmetry to solve higher order polynomials. In Part I, he compares and contrasts two algorithms based on algebra, symmetry, geometric structures, and dynamical processes to solve certain polynomials up to degree 6. Though the text doesn’t assume a reader has seen group theory, a course in abstract algebra would be helpful prior to working through this book. In Part II, Crass explains the effect of symmetry and choice in our everyday decision making. This book would make a good independent study text for an advanced undergraduate or could be used in an introduction to geometry or dynamics graduate course. Alternatively, a group of graduate students could work through the book as part of a reading group, seminar, or independent study, especially if they use the works cited to recreate the algorithms. Be warned there are no exercises included; however, Crass presents plenty of examples. The This Bookshelf was prepared by Notices Associate Editor Emily J. Olson. Appearance of a book in the Notices Bookshelf does not represent an endorsement by the Notices or by the AMS. Suggestions for the Bookshelf can be sent to [email protected]. DOI: https://doi.org/10.1090/noti2794

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Mathematical Tools for Real-World Applications A Gentle Introduction for Students and Practitioners

By Alexandr Draganov. MIT Press, 2022, 306 pp.

Courtesy of MIT Press.

Reproduced by permission of Taylor & Francis Group.

Polynomials, Dynamics, and Choice

Many mathematics students will end up in industry, but math courses do not always contain the techniques professionals need to use. Instead of being required to regurgitate an algorithm or apply the correct equation, students will need to analyze a situation and have an idea of how to evaluate models and improve them. The real world does not always require “the right answers,” and we need our students to be able to adapt and think critically about complex phenomena. The first six sections of Mathematical Tools for Real-World Applications illustrate the techniques of checking units, limiting cases, symmetry, scaling, order of magnitude estimates, and successive approximations. The last two sections present two real-world problems the author offers as an extra challenge. Throughout the book, problems are solved in more than one way to illustrate how various methods can be used to achieve or check a solution. Examples range from typical, like the intersection of a circle and a line, to unusual, such as designing satellite antenna. Every section includes key points, a summary, and a long list of exercises. I imagine one could use this text as part of an undergraduate mathematical modeling class or a course for future engineers. A mathematics department that offers a “getting to know your major” course or even an advanced high school student preparing for an engineering degree could find insight here. I think this text strikes a balance between teaching technical knowledge and building critical thinking skills that students desperately need upon graduation. I found the material in this book to be interesting and insightful; I think you and your students will, too.

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VOLUME 70, NUMBER 10

AMS BOOKSHELF The AMS Book Program serves the mathematical community by publishing books that further mathematical research, awareness, education, and the profession while generating resources that support other Society programs and activities. As a professional society of mathematicians and one of the world’s leading publishers of mathematical literature, we publish books that meet the highest standards for their content and production. Visit bookstore.ams.org to explore the entire collection of AMS titles.

Mathematics 2023: Your Daily Epsilon of Math By Rebecca Rapoport and Dean Chung. MBK/144, 2022, 14 pp. AMS Page a Day Calendar By Evelyn Lamb. MBK/128, 2019, 372 pp. With the development of new tools to produce and print high-quality images, wall calendars have become even more visually appealing than, say, 20 years ago. In addition to their direct functions of reminding you about a dentist’s appointment or a visit from your in-laws, many calendars serve educational or entertainment purposes. A well-thought-of and nice-looking calendar has become a popular gift for family members, friends, and coworkers. Mathematicians could not be left behind in this trend and a number of mathematical institutes, societies, and interest groups have joined the calendar bandwagon. In particular, the AMS recently published two mathematically inspired calendars. The first calendar was prepared and designed by Rebecca Rapoport and Dean Chung. The authors selected mathematical problems (365 of them) and put them in date boxes of the calendar, one problem per day. However, there is a special twist, which is quite unique and distinguishes this calendar from others. Before telling you what it is, here are two daily problems: 3

The problem for July 27: Find 9 2 The problem for November 14: How many different rectangles can be formed from 1400 identical squares? This AMS Bookshelf is by AMS Publisher Sergei Gelfand. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2792

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Try to solve them and then, looking at the answers and using Sherlock Holmes type thinking, you should be able to guess what the special twist is. Got it? Right, the twist is that the answer to the problem for each day is the date, so the answer to the August 1 problem is 1, and the January 29 problem yields 29. The problems in the calendar vary significantly in difficulty and required prerequisites, from elementary problems that can be successfully tackled by a middle school student to quite difficult ones that can pose a challenge to a college math major or even a math graduate student. And it goes without saying that the calendar is beautifully designed and produced. Intricate colorful mathematically inspired graphics strengthen the main message of the calendar: “Math is beautiful”. The other calendar published by the AMS is called the AMS Page a Day Calendar, by Evelyn Lamb. While also being a mathematically motivated calendar, it differs in many respects from the other. The Page a Day calendar is a desk calendar, and each of its 366 pages is marked by the date (day and month). However, contrary to most calendars, the page does not mention a particular day of the week, making the calendar not tied to a particular year, so that if you didn’t use it in one year, you may start using it in the next year and the calendar will still serve you well! Each page of the calendar contains a small morsel about mathematics, such as a fun fact, a piece of history, a piece of art made using mathematics, a mathematical puzzle, and more. A morsel is often related to the particular date where it appears. For example, the February 7 (i.e., 2.7) page talks about the number 𝑒 ≈ 2.7. The text on the March 16 page is about 𝐾3 surfaces so named for three mathematicians, Kummer, Kähler, and Kodaira (who was born on this date in 1915). Almost every daily piece in the calendar is the tip of a bigger story, and for the reader who wants to learn more, the page offers a reference for accessible further reading. Overall, if you like mathematics, each day with this calendar will bring you a piece of fun, and sometimes a challenge. By the time the year is over you may forget some of the stories you saw earlier and the design of the calendar allows you to start using it again, assuming you didn’t tear the pages off each day. Otherwise, just get a new copy.

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Applications Open for

AMS CONGRESSIONAL FELLOWSHIP 2024–2025

Apply your mathematics knowledge toward solutions to societal problems. The American Mathematical Society will sponsor a Congressional Fellow from September 2024 through August 2025. The Fellow will spend the year working on the staff of either a member of Congress or a congressional committee, working in legislative and policy areas requiring scientific and technical input. The Fellow brings his/her/their technical background and external perspective to the decision-making process in Congress. Prospective Fellows must be cognizant of and demonstrate sensitivity toward political and social issues and have a strong interest in applying personal knowledge toward solutions to societal problems.

“Every day on the Hill is new and exciting! Whether I am writing press releases, synthesizing complicated ideas into one-pagers, or meeting influential individuals, I learn something new every day. “ —Duncan Wright, AMS Congressional Fellow 2022–2023

Now in its 19th year, the AMS Congressional Fellowship provides a unique public policy learning experience, and demonstrates the value of science– government interaction. The program includes an orientation on congressional and executive branch operations, and a year-long seminar series on issues involving science, technology, and public policy. Applicants must have a PhD or an equivalent doctoral-level degree in the mathematical sciences by the application deadline (February 1, 2024). Applicants must be US citizens. Federal employees are not eligible. The Fellowship stipend is US$100,479 for the Fellowship period, with additional allowances for relocation and professional travel, as well as a contribution toward health insurance. Applicants must submit a statement expressing interest and qualifications for the AMS Congressional Fellowship as well as a current curriculum vitae. Candidates should have three letters of recommendation sent to the AMS by the February 1, 2024 deadline.

For more information and to apply, please go to www.ams.org/ams-congressional-fellowship. Deadline for receipt of applications: February 1, 2024

2023

Learn more at the JMM 2024 session on DC-based Policy and Communications Opportunities to be held Friday, January 5, 2024 at 4:30 p.m.

Walter Rudin Meets Elias M. Stein Odysseas Bakas, Valentina Ciccone, and James Wright Walter Rudin and Elias M. Stein were giants in the world of mathematics. They were loved and admired by students, researchers, teachers, and academics, both young and old. They touched many of us through their inspiring books at the undergraduate and postgraduate level. Although they were leading researchers in both harmonic analysis and several complex variables, we are not aware whether they interacted and discussed mathematics. In this article, Rudin and Stein meet mathematically through a reformulation of the beautiful theory of Fourier series with gaps that Rudin developed in the 1950s as an equivalent Fourier restriction problem from the 1970s, a problem Stein proposed and which remains a fundamental, central problem in Euclidean harmonic analysis today. Walter Rudin was born in Vienna on May 2, 1921 and emigrated to the US in 1945, completing his PhD at Duke University in 1949. While a C.L.E. Moore Instructor at MIT in the early 1950s, Walter was asked to teach a real analysis course but he could not find a textbook that he liked so he decided to write Principles of Mathematical Analysis which despite its age, has remained the paragon of high quality. After a stint of teaching at the University of Rochester, he took up a position at the University of Odysseas Bakas is a postdoc fellow at the Basque Center for Applied Mathematics - BCAM. His email address is [email protected]. Valentina Ciccone is a PhD student at the University of Bonn, currently visiting the Basque Center for Applied Mathematics - BCAM. Her email address is [email protected]. James Wright is a professor of mathematics at the University of Edinburgh. His email address is [email protected]. Communicated by Notices Associate Editor Daniela De Silva. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2804

NOVEMBER 2023

Figure 1. At a complex analysis conference in Madison, WI, March 16–19, 2006. E.M. Stein (third from left) and W. Rudin (far right).

Wisconsin–Madison in 1959 where he remained until his retirement as Vilas Professor in 1991. He died at his home in Madison on 20 May, 2010. Elias M. Stein (known to friends and colleagues as Eli) was born in Antwerp on January 13, 1931 and emigrated with his family to the US in 1941, settling in New York where Eli attended high school. He went to the University of Chicago, received his PhD in 1955, and then went to MIT as a C.L.E. Moore Instructor before Antoni Zygmund told Eli “it’s time to return to Chicago.” In 1963, Stein moved to Princeton University as a full professor where he remained until he died on December 23, 2018. Between 2003 and 2011, Eli (together with Shakarchi) expanded the presentation of Walter’s Principles and published a series of four books aimed at advanced undergraduates. This series is quickly becoming an important part of any young analyst’s education. However the majority of books written by Rudin and Stein are postgraduate textbooks and research monographs (too many to list here),

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mainly in the areas of harmonic analysis and several complex variables where both men were central figures. In this article, these two luminaries meet in the world of mathematical analysis. We look back at some important work Rudin did in the 1950s and recast it in terms of a farreaching problem from the 1970s that Stein gave us.

1950s: A Golden Age for Fourier Analysis At a 1946 conference in Princeton,1 Zygmund gave a scathing review of the post-world war state of harmonic/Fourier analysis, describing the area as fettered with unsolved problems with no guiding theme or programme. This all changed in the 1950s in two profound ways. First, the 1950s represented a convergence of the point of view that the most appropriate setting for Fourier analysis is furnished by the class of locally compact abelian groups. This abstraction is not done for the sake of mere generalisation. It not only gives conceptual clarification to classical problems, it also leads to the introduction of new, interesting analytical problems. A beautiful example is the theory Rudin developed in his paper Trigonometric series with gaps [Rud60] which we will turn to momentarily. The second profound change from the 1950s is the advent of the real variable method which emerged from the seminal paper of Calderon ´ and Zygmund [CZ52], freeing us from the complex method which tied us to one dimension in the study of Fourier series and the Fourier transform. This led Stein to propose a series of fundamental problems addressing basic properties of Fourier series and the Fourier transform in higher dimensions. These problems are interconnected and the core conjectures underpinning each problem still remain unsolved today despite the heroic efforts of many eminent mathematicians. One of these problems is the Fourier restriction problem which Stein introduced in the mid to late 1960s and which we will discuss in more detail below.

Trigonometric Series with Gaps This paper [Rud60] of Rudin introduced several kinds of sparse sets of integers with interesting properties. In the 1920s Sidon showed that a continuous function on the circle 𝕋 = ℝ/ℤ, with a lacunary2 Fourier series 𝑓 ∼ 𝑘 ∑ 𝑐𝑘 𝑒2𝜋𝑖2 𝜃 , automatically has an absolutely convergent Fourier series; i.e., ∑𝑘 |𝑐𝑘 | < ∞. Giving us a glimpse of how he thinks, Rudin took this isolated result and realised that there is a rich theory of sets Λ ⊂ ℤ with the property that 𝐶Λ (𝕋) ⊂ 𝐴(𝕋). Here 𝐴(𝕋) is the space of absolutely convergent Fourier series and 𝐶Λ (𝕋) is the closed subspace of continuous functions on the circle 𝕋 which are Fourier 1

Problems of mathematics, Princeton University bicentennial conferences, series 2, conference 2, Princeton, New Jersey, 1947. 2More generally, the sequence {2𝑘 } can be replaced by any sequence of positive integers {𝑏𝑘 } satisfying inf𝑘 𝑏𝑘+1 /𝑏𝑘 > 1.

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ˆ supported in Λ; that is, the Fourier coefficients 𝑓(𝑛) = 0 for all 𝑛 ∉ Λ. Rudin called such spectral sets Sidon sets and observed they have interesting arithmetic properties. He first developed the theory on the circle 𝕋 and then extended it to any compact, abelian group 𝐺, detailing a programme to characterise Sidon sets in terms of their arithmetic properties. He intimated that the key to unlock this arithmetic characterisation is the following improving bound: for 𝐹 = exp(𝐿2 ), there is a constant 𝐶 > 0 such that ‖𝑓‖𝐹 ≤ 𝐶‖𝑓‖𝐿2 for all 𝑓 ∈ 𝐶Λ .

(F)

The function space exp(𝐿2 ) lies near 𝐿∞ ; in fact, 𝐿∞ ⊂ exp(𝐿2 ) ⊂ 𝐿𝑝 for all finite 𝑝 < ∞. In the 1930s, Zygmund established (𝐹) with 𝐹 = exp(𝐿2 ) for lacunary sequences Λ and Rudin extended this to any Sidon set on any compact abelian group 𝐺. Pisier established the reverse implication, showing that (𝐹) with 𝐹 = exp(𝐿2 ) characterises when Λ is a Sidon set and he did this on any compact abelian group; see [Pis78a, Pis78b]. This led Pisier to his arithmetic characterisation of Sidon sets, the definitive result in the theory of Sidon sets; see [Pis81]. There are many sparse families of spectral sets Λ which are defined by or characterised by (𝐹) for some endpoint function space 𝐹 near 𝐿∞ . In his gaps paper, Rudin introduced and developed the theory of Λ(𝑝) sets which are defined by (𝐹) for 𝐹 = 𝐿𝑝 when 𝑝 > 2. Rudin conjectured that the squares Λ = {𝑛2 } is a Λ(𝑝) set for all 𝑝 < 4 but this seems difficult and remains unsolved (see [Bou89b]). A deep result of Bourgain [Bou89a] established that for every 𝑝 > 2, there is a Λ(𝑝) set which is not a Λ(𝑝 + 𝜖) set for any 𝜖 > 0. In a different paper [Rud57] from the 1950s, Rudin introduced Paley sets Λ ⊂ ℤ and showed that the bound (𝐹) holds for 𝐹 = 𝐵𝑀𝑂(𝕋) if and only if sup𝐼∈𝒟 #[Λ ∩ 𝐼] < ∞ where 𝒟 is the set of dyadic intervals {±[2𝑘 , 2𝑘+1 ] ∶ 𝑘 ∈ ℕ}; in other words, Λ is a finite union of lacunary sequences. Similar to exp(𝐿2 ), the function space 𝐵𝑀𝑂 of bounded mean oscillation lies near 𝐿∞ , again 𝐿∞ ⊂ 𝐵𝑀𝑂 ⊂ 𝐿𝑝 for all finite 𝑝 < ∞.

The Fourier Restriction Problem The Fourier transform ˆ 𝑓(𝜉) = ∫ 𝑓(𝑥)𝑒−2𝜋𝑖𝜉⋅𝑥 𝑑𝑥, ℝ𝑛

defined initially for Lebesgue integrable functions, is a fundamental object in many different areas. In the 1960s, Stein introduced the Fourier restriction problem which seeks to understand the singularities that arise when one computes the Fourier transform of 𝐿𝑝 (ℝ𝑛 ) functions. When 𝑝 = 1, the Fourier transform is well behaved. A basic fact is that the Fourier transform of 𝑓 ∈ 𝐿1 (ℝ𝑛 ) is a

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continuous, bounded function and so restricting 𝑓ˆ to any set 𝑆 ⊂ ℝ𝑛 defines a function on 𝑆, continuous with respect to the induced topology. On the other hand, the Fourier transform defines a unitary operator from 𝐿2 onto 𝐿2 and so the singularities of 𝑓ˆ for 𝑓 ∈ 𝐿2 (ℝ𝑛 ) that arise are those of an arbitrary 𝐿2 function. Hence it can be identically ∞ on any set 𝑆 of measure zero. By interpolation, one can define the Fourier transform ′ for 𝑓 ∈ 𝐿𝑝 (ℝ𝑛 ) when 1 ≤ 𝑝 ≤ 2 and 𝑓ˆ ∈ 𝐿𝑝 where 𝑝′ is the conjugate exponent to 𝑝, satisfying 1/𝑝 + 1/𝑝′ = 1. However when 1 ≤ 𝑝 < 2, the mapping 𝑓 → 𝑓ˆ is not onto ′ 𝐿𝑝 (we have already seen this when 𝑝 = 1; the function 𝑓ˆ is not a general 𝐿∞ function, it is also continuous). Stein observed that there is a range 1 ≤ 𝑝 < 𝑝0 for some 𝑝0 (𝑛) > 1 so that for any 𝑓 ∈ 𝐿𝑝 (ℝ𝑛 ), one can make sense of 𝑓ˆ as an 𝐿2 density on the unit sphere 𝕊𝑛−1 = {|𝑥| = 1}. More precisely, he showed the Fourier restriction operator ˆ 𝕊𝑛−1 , defined initially on test functions, extends to ℛ𝑓 = 𝑓| a bounded operator from 𝐿𝑝 (ℝ𝑛 ) to 𝐿2 (𝕊𝑛−1 ) when 1 ≤ 𝑝 < 𝑝0 and so the singularities of 𝑓ˆ on 𝕊𝑛−1 can be no worse than an 𝐿2 function on the sphere. The fact that the unit sphere 𝕊𝑛−1 has curvature is crucial for Stein’s observation; if 𝑆 is a compact piece of a hyperplane, then for any 𝑝 > 1, ˆ 𝑆 ≡ ∞. See [Tao04] there is an 𝑓 ∈ 𝐿𝑝 (ℝ𝑛 ) such that 𝑓| for this example and a general discussion of the Fourier restriction phenomenon. The Fourier restriction problem is to determine the complete 𝐿𝑝 → 𝐿𝑞 mapping properties of ℛ and the conjecture is that ‖ℛ𝑓‖𝐿𝑞 (𝕊𝑛−1 ) ≤ 𝐶‖𝑓‖𝐿𝑝 (ℝ𝑛 ) (1) holds if and only if 1 ≤ 𝑝 < 2𝑛/(𝑛 + 1) and (𝑛 + 1)𝑞 ≤ (𝑛 − 1)𝑝′ . Here the space 𝐿𝑞 (𝕊𝑛−1 ) is defined with respect to surface measure 𝑑𝜎. In two dimensions 𝑛 = 2, the conjecture was solved by Fefferman and Stein [Fef70] in 1970 and independently by Zygmund [Zyg74] in 1974 but it remains open in dimensions 𝑛 ≥ 3. One can formulate conjectures for the Fourier restriction problem associated to other varieties of varying dimension with nonvanishing curvature. Remarkably the Fourier restriction problem has profound applications in disparate areas of mathematics, from PDEs in the form of fundamental Strichartz estimates to the recent solution of decoupling conjectures which led to Bourgain, Demeter and Guth’s [BDG16] complete resolution of the Vinogradov mean value theorem, a central problem in analytic number theory from the 1930s.

Duality The golden age of the 1950s brought to the fore the central role that the principle of duality plays in Fourier analysis. By duality, we can give an equivalent formulation of the NOVEMBER 2023

improving bound (𝐹): if 𝐸 is a Banach space of functions near 𝐿1 such that 𝐸 ∗ = 𝐹 (the Banach space dual of 𝐸 is 𝐹 with norms ‖ ⋅ ‖𝐸 and ‖ ⋅ ‖𝐹 , respectively), then 1/2

ˆ 2) ( ∑ |𝑓(𝑛)|

≤ 𝐶‖𝑓‖𝐸

(E)

𝑛∈Λ

with the same constant 𝐶 appearing in (𝐹). The bound in (𝐸) holds for all functions 𝑓 ∈ 𝐸 and we are restricting the ˆ Fourier coefficients {𝑓(𝑛)} to the given sequence Λ (for general compact abelian groups 𝐺, the spectral set Λ lies in the ˆ Here we see a connection discrete Fourier dual group 𝐺). between trigonometric series with gaps and certain discrete variants of the Fourier restriction problem. Our discussion of Rudin’s 1950s theory can be rephrased as follows: Pisier’s deep result says that Λ is a Sidon set if and only if (𝐸) holds with 𝐸 = 𝐿√log 𝐿 and this holds on any compact abelian group. A spectral set is ′ a Λ(𝑝) set for 𝑝 > 2 if and only if (𝐸) holds with 𝐸 = 𝐿𝑝 . Furthermore a set Λ ⊂ ℤ is a finite union of lacunary sequences if and only if (𝐸) holds for 𝐸 = 𝐻 1 (𝕋), the Hardy space on the circle whose Banach space dual is 𝐵𝑀𝑂(𝕋). Also by duality, we can reformulate the Fourier restriction problem in terms of the extension operator 𝑔(𝜔)𝑒−2𝜋𝑖𝜉⋅𝜔 𝑑𝜎(𝜔);

ℰ𝑔(𝜉) ∶= ∫ 𝕊𝑛−1

the Fourier restriction conjecture (1) is equivalent to ‖ℰ𝑔‖𝐿𝑝′ (ℝ𝑛 ) ≤ 𝐶‖𝑔‖𝐿𝑞′ (𝕊𝑛−1 ) holding if and only if 1 ≤ 𝑝 < 2𝑛/(𝑛 + 1) and (𝑛 + 1)𝑞 ≤ (𝑛 − 1)𝑝′ , Here 𝑝′ and 𝑞′ are the conjugate exponents of 𝑝 and 𝑞, respectively. At the endpoint, the exponents 𝑝 = 𝑞 = 2𝑛/(𝑛 + 1) agree but the bound (1) is known to fail ′ (in fact it suffices to take 𝑔 ≡ 1 and check that ℰ1 ∈ 𝐿𝑝 precisely when 𝑝′ > 2𝑛/(𝑛 − 1)). In [Tom80] Tomas proved a local Fourier restriction estimate in two dimensions (the one dimensional sphere 𝕊1 = 𝕋 is the circle) at the endpoint 𝑝′ = 𝑞′ = 4; ‖ℰ𝑔‖𝐿4 (𝐵𝑅 ) ≤ 𝐶[log 𝑅]1/4 ‖𝑔‖𝐿4 (𝕋) where 𝐵𝑅 = {|𝑥| ≤ 𝑅}. Nowadays it is known that this local estimate implies the Fefferman-Stein/Zygmund result that (1) holds when 𝑛 = 2. In Zygmund’s paper [Zyg74], a connection was almost made between bounds (𝐸) or (𝐹) for trigonometric series with gaps and Euclidean Fourier restriction bounds. Zygmund proved two theorems in [Zyg74]. His second result established (1) for 𝑛 = 2 but his first result, answering a question by Fefferman, showed 1/2

ˆ 2) ( ∑ |𝑓(𝑛)|

≤ 51/4 ‖𝑓‖𝐿4/3 (𝕋2 ) ,

(2)

|𝑛|=𝑅

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the sum being taken over all lattice points in 𝑆𝑅 = {𝑛 = (𝑚, 𝑛) ∈ ℤ2 ∶ 𝑚2 + 𝑛2 = 𝑅2 }. He remarks that the bounds (1) and (2) are analogous and his proof of (1) for 𝑛 = 2 is modelled on the proof he gave for (2). Furthermore he comments that his proof of (2) is quite general and works for any spectral set 𝑆 which has the property that #{(𝑟, 𝑠) ∈ 𝑆 2 ∶ 𝑟 ± 𝑠 = 𝑡} is bounded for all 𝑡. The bound (2) simply says 𝑆𝑅 is a Λ(4) set. It is the bound (𝐸) for the compact group 𝐺 = 𝕋2 , the Banach space 𝐸 = 𝐿4/3 (𝕋2 ) and the spectral set 𝑆𝑅 . In Rudin’s gaps paper the connection between Λ(4) sets and the number of representations as the sum of two elements from the spectral set is made explicit (see Theorem 4.5 in [Rud60]). In what follows, we will show that (2) is equivalent to a Fourier restriction bound for the two dimensional torus 𝕋2 .

W. Rudin Meets E. M. Stein Here Rudin’s theory of trigonometric series with gaps meets Stein’s theory of Fourier restriction. We will work on the compact group 𝕋𝑛 instead of the circle. Continuous functions 𝑓 ∈ 𝐶(𝕋𝑛 ) can be identified with (multiply) periodic functions on ℝ𝑛 ; they have a Fourier series 2𝜋𝑖𝑛⋅𝜃 ˆ ˆ = ∫ 𝑓(𝜃)𝑒−2𝜋𝑖𝜃⋅𝑛 𝑑𝜃. ∑ 𝑓(𝑛)𝑒 where 𝑓(𝑛) 𝕋𝑛

𝑛∈ℤ𝑛

The result is a general statement about spectral sets Λ that lie in the 𝑛-dimensional lattice ℤ𝑛 and Banach function spaces 𝐸 = 𝐸(𝕋𝑛 ) with norm ‖ ⋅ ‖𝐸 which are continuously embedded in 𝐿1 (𝕋𝑛 ). We will also need to assume 𝐸 continuously includes 𝐿𝑝 (𝕋𝑛 ) for some 1 < 𝑝 ≤ 4/3; thus we will assume there are two continuous embeddings 𝐿𝑝 (𝕋𝑛 ) ↪ 𝐸 ↪ 𝐿1 (𝕋𝑛 ) for some 𝑝 ≤ 4/3. This implies that there is a constant 𝐶 such that the following norm inequalities (for some 𝑝 ≤ 4/3) hold: 𝐶 −1 ‖𝑓‖𝐿1 (𝕋𝑛 ) ≤ ‖𝑓‖𝐸 ≤ 𝐶‖𝑓‖𝐿𝑝 (𝕋𝑛 ) .

(3)

∗

Therefore if 𝐹 = 𝐸 is the Banach space dual of 𝐸, then ′ 𝐿∞ (𝕋𝑛 ) ↪ 𝐹 ↪ 𝐿𝑝 (𝕋𝑛 ); or equivalently, 𝐶 −1 ‖𝑓‖𝐿𝑝′ (𝕋𝑛 ) ≤ ‖𝑓‖𝐹 ≤ 𝐶‖𝑓‖𝐿∞ (𝕋𝑛 ) .

(4)

We will consider the extension operator for 𝕋𝑛 : let 𝑧 = (𝑧1 , … , 𝑧𝑛 ) ∈ ℂ𝑛 ≃ ℝ2𝑛 and define

(W. Rudin) There is a constant 𝐴Λ such that ‖𝑓‖𝐹(𝕋𝑛 ) ≤ 𝐴Λ ‖𝑓‖𝐿2 (𝕋𝑛 ) , ∀𝑓 ∈ 𝐶Λ (𝕋𝑛 ). (E. M. Stein) There is a constant 𝐵Λ such that ‖ℰ𝑔‖𝐿4 (𝐵𝑅 ) ≤ 𝐵Λ (log 𝑅)𝑛/4 ‖𝑔‖𝐸(𝕋𝑛 ) , for all 𝑅 ≥ 2 and for all 𝑔 ∈ 𝐶Λ (𝕋𝑛 ). The (W. Rudin) statement is simply the bound (𝐹) (or equivalently (𝐸)) in the setting of the compact group 𝕋𝑛 . The theorem does not recover the local Fourier restriction endpoint bound of Tomas since 𝐸 cannot be taken to be 𝐿4 as we are assuming that 𝐸 continuously includes 𝐿𝑝 for some 1 < 𝑝 ≤ 4/3 (see (3)). The (E. M. Stein) statement is an extension of the Tomas local endpoint Fourier restriction bound to functions in some function space 𝐸(𝕋𝑛 ) with Fourier support in Λ ⊂ ℤ𝑛 . Applying the theorem to 𝐸 = 𝐿4/3 (𝕋2 ) and Λ = 𝑆𝑅 , the lattice points on a circle of radius 𝑅, we see that Zygmund’s result (2) is in fact equivalent to an endpoint Euclidean Fourier restriction bound for 𝕋2 . Again if Λ ⊂ ℤ𝑛 is a Sidon set (that is, if 𝐶Λ (𝕋𝑛 ) ⊂ 𝐴(𝕋𝑛 ), the space of absolutely convergent Fourier series), then we can take 𝐸 = 𝐿√log 𝐿(𝕋𝑛 ) so that 𝐸 ∗ = 𝐹 = exp(𝐿2 )(𝕋𝑛 ). Interestingly, the product of Sidon sets is not a Sidon set (for example, {2𝑘 } × {2ℓ } ⊂ ℤ2 is not a Sidon set). It was shown in [Bak19] that if Λ = Λ1 × ⋯ × Λ𝑛 ⊆ ℤ𝑛 is an 𝑛-fold product of (countably infinite) spectral sets Λ𝑗 ∈ ℤ, then each Λ𝑗 ⊂ ℤ is a Sidon set if and only if (𝐸) holds for 𝐸 = 𝐿(log 𝐿)𝑛/2 (𝕋𝑛 ). We have 𝐸 ∗ = 𝐹 = exp(𝐿2/𝑛 )(𝕋𝑛 ). See also [Pis78b]. Rudin’s theorem for Paley sets in ℤ was extended by Oberlin [Obe79] to the 𝑛-dimensional torus 𝕋𝑛 . Specifically, the bound (𝐸) holds for 𝐸 = 𝐻 1 (𝕋𝑛 ) if and only if sup𝑅 #[Λ ∩ 𝑅] < ∞ where the supremum is taken over all dyadic rectangles 𝑅 ⊂ ℤ𝑛 . In this case, 𝐸 ∗ = 𝐹 = 𝐵𝑀𝑂(𝕋𝑛 ).

The Proof First we introduce some notation. We write points 𝑧 ∈ ℂ𝑛 ≃ ℝ2𝑛 in polar form 𝑧𝑗 = 𝑟𝑗 𝑒𝑖𝜃𝑗 for each component of 𝑧. We think of 𝑟 = (𝑟1 , … , 𝑟𝑛 ) as the 𝑛 radii for 𝑧 and 𝜃 = (𝜃1 , … , 𝜃𝑛 ) as parametrising points on 𝕋𝑛 so that points in ℝ2𝑛 can be expressed as 𝑟𝑒𝑖𝜃 = (𝑟1 𝑒𝑖𝜃1 , … , 𝑟𝑛 𝑒𝑖𝜃𝑛 ) ∈ ℝ2𝑛 and functions on ℝ2𝑛 can be formally represented as

ℰ𝑔(𝑧) ∶= ∫ 𝑔(𝜃)𝑒−2𝜋𝑖⟨𝑧,𝜃⟩ 𝑑𝜃.

𝐹(𝑟𝑒𝑖𝜃 ) = ∑ 𝑓𝑘 (𝑟)𝑒𝑖𝑘⋅𝜃 .

𝕋𝑛

𝑘∈ℤ𝑛

𝑛

Finally we set 𝐵𝑅 = {𝑧 ∈ ℂ ∶ |𝑧𝑗 | ≤ 𝑅, ∀𝑗}, a polydisc of radius 𝑅 sitting in ℝ2𝑛 .

We define the following mixed norms for functions 𝐹 ∶ ℝ2𝑛 → ℂ,

Theorem 1. Let Λ and 𝐸 be as above. Then the following two statements are equivalent:

𝑝/2 ‖𝐹 ‖ 𝑖𝜃 2 ‖ ‖𝐿𝑝 𝐿2𝑛 ∶= ∫𝑛 (∫𝑛 |𝐹(𝑟𝑒 )| 𝑑𝜃) 𝑟1 ⋯ 𝑟𝑛 𝑑𝑟.

1700

𝑝

+ 𝕋

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

ℝ+

𝕋

VOLUME 70, NUMBER 10

With this notation the polydisc in the theorem can be expressed as 𝐵𝑅 = {𝑟𝑒𝑖𝜃 ∶ ∀𝑗, 0 ≤ 𝑟𝑗 ≤ 𝑅}. The proof factors through a variant of a result of L. Vega [Veg92]. Namely, ‖ℰ𝑔‖𝐿4+ 𝐿2𝑛 (𝐵𝑅 ) ≤ 𝐶(log 𝑅)𝑛/4 ‖𝑔‖𝐿2 (𝕋𝑛 ) .

(5)

𝕋

𝑝 𝐿rad 𝐿2ang

The theorem of L. Vega is a global mixed norm estimate for the Fourier extension operator on the sphere 𝕊𝑑−1 . See also [COeSS19]. The proof of (5) follows from classical bounds for Bessel functions which give the (log 𝑅)𝑛/4 bound, see e.g. [BC89]. If the (W. Rudin) statement holds (that is, (𝐹) and hence (𝐸) holds for Λ on 𝕋𝑛 ), then our spectral set is a Λ(4) set. To see this, we use (4) with 𝑝′ ≥ 4, together with (𝐹) for Λ, to conclude ‖𝑓‖𝐿4 (𝕋𝑛 ) ≤ ‖𝑓‖𝐿𝑝′ (𝕋𝑛 ) ≤ 𝐶‖𝑓‖𝐹 ≤ 𝐴Λ ‖𝑓‖𝐿2 (𝕋𝑛 )

∫ |ℰ𝑔(𝑟𝑒 )| 𝑑𝜃 ≤ 𝕋𝑛

𝐴4Λ (∫ 𝕋𝑛

𝑖𝜃 2

ˆ 𝑖𝜃 ) = 𝐻(𝑒

𝐽𝑘 (2𝜋𝑟) = √

where, importantly, the implicit constant in 𝑂(𝑟−3/2 ) is uniform in 𝑘 for 𝑟 ≥ 5𝑘. See [BC89]. We now construct a function ℎ𝑘,𝑁 (𝑟) which depends on 𝑁 and 𝑘. For every 𝑚 ∈ ℕ satisfying

𝑚 + (2𝑗 + 1)/8 ≤ 𝑟 ≤ 𝑚 + (2𝑗 + 1)/8 + 10−10 and we set ℎ𝑘,𝑁 (𝑟) = 0 otherwise. Hence for 5𝑘 ≤ 𝑁, 𝑁2

∫

‖ℰ𝑔‖𝐿4 (𝐵𝑅 ) ≤ 𝐴Λ ‖ℰ𝑔‖𝐿4+ 𝐿2𝑛 (𝐵𝑅 )

ℎ𝑘,𝑁 (𝑟)𝐽𝑘 (2𝜋𝑟)𝑟𝑑𝑟 𝑁2

for 𝑔 ∈ 𝐶Λ (𝕋𝑛 ). Therefore by (5),

= ∑ 𝑚=5𝑁

‖𝑔‖𝐿2 (𝕋𝑛 )

𝑚+(2𝑗+1)/8+10−10

2 √𝜋

∫ 𝑚+(2𝑗+1)/8

cos(2𝜋𝑟 − 𝑗𝜋/2 − 𝜋/4)) 𝑑𝑟 𝑟

+ 𝑂(𝑁 −1 ) = 𝑐 log 𝑁 + 𝑂(𝑁 −1 )

and 1/2

‖𝑔‖𝐿2 (𝕋𝑛 ) = ( ∑ |𝑔(𝑛)| ̂ )

(8)

0

𝕋

2

2 cos(2𝜋𝑟 − 𝑘𝜋/2 − 𝜋/4) + 𝑂(𝑟−3/2 ) 𝜋𝑟

we set ℎ𝑘,𝑁 (𝑟) = √2𝑟−3/2 for

4/2

whenever 𝑔 ∈ 𝐶Λ (𝕋𝑛 ).3 Integrating over ℬ𝑅 ∶= {𝑟 ∶ ∀𝑗, 0 ≤ 𝑟𝑗 ≤ 𝑅} shows that

‖ℰ𝑔‖𝐿4 (𝐵𝑅 ) ≤ 𝐵Λ (log 𝑅)

ℝ+

where 𝐽𝑘 is the classical Bessel function of order 𝑘. We use the asymptotic formula

|ℰ𝑔(𝑟𝑒 )| 𝑑𝜃)

𝑛/4

∑ [∫ 𝔥𝑘,𝑁 (𝑟)𝐽𝑘 (2𝜋𝑟)𝑟 𝑑𝑟]𝑒𝑖𝑘𝜃 𝑘∈Λ𝑁

5𝑁 ≤ 𝑚 ≤ 𝑁 2 , if 𝑘 ≡ 𝑗 mod4, 𝑗 = 0, 1, 2, 3,

for every 𝑓 ∈ 𝐶Λ (𝕋𝑛 ). Hence, for every 𝑟 ∈ ℝ𝑛+ , 𝑖𝜃 4

with certain properties, including that each 𝔥𝑘,𝑁 is supported in {𝑟 ≤ 𝑁 2 } and so 𝐻 is supported in the disc 𝐵𝑁 2 . To construct 𝔥𝑘,𝑁 , we first note that

≤ 𝐶‖𝑔‖𝐸(𝕋𝑛 )

𝑛∈Λ

for some 𝑐 = 𝑐𝑘,𝑁 > 0 with 𝑐 ∼ 1. Let us denote the integral in (8) by 𝐴𝑘,𝑁 . Hence |𝐴𝑘,𝑁 | ∼ log 𝑁. With {𝑎𝑘 }𝑘∈Λ𝑁 in (6), we define {𝑔𝑘 }𝑘∈Λ𝑁 by the relation

for 𝑔 ∈ 𝐶Λ (𝕋𝑛 ) by (𝐸). This shows that the (E. M. Stein) statement holds. The reverse implication is more involved and so, for clarity, we only give the proof when 𝑛 = 1. The extension to general 𝑛 ≥ 1 is straightforward. Suppose that the (E. M. Stein) statement holds for 𝑛 = 1 and fix a sequence {𝑎𝑘 }𝑘∈Λ . Our goal is to show

We finally arrive at our 𝔥𝑘,𝑁 (𝑟) ∶= 𝑔𝑘 ℎ𝑘,𝑁 (𝑟) which defines 𝐻 in (7). Note that 𝔥𝑘,𝑁 is supported in {𝑟 ≤ 𝑁 2 } since the same is true for ℎ𝑘,𝑁 . The dual formulation of the (E. M. Stein) statement with 𝑅 = 𝑁 2 implies (since 4/3 ≤ 2)

‖ ∑ 𝑎 𝑒𝑖𝑘(⋅) ‖ ≤ 𝐴 ( ∑ |𝑎 |2 )1/2 𝑘 Λ 𝑘 ‖ ‖𝐹

ˆ 𝕋 ‖𝐹 ≤ 𝐵Λ (log 𝑁)1/4 ‖𝐻‖ 4/3 2 , ‖𝐻| 𝐿 𝐿

𝑘∈Λ𝑁

(6)

𝑘∈Λ𝑁

with a constant 𝐴Λ which is independent of 𝑁. Here Λ𝑁 = {𝑘 ∈ Λ ∶ |𝑘| ≤ 𝑁}. This will establish (𝐹), the (W. Rudin) statement. From {𝑎𝑘 }, we will construct a function 𝑖𝜃

𝐻(𝑟𝑒 ) ∶= ∑ 𝔥𝑘,𝑁 (𝑟)𝑒

𝑖𝑘𝜃

𝑔𝑘 𝐴𝑘,𝑁 = 𝑐 log 𝑁𝑎𝑘 so that |𝑔𝑘 | ≲ |𝑎𝑘 |.

(7)

+

where ∞

‖𝐻‖𝐿4/3 𝐿2 = (∫ ( ∑ |𝔥𝑘,𝑁 (𝑟)|2 ) + 𝕋

NOVEMBER 2023

0

1 4 2 3

3/4

𝑟 𝑑𝑟)

.

𝑘∈Λ𝑁

Since each 𝔥𝑘,𝑁 is supported in [𝑁, 𝑁 2 ] and |𝔥𝑘,𝑁 (𝑟)| ≲ 𝑟−3/2 |𝑎𝑘 |, we see that

𝑘∈Λ𝑁

𝑁2

3Note that 𝑔 ∈ 𝐶 (𝕋𝑛 ) implies that ℰ𝑔(𝑟⋅) ∈ 𝐶 (𝕋𝑛 ). Λ Λ

(9)

𝕋

‖𝐻‖𝐿4/3 𝐿2 ≤ 𝐶(∫ +

𝕋

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

𝑁

[

1

] 𝑟3/2

4/3

3/4

𝑟𝑑𝑟)

1/2

( ∑ |𝑎𝑘 |2 ) 𝑘∈Λ𝑁

1701

and so

References

‖𝐻‖𝐿4/3 𝐿2 ≤ 𝐶(log 𝑁) +

𝕋

3/4

2

1/2

( ∑ |𝑎𝑘 | )

.

𝑘∈Λ𝑁

Also ˆ 𝑖𝜃 ) = [𝑐 log 𝑁] ∑ 𝑎𝑘 𝑒𝑖𝑘𝜃 , 𝐻(𝑒 𝑘∈Λ𝑁

implying ˆ 𝕋 ‖𝐹 = [𝑐 log 𝑁]‖ ∑ 𝑎𝑘 𝑒𝑖𝑘(⋅) ‖ . ‖𝐻| ‖ ‖ 𝑘∈Λ𝑁

𝐹

Therefore (9) implies 1/2 [𝑐 log 𝑁] ‖‖ ∑ 𝑎𝑘 𝑒𝑖𝑘(⋅) ‖‖ ≤ 𝐴Λ log 𝑁( ∑ |𝑎𝑘 |2 ) , 𝐹 𝑘∈Λ𝑁

𝑘∈Λ𝑁

showing that (6) holds, establishing the (W. Rudin) statement.

Conclusion Here we tried to draw some parallels between two great men; similar life stories, similar research interests and similar influence through their books and monographs. And although they may not have interacted mathematically as working researchers, their mathematics is nonetheless intimately connected. This is the beauty of mathematics. ACKNOWLEDGMENTS. The authors would like to thank Lillian Pierce for her interest in this project and for her helpful comments on an earlier draft of this paper. O. Bakas was partially supported by the grant KAW 2017.0425, financed by the Knut and Alice Wallenberg Foundation, by the projects CEX2021001142-S, RYC2018-025477-I, PID2021-122156NBI00/AEI/10.13039/501100011033 funded by Agencia Estatal de Investigacion ´ and acronym “HAMIP,” Juan de la Cierva Incorporacion ´ IJC2020-043082-I and grant BERC 2022-2025 of the Basque Government. V. Ciccone is grateful to Paolo Ciatti for many inspiring discussions on the topics of Fourier restriction estimates and lacunary Fourier series. O. Bakas and J. Wright would like to thank Sandra Pott and the Centre for Mathematical Sciences at Lund University for their hospitality during their visit in May 2022, where part of this work was carried out and funded in part by a Knut and Alice Wallenberg Foundation guest professorship, KAW 2020.0289.

1702

[Bak19] Odysseas Bakas, Variants of the inequalities of Paley and Zygmund, J. Fourier Anal. Appl. 25 (2019), no. 3, 1113– 1133, DOI 10.1007/s00041-018-9605-7. MR3953499 [BC89] Juan A. Barcelo´ and Antonio Cordoba, ´ Bandlimited functions: 𝐿𝑝 -convergence, Trans. Amer. Math. Soc. 313 (1989), no. 2, 655–669, DOI 10.2307/2001422. MR951885 [Bou89a] J. Bourgain, Bounded orthogonal systems and the Λ(𝑝)set problem, Acta Math. 162 (1989), no. 3-4, 227–245, DOI 10.1007/BF02392838. MR989397 [Bou89b] J. Bourgain, On Λ(𝑝)-subsets of squares, Israel J. Math. 67 (1989), no. 3, 291–311, DOI 10.1007/BF02764948. MR1029904 [BDG16] Jean Bourgain, Ciprian Demeter, and Larry Guth, Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633–682, DOI 10.4007/annals.2016.184.2.7. MR3548534 [CZ52] A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139, DOI 10.1007/BF02392130. MR52553 [COeSS19] Emanuel Carneiro, Diogo Oliveira e Silva, and Mateus Sousa, Sharp mixed norm spherical restriction, Adv. Math. 341 (2019), 583–608, DOI 10.1016/j.aim.2018.10.043. MR3873546 [Fef70] Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36, DOI 10.1007/BF02394567. MR257819 [Obe79] Daniel M. Oberlin, Two multiplier theorems for 𝐻 1 (𝑈 2 ), Proc. Edinburgh Math. Soc. (2) 22 (1979), no. 1, 43–47, DOI 10.1017/S0013091500027796. MR536591 [Pis78a] Gilles Pisier, Ensembles de Sidon et processus gaussiens (French, with English summary), C. R. Acad. Sci. Paris S´er. A-B 286 (1978), no. 15, A671–A674. MR511046 [Pis78b] G. Pisier, Sur l’espace de Banach des s´eries de Fourier al´eatoires presque sûrement continues (French), S´eminaire sur la G´eom´etrie des Espaces de Banach (1977–1978), ´ Ecole Polytech., Palaiseau, 1978, pp. Exp. No. 17–18, 33. MR520216 [Pis81] Gilles Pisier, De nouvelles caract´erisations des ensembles de Sidon (French, with English summary), Mathematical analysis and applications, Part B, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 685–726. MR634264 [Rud57] Walter Rudin, Remarks on a theorem of Paley, J. London Math. Soc. 32 (1957), 307–311, DOI 10.1112/jlms/s132.3.307. MR94650 [Rud60] Walter Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227, DOI 10.1512/iumj.1960.9.59013. MR0116177 [Tao04] Terence Tao, Some recent progress on the restriction conjecture, Fourier analysis and convexity, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004, pp. 217–243, DOI 10.1198/106186003321335099. MR2087245

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VOLUME 70, NUMBER 10

NOW AVAILABLE FROM [Tom80] Peter A. Tomas, A note on restriction, Indiana Univ. Math. J. 29 (1980), no. 2, 287–292, DOI 10.1512/iumj.1980.29.29020. MR563213 [Veg92] Luis Vega, Restriction theorems and the Schrödinger multiplier on the torus, Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990), IMA Vol. Math. Appl., vol. 42, Springer, New York, 1992, pp. 199–211, DOI 10.1007/978-1-4612-2898-1_18. MR1155865 [Zyg74] A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189–201, DOI 10.4064/sm-50-2-189-201. MR387950

Odysseas Bakas

Valentina Ciccone

Analysis II Fourth Edition Terence Tao, University of California Los Angeles, CA

James Wright Credits

Figure 1 is courtesy of Alexander Nagel. Photo of Odysseas Bakas is courtesy of Odysseas Bakas. Photo of Valentina Ciccone is courtesy of Valentina Ciccone. Photo of James Wright is courtesy of James Wright.

This is part two of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning—the construction of the number systems and set theory—then goes on to the basics of analysis, through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. Hindustan Book Agency; 2022; 242 pages; Hardcover; ISBN: 978-81-95196-12-8; List US$52; AMS members US$41.60; Order code HIN/83 Titles published by the Hindustan Book Agency (New Delhi, India) include studies in advanced mathematics, monographs, lecture notes, and/or conference proceedings on current topics of interest.

Discover more books at bookstore.ams.org/hin. Publications of Hindustan Book Agency are distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for commercial channels.

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SACNAS Turning 50 Fabio Augusto Milner

It is a time for celebration: The Society for Advancement of Chicanos and Native Americans in Science (SACNAS) is 50 years old! That is half a century of remarkable work on behalf of, and performed for and by, dedicated scientists of diverse backgrounds with the common goal of bringing social justice to groups of outstanding individuals who had been ignored at best, and blatantly discriminated at worst, for much too much time.1 In 1973 Dr. Alonzo Atencio, a biochemist at the University of New Mexico, was bothered by this apparent systemic disregard for inclusion of Chicanos in scientific research and in most decision-making. He obtained NIH funds to sponsor a meeting in Albuquerque, where he lived and worked, for planning the organization of a group that would somehow address the scarcity of Chicanos and Fabio Augusto Milner is a professor of mathematics and a member of the SACNAS board of directors. His email address is [email protected]. Communicated by Notices Associate Editor William McCallum. For permission to reprint this article, please contact: [email protected].

2

DOI: https://doi.org/10.1090/noti2809 1 Joe R. Feagin, Jos´e A. Cobas (2014). Latinos Facing Racism, Discrimination, Resistance, and Endurance (1st Edition). Routledge: New York. DOI https://doi.org/10.4324/9781315633749

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Native Americans in academia and government. He invited the few Chicano and Native American scientists he either knew or had heard of, including a stochastic processes specialist, Dr. Richard Griego, who made a call to action using the blunt expression “We need to get our caca together.”2 The time had come to set up a mechanism to develop future leaders from the Hispanic and Indigenous communities, who would join the tables where decisions are made, and for students from these neglected groups to be trained in the sciences. A total of 17 PhDs attended the meeting, 3 chemists, 2 biochemists, 3 biologists, 3 physicists, 2 mathematicians—Richard Griego and Richard Tapia—and 4 in other STEM areas. This group is considered as the founders of SACNAS. They are all men because, quite unfortunately, that is what the face of STEM was like in the early 1970s and for some time thereafter. This underrepresentation and discrimination of women in many STEM fields has been an ongoing concern of mine and of SACNAS, and we continually work towards helping mitigate it and, ultimately, making it disappear. SACNAS is a remarkable organization, both in its history and its modus operandi. According to its Mission Statement, “SACNAS is an inclusive organization dedicated to fostering the success of Chicanos/Hispanics and Native Americans, from college students to professionals, in attaining advanced degrees, careers, and positions of leadership in STEM.”3 It was always driven by the selfless commitment and devoted work of an ever-increasing group of diverse scientists who act as mentors for students, early career scientists, and STEM professionals. As of today, SACNAS has over 9,000 members and 126 chapters in nearly all US states, Puerto Rico, and Guam. The first 50 years of SACNAS mathematics. In its first 50 years there were several well-known mathematicians https://www.sacnas.org/history-of-sacnas. This history focuses on how SACNAS fits in with the Chicano Movement in the United States that predates SACNAS and provides details on the early meetings of SACNAS, with some closing thoughts on the SACNAS of today. 3

https://www.sacnas.org/mission-impact

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Figure 1. SACNAS Secretary Fabio Augusto Milner, with Robert E. Megginson and Rodrigo Banuelos ˜ at the 2015 Annual Conference in Washington, DC.

who played major roles in leading SACNAS’s efforts toward fulfilling its mission. Among them we should mention Drs. Rodrigo Bañuelos, Richard Griego, Robert Megginson, Richard Tapia, and William Yslas V´elez. I met Rodrigo Banuelos ˜ in 1987, when he joined the Mathematics Department at Purdue University that I had joined four years earlier. His rise in the profession was meteoric: the typical promotion clock from assistant professor to associate professor and tenure is six years, but Rodrigo was promoted to full professor after five years at Purdue. He was later associate director, interim director, and director of the Math Department. Only around that time did he first tell me the fascinating and inspirational story of how he arrived there. He was born in Zacatecas, Mexico, and had no formal education as a child, working primarily in farming—like most kids in the region. He reflected that life then was primitive and hard, but his childhood memories were those of a campesino community, where everyone is always there to help each other. Banuelos ˜ and his mother, grandmother, five brothers, and one sister moved to Pasadena, California, when he was 15. His father had lived in the US most of his life, commuting from Pasadena to Zacatecas once or twice a year. They were poor, they were sick, and people mistreated them or took advantage of them—all because of a lack of education, according to his mother: “Eso nos pasa por falta de educacion,” ´ she would often say. When he was young, Rodrigo did not quite understand or fully appreciate the depth of the NOVEMBER 2023

statement, “This happens to us because of lack of education.” He was the first in his family to attend and graduate from college, or even from high school! Rodrigo feels indebted to Juan Francisco Lara, whom he met in 1973 while working at a car wash in Pasadena where Juan would stop to wash his car and encouraged him to enroll in his course on Chicano studies at Pasadena City College, which he eventually did. After a year and two summers at Pasadena City College, with the help of Juan Lara and Rub´en Ruvalcaba, Rodrigo transferred to UC Santa Cruz. He, like most of us, was unwittingly benefiting from the selfless support of silent mentors who helped shape his later commitment to become himself a strong mentor and supporter of others. Before UC Santa Cruz, he had applied to and been rejected from several other campuses of the UC System, including UCLA. During his first year at UC Santa Cruz, Rodrigo met two of the earliest SACNAS members, Eugene Cota-Robles and Frank Talamantes. He remembers that he and many of his peers at UC Santa Cruz received tremendous support and encouragement from those two original SACNISTAS. Banuelos ˜ credits them, as well as Lara and others, with exemplifying at its very best C´esar Ch´avez’s statement that “We cannot seek achievement for ourselves and forget about progress and prosperity for our community. Our ambitions must be broad enough to include the aspirations and needs of others, for their sakes and for our own.” Rodrigo received his three university degrees in mathematics in California: a BS from UC Santa Cruz in 1978, an MA in mathematics education from UC Davis in 1980, and a PhD from UCLA in 1984. He was then a postdoc at Caltech and, two years later, he received an NSF Postdoctoral Fellowship to go to the University of Illinois at Urbana-Champaign for one year, after which he settled at Purdue University, where he would remain for the rest of his career. Rodrigo has given scores of invited lectures at conferences and universities in many countries around the world. His research has been continuously funded by the NSF that also awarded him, in 1989, the Presidential Young Investigator Award. He has participated in many efforts to increase the number of minority students in sciences and engineering, as a recognition for which he became, in 2004, the second recipient of the Blackwell–Tapia Prize in Mathematics, presented to him at the Institute for Pure and Applied Mathematics (IPAM) at UCLA—the same institution that had previously rejected his application for admission and later awarded him his PhD.4 Rodrigo is a kind, generous, and humble human being, and a very long-time SACNISTA who introduced me and many others to SACNAS, whose annual conferences we attended together many times. At the Long Beach Conference in 4

https://www.sacnas.org/sacnas-biography-project

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2016, we organized together a Professional Development Session titled What do I do with my Bachelor’s? It attracted such a large audience that there was standing-room only in a meeting room with 180 seats. I never had the pleasure of meeting Richard Griego in person, though I have heard plenty about him and wish I had. He was very direct and to the point, and I like very much his 1973 call-to-action phrase “We need to get our caca together.” He was born in Albuquerque, New Mexico, and received his PhD in mathematics from the University of Illinois in 1965. His academic field is probability theory and stochastic processes, and he is recognized as one of the initiators of the theory of random evolutions. He has taught mathematics and statistics at UC Riverside, Instituto Polit´ecnico Nacional in Mexico City, UC Berkeley, the University of Utah, and the University of New Mexico where he retired as Presidential Professor of Mathematics. Professor Griego was the Chairman of the Mathematics Departments at the University of New Mexico and at Northern Arizona University in Flagstaff. He worked as a research scientist for projects on space medicine and spent time as a visiting scientist at Los Alamos National Laboratory in New Mexico. He also served in numerous other administrative duties in academic settings. Richard made valuable contributions to theoretical and applied statistics, one of which is cited in the Encyclopedia Britannica. Griego has been very active in conferences and projects at the national and international levels and has collaborated extensively on statistics research with mathematicians in Mexico. He received over $5 million in grants for research and intervention programs. He was also the recipient of numerous research fellowships and honorable mentions in recognition of his academic accomplishments and community service. Dr. Griego is a member of the AMS, the AAAS, the Sociedad Matemática Mexicana, SACNAS, and other professional organizations. Dr. Griego has published in many journals, including the Proceedings of the National Academy of Sciences, Transactions of the American Mathematical Society, and Scientific American. He has also published the books Conceptos de Probabilidad (Fondo de Cultura Economica, ´ M´exico) and Lenguaje: A Cultural History of the Spanish Language of New Mexico (via Amazon and Kindle). He has been a director of many science and other programs for enhancing the educational opportunities of underrepresented groups.5 Richard Griego exemplifies the commitment and drive of SACNAS’s founders to “walk the walk” rather than just “talk the talk.” Dr. Robert “Bob” Eugene Megginson (Lakota) is one of the very few Native Americans to hold a PhD in

mathematics.6 He earned a BS in physics from the University of Illinois at Urbana–Champaign and became a software specialist for the Roper Corporation7 until 1977. He returned to graduate school after more than a decade and earned an MS in statistics and a PhD in mathematics at the University of Illinois. Bob is the Arthur F. Thurnau Professor of Mathematics at the University of Michigan in Ann Arbor, Michigan. He took great interest in supporting and promoting underrepresented minorities in mathematics, which he made a focal point of his career. I had the pleasure of meeting Bob many years ago at the SACNAS Annual Conference and vividly remember how tall he seemed. We shared many endeavors since with the zest of promoting underrepresented groups in STEM, and I then began to see Bob’s true stature as being well beyond the physical. He spent many summers in the 1990s teaching mathematics in special programs at Turtle Mountain Community College in North Dakota. Dr. Megginson has personally mentored many minority students and has received numerous awards and recognitions for that work. He is a 1997 recipient of the US Presidential Award for Excellence in Science, Mathematics, and Engineering Mentoring, and the 1999 recipient of the Ely S. Parker Award for lifetime service to the Native American community from the American Indian Science and Engineering Society (AISES). Bob was named to the Native American Science and Engineering Wall of Fame in 2001, and he was elected Fellow of the AAAS in 2009. That same year he received the Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service from the Mathematical Association of America (MAA) for his work on underrepresented minorities. In 2012, Megginson became one of the inaugural Fellows of the American Mathematical Society, and he was honored by SACNAS in 2019 with the Distinguished Mentor of the Year Award. Each year, with these awards, SACNAS recognizes two scientists who have made significant contributions to their field and who have a long-standing commitment to diversity and inclusion in STEM. He was co-Principal Investigator for AISES’s Lighting the Pathway to Faculty Careers for Natives in STEM, funded by the single largest National Science Foundation grant ever received by AISES.8 I met Richard Tapia during the Sixth International Joint Meeting of the AMS and the Sociedad Matemática Mexicana at the University of Houston in 2004. I was pleasantly taken by his good humor combined with incisiveness and lack of embellishment in talking about serious structural problems in the promotion and advancement of 6

https://aimathcircles.org/robert-eugene-megginson/ https://en.wikipedia.org/wiki/Roper_Technologies 8 https://www.maa.org/programs-and-communities/outreach -initiatives/summa/summa-archival-record/robert-eugene -megginson 7

5 https://www.pps.net/cms/lib/OR01913224/Centricity/Domain /179/pdfs/be-hi-ma.pdf

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Hispanic scientists’ careers. He was born and raised by Mexican immigrant parents in a poor white area of Los Angeles. Richard recalls hearing at times “Mexican, go back where you came from.” That would lead him to think “but I am from here, I am not Mexican, my mother and father are Mexican.” He experienced firsthand the racism still waiting to be eradicated. His mother told him that he would understand it better when he visited Mexico. Around the age of 13 he went to Mexico with his family and there he was told “Gringo, go back where you came from.” That led Richard to an identity crisis because he felt that he did not fit as Mexican or as American. Richard remembers fondly the year 1968 as “an amazingly impactful year in the United States.” It was the year that Martin Luther King and Robert Kennedy were assassinated, the year that saw the height of the Vietnam War protests leading to the Mothers’ March on Washington, and it was a very successful year for César Chávez and the United Farm Workers. It was also the year he received his PhD in mathematics from UCLA, as Bañuelos would 16 years later. Moreover, in 1968 the Chicano movement in Los Angeles and at UCLA was alive and strong. It was then that Tapia found his identity and became a proud Chicano. He recalls the beginnings of SACNAS as a multidisciplinary group whose members came together to support each other through the tenure process, because only they, rather than their university colleagues, understood the challenge of dealing with the extra baggage that underrepresented minorities growing up in the US faced in their professional lives. He feels that SACNAS’s leitmotiv has always been to put the cause above the individual. He remembers how everyone at SACNAS bonded and tried to guide and help each other and mentions fondly his lifelong bond with Jos´e Martinez who played a major role in Richard’s nomination for the National Medal of Science awarded to him in 2011.9 Dr. Tapia is a University Professor of Mathematics, Maxfield & Oshman Professor of Engineering, and Director of the Center for Excellence and Equity in Education at Rice. He has received numerous honorary degrees, awards and recognitions, including AAAS and AMS Fellowships; an appointment to the National Science Board by President Clinton; NSF’s Presidential Award for Excellence in Science, Mathematics, and Engineering Mentoring; AAAS’s Lifetime Mentor Award; induction into the Hispanic Engineer National Achievement Awards Conference Hall of Fame; SACNAS’s Distinguished Scientist Award; SIAM’s Distinguished Service to the Profession Prize; and AMS’s Distinguished Public Service Award.10 9

https://www.cmor-faculty.rice.edu/~rat/2018.Sacnas.Tapia .talk.pdf 10 https://www.cmor-faculty.rice.edu/~rat/

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My first encounter with William “Bill” Yslas Vélez was in Tucson, Arizona, in 2007, where he was born and has lived for most of his life. I immediately noticed his welcoming warm smile and attitude. Bill grew up in the Spanishspeaking part of town with parents from Sonora, Mexico. His father died when he was nine years old, and his mother had to work three jobs to make ends meet. He and his siblings also helped by working at the family gas station after school, on weekends, and during the summers. Though his family experienced poverty, he came away with a sense of pride. His family valued education, so after graduating from high school, he attended the University of Arizona, where he received BS, MS, and PhD degrees in mathematics. He served in the US Navy from 1968–1969. After receiving his doctorate, he accepted a position at Sandia National Laboratories in Albuquerque, New Mexico, from which he moved to the University of Arizona as assistant professor of mathematics and went on to the ranks of associate and full professor. His main research areas were algebra and number theory. Bill traveled extensively, giving lectures throughout the US and around the world, including a three-week trip to China, where he lectured in Beijing, Sichuan, and Shanghai. He also worked as a consultant to the Naval Ocean Systems Center and served as SACNAS’s president from 1994 to 1996. Bill is a Fellow of the AAAS and a professor emeritus of mathematics at the University of Arizona.11 We can immediately see the common thread these remarkable individuals followed, rising up from underprivileged communities through strong and committed mentors, their own families, and a personal drive to help lessen the injustices and discrimination that their peoples had suffered for a long time.12 13 14 They became distinguished scientists and great mentors who founded SACNAS as the place where they and over 9,000 others today find a safe place to bond with peers, to network and make friends and professional connections, to sing and dance, and to continue together the hard work of simultaneously fighting for social justice and making discoveries significant enough to help a rigid academic and power system accept the fact that Chicano/Hispanic and Native American/Indigenous

11 https://www.sacnas.org/diversity-news/honoring-dr-william -yslas-velez 12 https://blog.ucsusa.org/derrick-jackson/a-new-study -confirms-structural-racism-in-stem-programs-needs-fixing/ 13 https://www.whitehouse.gov/briefing-room/presidential -actions/2021/06/25/executive-order-on-diversity-equity -inclusion-and-accessibility-in-the-federal-workforce/ 14 https://www.eeoc.gov/federal-sector/reports/report -hispanic-employment-challenge-federal-government-federal -hispanic

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scientists are as good as any and deserve to be included as equals in positions of power and in academia.15 16 What the next 50 years may have in store. The next 50 years are looking bright for mathematical SACNISTAS. Some of the new generation of SACNISTAS already making great contributions to the SACNAS mission in similar ways to the founders and their successors include Drs. Mario Banuelos, ˜ Minerva Cordero Brana, ˜ Ivelisse Rubio, Adriana Salerno, and Kamuela Yong, among many others. Mario Banuelos ˜ is from the small, agricultural town of Delano, California, and a first-generation college student. He received a PhD in applied mathematics in 2018 and is an assistant professor at Fresno State University, where he is already associate chair of the Department of Mathematics, as well as recipient of Fresno State College of Science and Mathematics Promising New Faculty Award, Scholarly and Creative Activity Award, and Professional Development Award. Mario was also program director and is presently vice chair of the SIAM Activity Group on Applied Mathematics Education, and a volunteer on SACNAS committees. His research focuses on mathematical biology, optimization, statistical models for genome evolution, and data science.17 Minerva Cordero Brana ˜ grew up in Bayamon, ´ Puerto Rico, and received a BS in mathematics from the Universidad de Puerto Rico in Rio Piedras. She then attended UC Berkeley with an NSF Minority Graduate Fellowship where she received an MS in mathematics and then went on to receive a PhD in mathematics from the University of Iowa. Cordero was an assistant and associate professor at Texas Tech University until 2001 and is now a full professor of mathematics and senior associate dean for the College of Science at the University of Texas at Arlington. Last year, President Biden awarded her the Presidential Award for Excellence in Science, Mathematics, and Engineering Mentoring (PAESMEM) and this year she was named Fellow of the Association for Women in Mathematics. Cordero’s research is in the general area of finite geometry on finite semifields and their associated planes. She served as the MAA’s governor-at-large for minority interests from 2008 to 2011 and received the Great Minds in STEM HENAAC Award Education Distinction. She was the Principal Investigator for a multimillion-dollar NSF grant to the University of Texas at Arlington for a project that placed mathematics graduate students in Arlington public schools to enhance learning in the classroom and to inspire students to pursue careers in STEM. Minerva received 15

https://www.doi.gov/pmb/eeo/doi-minority-serving -institutions-program 16 https://www.pewresearch.org/social-trends/2018/01/09 /diversity-in-the-stem-workforce-varies-widely-across-jobs/ 17 https://www.mbgmath.com

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numerous awards and honors, including: Professor of the Year from the Texas Tech MAA student chapter; President’s Excellence in Teaching Award, Texas Tech University; University of Texas Board of Regents’ Outstanding Teaching Award; a Certificate of Meritorious Service from the MAA; and she was named IF/THEN Ambassador by the AAAS.18 Ivelisse “Ive” Rubio has a PhD in applied mathematics from Cornell University and is a professor in the Computer Science Department of the University of Puerto Rico, R´ıo Piedras. Her research interests are finite fields and their applications. She has directed the undergraduate research projects in computational mathematics of numerous minority students and has been involved in many activities to promote undergraduate research in mathematics. She cofounded and codirected the REU Summer Institute in Mathematics for Undergraduates (SIMU) (1998–2002) and the REU MSRI-UP (2007–2015). In 2006, SIMU received the American Mathematical Society’s award to “Programs That Make a Difference,” this being the first time that this award was given by the AMS. In recognition of her outstanding contributions, Ivelisse received a 2006 SACNAS Presidential Service Award. In 2010 she received the Dr. Etta Z. Falconer Award for Mentoring and Commitment to Diversity. She is currently a member of the US National Committee for Mathematics and of the American Mathematical Monthly Editorial Board. Her paper with Francis Castro on an elementary method for establishing the solvability of general diagonal polynomial equations over prime fields was selected as one of the best papers published in the Journal of Algebra and its Applications in 2014.19 I had the good fortune of meeting Ive in 2004 in R´ıo Piedras, Puerto Rico; have met with her several times since then; and have always appreciated her unwavering dedication to her students and her tireless work in research, mentoring, and promoting diversity, all while remaining a warm and positive human being. Adriana Salerno grew up in Caracas, Venezuela, where she received her undergraduate degree in mathematics from the Universidad Simon Bolivar in 2001. She then went on to earn her PhD at the University of Texas. While completing her doctorate in mathematics, Salerno was also selected as the AMS-AAAS Mass Media Fellow in the summer of 2007; as such, she wrote articles for the Voice of America. Salerno’s main research area is number theory, in particular the intersections of number theory with geometry, physics, and cryptography. She is also very interested in the communication and teaching of mathematics to create a more inclusive and equitable STEM workforce. She is an alumna of the Linton-Poodry SACNAS Summer Leadership Institute, and the SACNAS-HHMI Advanced 18

https://en.wikipedia.org/wiki/Minerva_Cordero https://www.lathisms.org/calendar-2016/ivelisse-rubio

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Leadership Institute, and is committed to increasing the representation of minorities and women in the mathematical sciences. She is a proud member of AWM, SACNAS, MAA, and AMS. Salerno was a visiting mathematician at the MAA in Washington, D.C. in 2016 and program director at the NSF, in the Algebra and Number Theory Program in 2021–2022.20 Kamuela “Wela” Yong is the first Native Hawaiian to earn a PhD in applied mathematics and he is an associate professor of mathematics at the University of Hawai’i– West O‘ahu.21 His doctorate is from the University of Iowa—where he also received an MS—and was followed by a three-year postdoctoral appointment at Arizona State University, where I met him in 2015. His research interests are in mathematical modeling of biological, ecological, and epidemiological systems using diffusion. While at ASU, he was a great asset to the MTBI summer REU, as well as to many doctoral students in the Applied Mathematics in the Life and Social Sciences (AMLSS) program, and to the staff, and to the life and research of the Simon A. Levin Mathematical, Computational, and Modeling Sciences Center. Yong impressed me as an insightful and committed mentor, always willing to listen and make contributions, and to generously give his time for the benefit of others. Wela is the 2019 recipient of the Frances Davis Award for Excellence in Undergraduate Teaching and the 2020 recipient of the University of Hawai’i Regents’ Medal for Excellence in Teaching.22 The impact of SACNAS’s and other sister organizations efforts on the face of STEM in our country over the past 50 years is quite apparent in the much-increased gender and ethnic diversity among academicians, leaders in the government and private sectors, and quite large numbers of

Chicano/Hispanic and Native American/Indigenous students pursuing and obtaining advanced STEM degrees. We should not rest in our work, however, because we know there is still a long road to be covered to bring representation of many neglected groups to positions of power and high-stakes decision making. Our hope is that, when the next generations celebrate SACNAS’s 100th anniversary in 2073, they may cite examples of peers from our communities who have been US presidents, Supreme Court justices, CEOs of Forbes 500 corporations, Distinguished University Professors, deans, provosts and presidents, mentors and drivers of social justice, and mainly just great citizens who feel accomplished by improving their own and the lives of others and continue to spread the spirit of bonding and community that has always been at the heart of SACNISTAS and SACNAS endeavors and activities.

Fabio Augusto Milner Credits

The logo is courtesy of SACNAS. Figure 1 and author photo are courtesy of Fabio Augusto Milner.

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https://sites.google.com/view/asalerno/home https://westoahu.hawaii.edu/facultyprofiles/user/kamuelay 22 https://indigenousmathematicians.org/kamuela-e-yong/ 21

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The American Mathematical Society welcomes applications for the 2024

The 2024 summer conferences of the Mathematics Research Communities will be held at Beaver Hollow Conference Center, Java Center, NY, where participants can enjoy a private, distraction-free environment conducive to research. In support of the AMS’s continuing efforts to promote equity, diversity, and inclusion in the mathematical research enterprise, we strongly encourage and welcome applicants from diverse backgrounds and experiences. The application deadline is February 15, 2024.

TOPICS FOR 2024 Week 1: June 9 –15, 2024 Algebraic Combinatorics Organizers: Susanna Fishel, Arizona State University Rebecca Garcia, Colorado College Pamela Harris, University of Wisconsin–Milwaukee Rosa Orellana, Dartmouth College Stephanie van Willigenburg, University of British Columbia

Week 2: June 23–29, 2024 Mathematics of Adversarial, Interpretable, and Explainable AI Organizers: Karamatou Yacoubou Djima, Wellesley College Tegan Emerson, Pacific Northwest National Laboratory; Colorado State University; University of Texas El Paso Emily King, Colorado State University Dustin Mixon, The Ohio State University Tom Needham, Florida State University

Week 3a: June 30–July 6, 2024 Climate Science at the Interface Between Topological Data Analysis and Dynamical Systems Theory Organizers: Davide Faranda, Laboratoire des Sciences du Climat et de l’Environnement; London Mathematical Laboratory Théo Lacombe, Université Gustave Eiffel Nina Otter, Queen Mary University of London Kristian Strommen, Department of Physics, University of Oxford

Week 3b: June 30–July 6, 2024 Homotopical Combinatorics Organizers: Andrew Blumberg, Columbia University Michael Hill, University of California, Los Angeles Kyle Ormsby, Reed College Angélica Osorno, Reed College Constanze Roitzheim, University of Kent

This program is funded through a generous grant (funded by NSF award 1916439) from the National Science Foundation, the AMS, and private donors.

Learn more at: www.ams.org/mrc-24

MATH TALES

Some Colleges with Vulnerable Students Cull Math Programs Susan D’Agostino A young Black student beams1 in front of a chalkboard filled with math equations on the Chicago State University Mathematics Department webpage. Presumably she is pleased by the prospect of preparing for an advanced degree in mathematics or a career in banking, insurance, industry, or government. At least that is what the prose next to her image highlights as outcomes of the university’s BS in mathematics. But students at this 150-year-old, predominantly Black institution on the South Side of Chicago can no longer opt to major in mathematics. Current students will be allowed to finish, but the university has otherwise suspended enrollment in the math major program. This means that Chicago State students, approximately two-thirds (62 percent2) of whom receive Pell Grants (a marker of low income), will no longer be able to reap the intellectual or post-college career benefits that a math major often bestows. “You could draw about a two- to five-mile circle around our campus on the South Side of Chicago. That’s where Susan D’Agostino is a mathematician and Spencer Journalism Fellow at Columbia University. She may be reached through her website: https:// www.susandagostino.com. For permission to reprint this article, please contact: [email protected].

DOI: https://doi.org/10.1090/noti2806

80 percent of our students come from,” Mark Smith, the acting chair of the Computing, Information, Mathematical Sciences, and Technology Department at Chicago State University, said. Smith is a professor of music who sought to revitalize, not eliminate the math major. The incoming department chair will be a computer scientist. “The computer science program is running very, very healthily. They have upwards of 60 to 70 majors,” Smith added. Few institutions can be all things to all students. For this reason, colleges often identify programs that align with their values and target their specific student populations. At the same time, some financially struggling colleges have sought to improve their prospects by cutting academic programs in fields that include, but are not limited to, mathematics. But colleges that espouse missions of equity while eliminating math major programs may find their actions in conflict with their values. The mathematics major has a well-supported track record of providing students with an intellectually stimulating college pursuit that lays a foundation for a satisfying and potentially lucrative post-college career. Further, the US government has called for faster progress in providing opportunities for the “missing millions”3 of underrepresented individuals who might enter the mathematics workforce pipeline. This workforce includes those who have earned bachelor’s, master’s, or doctoral degrees in math.

1

https://www.csu.edu/cimst/math.htm https://scholarships360.org/colleges/illinois/chicago -state-university/affordability/ 2

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3

https://www.nsf.gov/nsb/publications/2020/nsb202015.pdf

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Math Tales Nonetheless, a small but noteworthy wave of mathematics major eliminations has swept across the United States in recent years. The hardest hit populations appear to be underrepresented students and those from low-income backgrounds. Leaders at colleges that face significant financial struggles have a duty to act, and acting often means cutting programs. But some college leaders, in justifying their choices to withdraw the math major as an option for their students, have put forth unsupported claims about math major outcomes. Some have argued that math program eliminations “revitalize academics” or protect students from dubious career outcomes, despite strong evidence to the contrary. Others have cited financial reasons, even when nearly all of the math major courses continue to be offered for students pursuing general education or science program requirements. “I didn’t feel that suspending admissions [in the math major] helps anything,” Smith said. “But I’ve not been able to prevail on that.”

“There are exactly two math courses that are only for math majors, and we don’t offer them both every year,” O’Donnol said in support of her position that savings from the cut would be minimal. In a given academic year, the department had offered either abstract algebra or real analysis, but not both. Those courses, which were predominantly pursued by math majors, will not be offered moving forward. But the net cut is one course per year. The National Science Board might contest the notion that cutting a math major at an Hispanic-serving institution better serves students. The number of individuals from underrepresented groups in the science, technology, engineering, and mathematics workforce has grown in the past decade. But the Board issued7 a call for a much faster increase so that the workforce better represents the population. To close the workforce talent gap, the number of Hispanic workers in these fields must triple from 2020 levels, while Black or African American workers must more than double.

Eliminating Math to “Better Serve Students”

Eliminating Math to “Maintain Financial Sustainability”

Marymount University, the first4 Hispanic-serving institution in Virginia, eliminated its math major this year, along with nine other majors in subjects including religion, philosophy, art, history, sociology, and English. Nick Munson, a spokesperson for the university, offered insight5 into the decision. “Overall, this is not because the university is suffering financially,” Munson told WUSA-TV. “It’s because these programs are all very low performing with low enrollment rates, and we’ve seen that over time. The student choices don’t lie. We’ve seen that year-to-year when you have a major with zero students in it. How can you sustain that? That wouldn’t be responsible.” The university plans to reallocate resources from the eliminated programs “to better serve our students and reflect their interests,” according to a statement.6 The institution does not offer a statistics or other quantitative major in its School of Science, Mathematics, and Engineering. But Danielle O’Donnol, associate professor of mathematics at Marymount, was unconvinced by the administration’s arguments. Students were pursuing the math major before it was cut, O’Donnol said. Also, given that math major courses support new majors in engineering and computer science, the program was poised to grow, she said.

Saint Xavier University, a private Roman Catholic university in Chicago, eliminated8 its math major in the spring of 2023. Saib Othman, Saint Xavier’s provost, declined to speak about the decision but offered a written statement. “Despite the elimination of the major, Saint Xavier University will continue to offer a math minor, general education courses, and service courses” for math education and computer science programs, Othman wrote. “We recognize math is a valuable subject in the age of technological advancements. However, enrollment in the general mathematics track has been declining in the state of Illinois while enrollment in applied fields like computer science has been growing.” When Saint Xavier eliminated the math major, six students were enrolled in the program. In contrast, nursing, biology, business, psychology, criminal justice, and education all enroll hundreds of majors, according to Othman. The institution’s list of academic programs on its website did not include statistics. The vast majority of Saint Xavier students (96 percent) receive financial aid, including more than half (53 percent9) who receive Pell Grants. Nearly half (49 percent10) of students at the institution are Latinx. Of the remaining, approximately one-third (32 percent) are White, 12

4 https://marymount.edu/blog/marymount-listed-as-first -hispanic-serving-institution-in-virginia/ 5 https://www.wusa9.com/article/news/education/marymount -university-approves-cut-several-liberal-arts-majors/65 -0d7b7c21-26ce-4b06-8ed0-da6c43518ada 6 https://wjla.com/news/local/marymount-university-major -cutbacks-minor-elimination-arlington-virginia-college -dmv-education-professors-students-stunned-9-nine-art -humanities-theology-catholic-sociology-math-english

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https://www.nsf.gov/nsb/publications/2020/nsb202015.pdf https://sxustudentmedia.com/saint-xavier-university -eliminates-four-majors/ 9 https://www.univstats.com/colleges/saint-xavier-university /ask/?question=what-is-the-average-grants-amount-at-saint -xavier-university 10 https://www.sxu.edu/about/sxu-at-glance.aspx 8

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Math Tales percent are African American, 2 percent are Asian, and 2 percent are multiracial. “[Saint Xavier students] view their education as an investment and are keenly interested in their return on this investment,” Othman wrote as part of a response explaining why math was among those majors that were cut. But college students who major in math are poised to pursue financially secure post-college pathways. That’s because the average base salary for those who earn an undergraduate math degree was approximately $83,000, according11 to PayScale. An undergraduate degree in nursing may offer a higher annual base salary ($92,00012) but an undergraduate degree in criminal justice is lower ($65,00013), according to PayScale. No doubt Henderson State University, which is part of Arkansas State University System, had finances in mind last year when Henderson State Chancellor Chuck Ambrose told14 the Arkansas Democrat Gazette of the institution’s $78 million debt. To stave off the threat of closure, Henderson eliminated15 25 degree programs, including mathematics, in 2022. Nearly half (43 percent) of students at the university are Pell Grant recipients. “We are a blue-collar school,” Carolyn Eoff, chair of the mathematics department at Henderson State University, said, adding that the institution serves a region of Arkansas that is poorer than many others. Majors such as education, aviation, and computer science survived the cuts, she said. “They did keep psychology . . . But my understanding is that [the psychology department’s] graduates don’t get the greatest jobs straight from the bachelor’s degree. . . . Our math majors in the statistics track have been very successful. They’ve commanded high salaries with bachelor’s degrees.” Eoff was concerned when the chancellor described college algebra as an “obstacle” to student success. She would have preferred for the department to downsize into a service department that supported students’ math literacy. But the entire math department was eliminated. As this story went to press, West Virginia University, a public university where nearly all students (95 percent16)

11 https://www.payscale.com/research/US/Degree=Bachelor_of _Science_(BS_%2F_BSc)%2C_Mathematics/Salary 12 https://www.payscale.com/research/US/Degree=Bachelor_of _Science_in_Nursing_(BSN)/Salary 13 https://www.payscale.com/research/US/Degree=Bachelor_of _Science_(BS_%2F_BSc)%2C_Criminal_Justice/Salary 14 https://www.arkansasonline.com/news/2022/may/12/cuts-to -degree-programs-and-faculty-at-henderson-state-university -explained 15 https://www.arkansasonline.com/news/2022/may/12/cuts-to -degree-programs-and-faculty-at-henderson-state-university -explained

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receive financial aid, announced that 25 programs,17 including mathematics, were under review for possible elimination. The review was necessary due to a $75 million budget shortfall, according to the institution’s president E. Gordon Gee. The university’s provost office stated that the goal18 of the review was to provide “a more focused academic program portfolio aligned with student demand, career opportunities, and market trends” along with efficient program delivery. Kevin Milans, associate professor of mathematics at West Virginia, in a communication to the AMS, deemed the timeline19 for programs to submit a self-study and possible appeal—essentially the summer months—“very short.” In a faculty senate meeting, Gee argued that the institution had “drifted away from student success” and that the eliminations would recommit the university to student success, as reported20 by Inside Higher Ed. But a faculty member who spoke with that newspaper contested Gee’s view. The cuts may be necessary due to the budget shortfall, but arguing that they benefit students was “disingenuous,” according to this faculty member.

Eliminating Math Because of “Hard-to-Discern” Career Outcomes In 2018, Benedict College, an historically Black institution in South Carolina where most students (83 percent21) are Pell grant recipients, also eliminated the mathematics major. History, religion and philosophy, sociology, political science, transportation and logistics engineering, and economics were also cut. The college also did not list a statistics major on its website. The seven majors that were cut all had “low numbers of student enrollment, low numbers graduating, and either hard-to-discern or nonexistent data as it relates to what happens to that student when they leave here,” Roslyn Artis, the college’s president, told22 the Post and Courier. “Unfortunately, many [Benedict students] will leave with some loan debt by virtue of the inability of their families to sustain them during college,” Artis said at the time in defense of the decision to cut mathematics and other majors. “So, we’re putting forth a full court press on 16 https://www.collegefactual.com/colleges/west-virginia -university/paying-for-college/financial-aid 17 https://provost.wvu.edu/academic-transformation/academic -program-portfolio-review 18 https://provost.wvu.edu/academic-transformation/academic -program-portfolio-review 19 https://transformation.wvu.edu/timeline 20 https://www.insidehighered.com/news/governance/executive -leadership/2023/06/23/distraught-west-virginia-u-faculty -push-back 21 https://www.raise.me/edu/benedict-college 22 https://www.postandcourier.com/free-times/news/local_and _state_news/benedict-to-eliminate-seven-majors-in-latest -big-change-at-college/article_97049aa7-3506-5b6a-88b5 -42b8c398a416.html

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Math Tales helping students make good decisions, helping them be placed in meaningful employment.” But the US Bureau of Labor Statistics projects23 that employment for all levels of math education will grow by 29 percent from 2021 to 2031. The chair of the Benedict College Computer Science, Physics, and Engineering Department did not respond to a request for comment. Despite Benedict’s decision to eliminate the math major, historically Black colleges and universities (HBCUs) have a strong track record of advancing equity in economic and educational opportunities for all of their students including, but not limited to, Black Americans. HBCUs contend with systemic barriers but produce one-quarter of Black graduates who earn science, technology, engineering, and mathematics degrees, according24 to remarks President Biden made in an August 2021 jobs report.

Eliminating Math with Scant Attention Paid At some institutions, math major programs appear to slip away with minimal public attention. “Cheyney University is Down to One Math Major,” a Philadelphia magazine headline broadcast in 2015. Cheyney, which draws half25 of its students from Philadelphia and where nearly threequarters of students (73%26) are Pell grant recipients, is part of the Pennsylvania State System of Higher Education. But the historically Black institution faced significant27 financial and enrollment struggles. Today, the university’s website28 lists no mathematics minor or major, and an online search for news about the decision turned up little information. The only mathematics professor listed on the institution’s faculty page29 did not respond to a request for comment.

Eliminating Math for “Academic Revitalization” Goucher College, where virtually all students (97 percent30) receive financial aid, offers a data science major for students “who want to delve into data-driven problems in the mathematical sciences.” But in 2018, the

23

https://www.bls.gov/ooh/math/home.htm https://www.whitehouse.gov/briefing-room/presidential -actions/2021/09/03/executive-order-on-white-house -initiative-on-advancing-educational-equity-excellence-and -economic-opportunity-through-historically-black-colleges -and-universities/ 25 https://www.phillymag.com/news/2015/08/03/cheyney -university-struggles/ 26 https://www.collegecalc.org/colleges/pennsylvania/cheyney -university-of-pennsylvania/ 27 https://www.phillymag.com/news/2015/08/03/cheyney -university-struggles/ 28 https://cheyney.edu/academics/programs/ 29 https://cheyney.edu/who-we-are/staff-and-faculty/ 30 https://www.goucher.edu/explore/facts-and-stats/ 24

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institution eliminated31 several liberal arts programs, including majors and minors in math, studio art, and religion. College leaders argued that the decision was not born from financial troubles32 and did not mark a shift away from the liberal arts. “We have long resisted the temptation to adopt more of the vocational programs currently in vogue with segments of the American public,” Jos´e Antonio Bowen, Goucher president, told33 Inside Higher Ed at the time, adding that dance was among the most popular majors on campus and that few students wanted to major in math. “Any new programs we offer will be interdisciplinary and in the liberal arts tradition. We have chosen this path carefully and strategically.” The college dubbed the program eliminations an “academic revitalization.”34 But the White House’s Office of Science and Technology Policy might contest the view that eliminating math revitalizes the academy. Last year, it broadcast35 a vision for enhanced mathematics pathways. “[O]ur science, technology, engineering, mathematics and medicine ecosystem shuts out and diverts away too many talented individuals, limiting opportunities for discovery and innovation, and our national potential for the greatest impact.” Some mathematics faculty, including those affected by the math program eliminations, indicated that reasons given for eliminating math programs do not always withstand scrutiny. But some have their own conjectures. “Colleges want students to pass,” Eoff of Henderson said, suggesting that academically challenging programs may be prime candidates for elimination. “If math departments uphold their rigor of mathematics, fewer students are able to get through the program. . . . We have a high bar, and we have done our best to help our students over that bar. But a lot of math departments are being pressured to lower standards to get more students through to be ‘successful.”’ Implicit bias, or unconscious beliefs, may contribute to persistent racial disparities in educational achievement,

31 https://www.insidehighered.com/news/2018/08/17/goucher -college-says-its-eliminating-liberal-arts-programs-such -math-physics-and 32 https://www.insidehighered.com/news/2018/08/17/goucher -college-says-its-eliminating-liberal-arts-programs-such -math-physics-and 33 https://www.insidehighered.com/news/2018/08/17/goucher -college-says-its-eliminating-liberal-arts-programs-such -math-physics-and 34 https://baltimorefishbowl.com/stories/goucher-college -cutting-nearly-a-dozen-paths-of-study-including-music-art -and-math-programs/ 35 https://www.whitehouse.gov/ostp/news-updates/2022/12/12 /equity-and-excellence-a-vision-to-transform-and-enhance -the-u-s-stemm-ecosystem/

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according36 to the Brookings Institution. Students may opt out of mathematics for a variety of reasons, including a sense that they do not belong or a lack of preparation, interest, or knowledge of career opportunities. Some of these reasons may disproportionately affect underrepresented and low-income students and could be addressed with interventions designed to benefit students. But without intervention, some colleges respond to a missing critical mass of math students by eliminating the mathematics major.

Susan D’Agostino Credits

Photo of Susan D’Agostino is courtesy of Chris Keeley.

36

https://www.brookings.edu/articles/educator-bias-is -associated-with-racial-disparities-in-student-achievement -and-discipline/

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FROM THE AMS SECRETARY

AMS Prizes and Awards Joint Prizes and Awards I. Martin Isaacs Prize for Excellence in Mathematical Writing The I. Martin Isaacs Prize is awarded for excellence in writing of a research article published in a primary journal of the AMS in the past two years.

About this Prize The prize focuses on the attributes of excellent writing, including clarity, grace, and accessibility; the quality of the research is implied by the article’s publication in Communications of the AMS, Journal of the AMS, Mathematics of Computation, Memoirs, Proceedings of the AMS, or Transactions of the AMS, and is therefore not a prize selection criterion. Professor Isaacs is the author of several graduate-level textbooks and of about 200 research papers on finite groups and their characters, with special emphasis on groups—such as solvable groups—that have an abundance of normal subgroups. He is a Fellow of the American Mathematical Society, and received teaching awards from the University of Wisconsin and from the School of Engineering at the University of Wisconsin. He is especially proud of his 29 successful PhD students. Next Prize: January 2025 Nomination Period: The deadline is March 31, 2024. Nomination Procedure: www.ams.org/isaacs-prize Nominations with supporting information should be submitted online. Nominations should include a letter of nomination, a short description of the work that is the basis of the nomination, and a complete bibliographic citation for the article being nominated.

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2024 MOS–AMS Fulkerson Prize The Fulkerson Prize Committee invites nominations for the Delbert Ray Fulkerson Prize, sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of US$1,500 each are presented at each (triennial) International Symposium of the MOS. The Fulkerson Prize is for outstanding papers in the area of discrete mathematics. The prize will be awarded at the 25th International Symposium on Mathematical Programming to be held in Montreal, Canada, in the summer of 2024. Eligible papers should represent the final publication of the main result(s) and should have been published in a recognized journal or in a comparable, well-refereed volume intended to publish final publications only, during the six calendar years preceding the year of the Symposium (thus, from January 2018 through December 2023). The prizes will be given for single papers, not series of papers or books, and in the event of joint authorship the prize will be divided. The term “discrete mathematics” is interpreted broadly and is intended to include graph theory, networks, mathematical programming, applied combinatorics, applications of discrete mathematics to computer science, and related subjects. While research work in these areas is usually not far removed from practical applications, the judging of papers will be based only on their mathematical quality and significance. Previous winners of the Fulkerson Prize are listed here: www.mathopt.org/?nav=fulkerson#winners. Further information about the Fulkerson Prize can be found at www.mathopt.org/?nav=fulkerson and https://www.ams.org/fulkerson-prize.

Notices of the American Mathematical Society

Volume 70, Number 10

Calls for Nominations & Applications

FROM THE AMS SECRETARY

The gift of connection to the mathematical community in three easy steps. Eligible purchasers will receive a special AMS blanket.

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ReConnect s sio Near na et ch w l

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AMS Membership

Pr of es

Give the Gift of

Please send your nominations (including reference to the nominated article and an evaluation of the work) by February 15, 2024 to the chair of the committee: Professor Daniel Spielman Email: [email protected]

Publ ducation ish

The Fulkerson Prize Committee consists of • Julia Böttcher (London School of Economics), MOS Representative • Rosa Orellana (Dartmouth College), AMS Representative • Dan Spielman (Yale University), Chair and MOS Representative

For more information:

www.ams.org/membership/givemembership

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Support for your research

APPLY FOR AMS FELLOWSHIPS Application period: August 15–November 8 The Joan and Joseph Birman Fellowship for Women Scholars

provides exceptionally talented women extra research support during their midcareer years. This fellowship program, established in 2017, is made possible by a generous gift from Joan and Joseph Birman. The most likely awardee is a midcareer woman, based at a US academic institution, with a well-established research record in a core area of mathematics. For more information see:

www.ams.org/birman-fellow

The AMS Claytor-Gilmer Fellowship is a fellowship created to further excellence in mathematical research and to help generate wider and sustained participation by Black mathematicians. The fellowship was established in 2021.

The most likely awardee is a midcareer Black mathematician based at a US institution whose achievements demonstrate significant potential for further contributions to mathematics.

The AMS Centennial Research Fellowship

provides research support for outstanding mathematical scientists who have held a doctoral degree for between three and twelve years. The primary selection criterion for the Centennial Fellowship is the excellence of the candidate’s research. For more information see:

www.ams.org/centfellow

For more information see:

www.ams.org/claytor-gilmer

Apply on MathPrograms.org

FROM THE AMS SECRETARY

Biennial Overview of AMS Honors

record, puby’s journal of et ci So e th as ecial s, the Notices, n, we list the sp io ct se is th Every two year In onors. riptions of view of AMS h give brief desc s; ip lishes an over sh ow ll fe st re, awards, and n, including pa io at rm fo lectures, prizes in er rth ide links for fu each; and prov cturers. cipients and le three vice rs (president, ce fi of e th of sts surer, and asso vernance consi cretaries, trea The Society go se e at ci l, so ci as n cretary, four tee of the Cou presidents, se utive Commit ec Ex l, ety ci ci n So ou t abou , the C information d ciate treasurer) n fi ay m u es links to Trustees. Yo which includ , ce and Board of an rn ve o /g ry www.ams.org entennial Histo AMS: A Semic governance at e th of ry re to la is C ond ronicle the h 938, by Raym –1 88 books that ch 18 y, et ci So ical Mathematical ican Mathemat of the American fty Years, Amer Fi nd co Se e th d History of Archibald, an t Pitcher. 988, by Everet Society, 1939–1 d secretaries. , treasurers, an ts en id es pr st a list of pa We begin with

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Boris Hasselb

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FROM THE AMS SECRETARY

Presidents 2022–2023 2021–2022 2019–2020 2017–2018 2015–2016 2013–2014 2011–2012 2009–2010 2007–2008 2005–2006 2003–2004 2001–2002 1999–2000 1997–1998 1995–1996 1993–1994 1991–1992 1989–1990 1987–1988 1985–1986 1983–1984 1981–1982 1979–1980 1977–1978 1975–1976 1973–1974 1971–1972 1969–1970 1967–1968 1965–1966 1963–1964 1961–1962 1959–1960 1957–1958 1955–1956

Bryna Kra Ruth Charney Jill C. Pipher Kenneth A. Ribet Robert L. Bryant David A. Vogan, Jr. Eric M. Friedlander George E. Andrews James G. Glimm James G. Arthur David Eisenbud Hyman Bass Felix E. Browder Arthur M. Jaffe Cathleen Synge Morawetz Ronald L. Graham Michael Artin William Browder George Daniel Mostow Irving Kaplansky Julia Bowman Robinson Andrew Mattel Gleason Peter David Lax R H Bing Lipman Bers Saunders Mac Lane Nathan Jacobson Oscar Zariski Charles Bradfield Morrey, Jr. Abraham Adrian Albert Joseph Leo Doob Deane Montgomery Edward James McShane Richard Dagobert Brauer Raymond Louis Wilder

1953–1954 1951–1952 1949–1950 1947–1948 1945–1946 1943–1944 1941–1942 1939–1940 1937–1938 1935–1936 1933–1934 1931–1932 1929–1930 1927–1928 1925–1926 1923–1924 1921–1922 1919–1920 1917–1918 1915–1916 1913–1914 1911–1912 1909–1910 1907–1908 1905–1906 1903–1904 1901–1902 1899–1900 1897–1898 1895–1896 1891–1894 1888–1890

Gordon Thomas Whyburn John von Neumann Joseph Leonard Walsh Einar Hille Theophil Henry Hildebrandt Marshall Harvey Stone Harold Calvin Marston Morse Griffith Conrad Evans Robert Lee Moore Solomon Lefschetz Arthur Byron Coble Luther Pfahler Eisenhart Earle Raymond Hendrick Virgil Snyder George David Birkhoff Oswald Veblen Gilbert Ames Bliss Frank Morley Leonard Eugene Dickson Ernest William Brown Edward Burr Van Vleck Henry Burchard Fine Maxime Bôcher Henry Seely White William Fogg Osgood Thomas Scott Fiske Eliakim Hastings Moore Robert Simpson Woodward Simon Newcomb George William Hill John Emory McClintock John Howard Van Amringe

Treasurers 2021–2025 2011–2021 1999–2010 1974–1998 1965–1973 1949–1964 1938–1948 1937 1930–1936 1921–1929 1908–1920 1900–1907 1897–1899 1895, 1896 1892–1894 1890, 1891

Douglas Ulmer Jane M. Hawkins John M. Franks Franklin P. Peterson W. T. Martin A. E. Meder, Jr. B. P. Gill P. A. Smith G. W. Mullins W. B. Fite J. H. Tanner W. S. Dennett Harold Jacoby R. S. Woodward Harold Jacoby T. S. Fiske

Secretaries 2021–2025 2013–2021 1999–2012 1989–1998 1967–1988 1957–1966 1951–1956 1941–1950 1921–1940 1896–1920 1888–1895

Boris Hasselblatt Carla D. Savage Robert J. Daverman Robert M. Fossum Everett Pitcher J. W. Green E. G. Begle J. R. Kline R. G. D. Richardson F. N. Cole T. S. Fiske

AMS Presidents: A Timeline AMS presidents play a key role in leading the Society and representing the profession. Browse through the timeline to see each AMS president’s page, which includes the institution and date of his/her doctoral degree, a brief note about his/her academic career and honors, and links to more extensive biographical information.

www.ams.org/presidents

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FROM THE AMS SECRETARY

In addition to the Invited Addresses at the Joint Mathematics Meetings and at the eight Sectional Meetings each year, the Society sponsors or co-sponsors several special lectures.

AMS Special Lectures Colloquium Lectures www.ams.org/meet-colloquium-lect The Colloquium Lectures have a long and prestigious history. Maxime Bôcher and James Pierpont delivered the first sets of Lectures in 1896. They continue today as a highlight of the Joint Mathematics Meetings. AMS Lecture on Education www.ams.org/education-lect The AMS Council established this Lecture in 2020. Its goal is to inform our mathematics community on evidence-based practices, developing trends, research, and scholarship in all educational settings that are of particular relevance to mathematicians and mathematics departments. Einstein Public Lectures in Mathematics www.ams.org/meet-einstein-lect The Einstein Lectures, created by the AMS in 2005 to celebrate the one-hundredth anniversary of Einstein’s annus mirabilis, are given at AMS Sectional Meetings. AMS Erdo˝s Lecture for Students www.ams.org/meet-erdos-lect This Lecture is named for the prolific mathematician Paul Erdo˝s (1913–1996) and was given at Sectional Meetings annually since 1999. From 2022 onward, the Erdo˝s Lecture for Students is an invited address at the Joint Mathematics Meetings. Josiah Willard Gibbs Lectures www.ams.org/meet-gibbs-lect These invited Lectures are of popular nature, directed at those who are not professional mathematicians. The Society established the Gibbs Lectures in 1923. The Maryam Mirzakhani Lectures www.ams.org/meet-mirzakhani-lect The AMS Council established this Lecture in 2018 to honor Maryam Mirzakhani (1977–2017), the first woman and the first Iranian to win a Fields Medal. November 2023

Arnold Ross Lectures www.ams.org/ross-lectures Created by the AMS at the encouragement of Paul Sally, these annual Lectures are aimed at talented high school mathematics students to stimulate their interest in mathematics beyond the traditional classroom. John von Neumann Lecture www.ams.org/von-neumann-lect The AMS Council established this Lecture in 2021 to honor John von Neumann (1903–1957). The inaugural Lecture was delivered by Anna Gilbert, Yale University, at the 2022 Joint Mathematics Meetings. This is supported in part by the C. V. Newsom Fund.

Joint Lectures NEW! AAAS–AMS Lecture www.ams.org/ams-aaas-lect Organized jointly by the AMS and Section A of AAAS, these Lectures feature mathematical scientists who embody the importance of mathematics in improving the human condition. AMS Invited Address at SIAM Annual Meeting www.ams.org/ams-siam-lect AMS selects a lecturer to deliver an address at the SIAM Annual Meeting. AMS–MAA Joint Lectures at MathFests www.ams.org/ams-mathfest These joint addresses, delivered annually at MathFest, are historical or expository in character. AMS–MAA–SIAM Gerald and Judith Porter Lectures www.ams.org/porter-lect The Porter Lecture on a mathematical topic accessible to the broader community is given each year at the Joint Mathematics Meetings.

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FROM THE AMS SECRETARY The AMS–NZMS Maclaurin Lectureship www.ams.org/maclaurin-lectures This Lectureship was a six-year reciprocal exchange between the New Zealand Mathematical Society and the American Mathematical Society. The Current Events Bulletin www.ams.org/current-events-bulletin Organized by David Eisenbud, this JMM event features speakers surveying some of the most interesting current developments in mathematics, pure and applied.

MAA–SIAM–AMS Hrabowski–Gates–Tapia–McBay Lecture and Panel at the JMM www.ams.org/hgtm-lect Through multiple mechanisms, these sessions expect to facilitate and accelerate the participation of scientists in the building of sustainable communities of mathematicians and mathematical scientists. AWM–AMS Noether Lectures www.awm-math.org/noether-lectures The Noether Lecture, given each year at the Joint Mathematics Meetings, honors women who have made fundamental and sustained contributions to the mathematical sciences.

AMS Prizes AMS Prizes recognize outstanding achievement in mathematics, exceptional public service in support of research and education in the mathematical sciences, and significant contributions to the public understanding of mathematics. The Society added new prizes starting in 2024: Ivo and Renata Babuška Thesis Prize and Elias M. Stein Prize for New Perspectives in Analysis; 2025: Elias M. Stein Prize for Transformative Exposition and I. Martin Isaacs Prize for Excellence in Mathematical Writing; and 2026: Elias M. Stein Mentoring Award. The prizes below are awarded at the Awards Celebration at the Joint Mathematics Meetings. Ivo and Renata Babuška Thesis Prize www.ams.org/babuska-prize Awarded annually to the author of an outstanding PhD thesis in mathematics, interdisciplinary in nature, possibly with applications to other fields. Elias M. Stein Prize for NEW! New Perspectives in Analysis www.ams.org/stein-prize Awarded for the development of groundbreaking methods in analysis which demonstrate promise to revitalize established areas or create new opportunities for mathematical discovery. Elias M. Stein Prize for NEW! Transformative Exposition Awarded for a written work or works, such as a book, survey, or exposition, that transforms the mathematical community’s understanding of the subject or reshapes the way it is taught. NEW!

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I. Martin Isaacs Prize for Excellence in Mathematical Writing www.ams.org/isaacs-prize Awarded for excellence in writing of a research article published in a primary journal of the AMS in the past two years. NEW! Elias M. Stein Mentoring Award Recognizes mathematicians who have demonstrated a sustained commitment to the training and advancement of future generations of mathematicians. Bôcher Memorial Prize www.ams.org/bocher-prize For a notable paper in analysis. Chevalley Prize in Lie Theory www.ams.org/chevalley-prize For notable work in Lie theory. Frank Nelson Cole Prize in Algebra www.ams.org/cole-prize-algebra For a notable paper in algebra. Frank Nelson Cole Prize in Number Theory www.ams.org/cole-prize-number-theory For a notable paper in number theory. NEW!

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FROM THE AMS SECRETARY Levi L. Conant Prize www.ams.org/conant-prize For an expository paper published in either the Notices of the AMS or the Bulletin of the AMS. Mary P. Dolciani Prize for Excellence in Research www.ams.org/dolciani-prize Recognizes a mathematician from a department that does not grant a PhD who has an active research program in mathematics and a distinguished record of scholarship. Joseph L. Doob Prize www.ams.org/doob-prize For a single, relatively recent, outstanding research book. Leonard Eisenbud Prize for Mathematics and Physics www.ams.org/eisenbud-prize For work that brings mathematics and physics closer together. Ciprian Foias Prize in Operator Theory www.ams.org/foias-prize For notable work in operator theory published during the preceding six years in a recognized, peer-reviewed venue. Ulf Grenander Prize in Stochastic Theory and Modeling www.ams.org/grenander-prize For theoretical and applied contributions in stochastic theory and modeling. E. H. Moore Research Article Prize www.ams.org/moore-prize For a research article appearing in one of the AMS primary research journals. David P. Robbins Prize www.ams.org/robbins-prize For a paper on novel research in algebra, combinatorics, or discrete mathematics. Bertrand Russell Prize of the AMS www.ams.org/russell-prize For research or service contributions of mathematicians or related professionals to promoting good in the world, recognizing the various ways that mathematics furthers human values. Ruth Lyttle Satter Prize in Mathematics www.ams.org/satter-prize For an outstanding contribution to mathematics research by a woman. Leroy P. Steele Prizes www.ams.org/steele-prize • Lifetime Achievement • Mathematical Exposition • Seminal Contribution to Research Oswald Veblen Prize in Geometry www.ams.org/veblen-prize For a notable research memoir in geometry or topology.

November 2023

Albert Leon Whiteman Memorial Prize www.ams.org/whiteman-prize For notable exposition and exceptional scholarship in the history of mathematics.

AMS Awards and Fellowships NEW! Stefan Bergman Fellowship www.ams.org/bergman-fellow Established in 2023 with the proceeds of the Stefan Bergman Trust to support the advancement of the research portfolio of a mathematician who specializes in the areas of real analysis, complex analysis, or partial differential equations. Joan and Joseph Birman Fellowship for Women Scholars www.ams.org/Birman-fellow Seeks to address the paucity of women at the highest levels of research in mathematics by giving exceptionally talented women extra research support during their mid-career years. Centennial Fellowship www.ams.org/centennial-fellow For outstanding mathematicians to help further their careers in research, with a focus on candidates who have not had extensive fellowship support in the past. Claytor-Gilmer Fellowship www.ams.org/claytor-gilmer Established to further excellence in mathematics research and to help generate wider and sustained participation by Black mathematicians. AMS Congressional Fellowship www.ams.org/ams-aaas-congressional-fellowship Selected Fellows spend a year working for a member of Congress or a congressional committee as a special legislative assistant in policy areas requiring scientific and technical input. AMS Mass Media Fellowship www.ams.org/massmediafellow In affiliation with the American Association for the Advancement of Science (AAAS), the AMS sponsors ten-week fellowships for graduate students in mathematics to work full time over the summer as reporters, researchers, and production assistants in US mass media organizations—radio and TV stations, newspapers, and magazines. Fellows of the American Mathematical Society www.ams.org/ams-fellows Recognizes members who have made outstanding contributions to the creation, exposition, advancement, and utilization of mathematics.

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FROM THE AMS SECRETARY Award for Distinguished Public Service www.ams.org/public-service-award For a research mathematician who has made a distinguished contribution to the mathematics profession. Award for an Exemplary Program or Achievement in a Mathematics Department www.ams.org/department-award For a department which has distinguished itself by undertaking an unusual or particularly effective program of value to the mathematics community. Award for Impact on the Teaching and Learning of Mathematics www.ams.org/impact For mathematicians who have made significant contributions of lasting value to mathematics education. Karl Menger Memorial Awards www.ams.org/menger-award For mathematically oriented projects presented at the International Science and Engineering Fair. Mathematics Programs that Make a Difference Award www.ams.org/make-a-diff-award Aims to bring more persons from underrepresented backgrounds into some portion of the pipeline beginning at the undergraduate level and leading to advanced degrees in mathematics and professional success, or retain them once in the pipeline. Undergraduate Opportunity Awards www.ams.org/trjitzinsky-award Provides assistance to students who have declared a major in mathematics at a college or university that is an institutional AMS member. Young Scholars Program www.ams.org/epsilon-award Supports existing summer programs for mathematically talented high school students.

Joint Prizes and Awards AMS–SIAM George David Birkhoff Prize in Applied Mathematics www.ams.org/birkhoff-prize Given jointly with the Society for Industrial and Applied Mathematics for an outstanding contribution to applied mathematics in the highest and broadest sense. Delbert Ray Fulkerson Prize www.ams.org/fulkerson-prize Given jointly with the Mathematical Optimization Society for outstanding papers in the area of discrete mathematics.

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AMS–EMS Mikhail Gordin Prize www.ams.org/gordin-prize Awarded to a mathematician working in probability or dynamical systems, with preference given to early career mathematicians from or professionally connected to an Eastern European country. JPBM Communications Award www.ams.org/jpbm-comm-award Given jointly with the Mathematical Association of America, the Society for Industrial and Applied Mathematics, and the American Statistical Association to reward and encourage communicators who, on a sustained basis, bring mathematical ideas and information to non-mathematical audiences. Mathematical Art Exhibition Award www.ams.org/art-exhibit-prize Given jointly with the Mathematical Association of America for aesthetically pleasing works that combine mathematics and art. Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student www.ams.org/morgan-prize Given jointly with the Mathematical Association of America and the Society for Industrial and Applied Mathematics for outstanding research in mathematics. Norbert Wiener Prize in Applied Mathematics www.ams.org/wiener-prize Given jointly with the Society for Industrial and Applied Mathematics for an outstanding contribution to applied mathematics in the highest and broadest sense.

Other Prizes and Awards Beal Prize www.ams.org/beal-prize This Prize, funded by D. Andrew Beal, is awarded for either a proof or a counterexample of the Beal Conjecture which is published in a refereed and respected mathematics journal. Maryam Mirzakhani Prize in Mathematics www.ams.org/nas-award Formerly The National Academy of Sciences Award in Mathematics, this Prize is awarded biennially for exceptional contributions to the mathematical sciences by a mid-career mathematician. Award for Outstanding Pi Mu Epsilon Student Paper Presentation www.ams.org/pme-award Made by the National Honorary Mathematics Society.

Notices of the American Mathematical Society Volume 70, Number 10

FROM THE AMS SECRETARY These prizes and awards have been discontinued: Stefan Bergman Prize www.ams.org/bergman-prize Awarded by the Bergman Trust in honor of his research in several complex variables, as well as the Bergman projection and the Bergman kernel function. This Prize has been discontinued. In 2023 the AMS allocated the Stefan Bergman Endowment to establish the Stefan Bergman Fellowship as the Society’s first fellowship specifically for early-career mathematicians. Leonard M. and Eleanor B. Blumenthal Award for the Advancement of Research in Pure Mathematics www.ams.org/blumenthal-award Presented to the individual deemed to have made the most substantial contribution in research in the field of pure mathematics, and who was deemed to have the potential for future production of distinguished research in such field.

Fredkin Foundation Prizes in Automatic Theorem Proving www.ams.org/atp-prizes The Fredkin Foundation asked the AMS to take over administration of these Prizes in the mid-1980s. Public Policy Award www.ams.org/profession/prizes-awards/ams-awards /public-policy-award Established in 2007 by the AMS to recognize a public figure for sustained and exceptional contributions to public policies that foster support for research, education, and innovation. Citation for Public Service www.ams.org/public-service-citation Created to provide encouragement and recognition for contributions to public service activities in support of mathematics.

American Mathematical Society

Policy on a Welcoming Environment (as adopted by the January 2015 AMS Council and modified by the January 2019 AMS Council) The AMS strives to ensure that participants in its activities enjoy a welcoming environment. In all its activities, the AMS seeks to foster an atmosphere that encourages the free expression and exchange of ideas. The AMS supports equality of opportunity and treatment for all participants, regardless of gender, gender identity or expression, race, color, national or ethnic origin, religion or religious belief, age, marital status, sexual orientation, disabilities, veteran status, or immigration status. Harassment is a form of misconduct that undermines the integrity of AMS activities and mission. The AMS will make every effort to maintain an environment that is free of harassment, even though it does not control the behavior of third parties. A commitment to a welcoming environment is

expected of all attendees at AMS activities, including mathematicians, students, guests, staff, contractors and exhibitors, and participants in scientific sessions and social events. To this end, the AMS will include a statement concerning its expectations towards maintaining a welcoming environment in registration materials for all its meetings, and has put in place a mechanism for reporting violations. Violations may be reported confidentially and anonymously to 855.282.5703 or at www.mathsociety.ethicspoint.com. The reporting mechanism ensures the respect of privacy while alerting the AMS to the situation. For AMS policy statements concerning discrimination and harassment, see the AMS Anti-Harassment Policy. Questions about this welcoming environment policy should be directed to the AMS Secretary.

November 2023

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Contact the AMS Development Office by phone: 401.455.4111 or email: [email protected]

AMS Reciprocity Agreements The American Mathematical Society (AMS) has reciprocity agreements with a number of mathematical organizations around the world. A current list of the reciprocating societies appears here; for full details of the agreements, see www.ams.org/membership/individual/mem-reciprocity. Allahabad Mathematical Society Argentina Mathematical Society Australian Mathematical Society Austrian Mathematical Society Azerbaijan Mathematical Society Balkan Society of Geometers Bangladesh Mathematical Society Belgian Mathematical Society Berliner Mathematische Gessellschaft Bharata Ganita Parisad Brazilian Mathematical Society Brazilian Society of Computational and Applied Mathematics Calcutta Mathematical Society Canadian Mathematical Society Catalan Society of Mathematicians Chilean Mathematical Society Colombian Mathematical Society Croatian Mathematical Society Cyprus Mathematical Society Danish Mathematical Society Dutch Mathematical Society Edinburgh Mathematical Society Egyptian Mathematical Society European Mathematical Society Finnish Mathematical Society German Mathematical Society German Society for Applied Maths & Mechanics Glasgow Mathematical Association Hellenic Mathematical Society Icelandic Mathematical Society November 2023

Indian Mathematical Society Indonesian Mathematical Society Iranian Mathematical Society Irish Mathematical Society Israel Mathematical Union Italian Mathematical Union János Bolyai Mathematical Society Korean Mathematical Society London Mathematical Society Luxembourg Mathematical Society Malaysian Mathematical Society Mathematical Society of France Mathematical Society of Japan Mathematical Society of the Philippines Mathematical Society of the Republic of China Mathematical Society of Serbia Mexican Mathematical Society Mongolian Mathematical Society Nepal Mathematical Society New Zealand Mathematical Society Nigerian Mathematical Society Norwegian Mathematical Society Palestine Society for Mathematical Sciences Parana’s Mathematical Society Polish Mathematical Society Portuguese Mathematical Society Punjab Mathematical Society Ramanujan Mathematical Society Romanian Mathematical Society Notices of the American Mathematical Society

Romanian Society of Mathematicians Royal Spanish Mathematical Society Saudi Association for Mathematical Sciences Singapore Mathematical Society Sociedad Matemática de la Republica Dominicana Sociedad Uruguaya de Matemática y Estadística Société Mathématiques Appliquées et Industrielles Society of Associations of Mathematicians and Computer Scientists of Macedonia Society of Mathematicians, Physicists, and Astronomers of Slovenia South African Mathematical Society Southeast Asian Mathematical Society Spanish Mathematical Society Swedish Mathematical Society Swiss Mathematical Society Tunisian Mathematical Society Turkish Mathematical Society Ukrainian Mathematical Society Union of Bulgarian Mathematicians Union of Czech Mathematicians and Physicists Union of Slovak Mathematicians and Physicists Vietnam Mathematical Society Vijñāna Parishad of India

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NEWS

AMS Updates AMS Mass Media Fellowship Open to Applicants Applications are open for the 2024 American Mathematical Society (AMS) Mass Media Fellowship. The deadline for applications is January 1, 2024. We urge mathematical sciences faculty to make their students and postdocs aware of this program. Each summer, the AMS sponsors a Mass Media Fellow through the Mass Media Science and Engineering Fellowship program organized by the American Association for the Advancement of Science (AAAS). The 10-week program is designed to improve public understanding of science and technology by placing advanced science, engineering, and mathematics students at media organizations nationwide. Mass Media Fellows use their academic training in the sciences to research, write, and report on today’s headlines. Fellows are embedded in national and local media outlets, where they learn how to communicate scientific topics clearly to a general audience and gain insight into newswriting and editorial decision-making. Fellows will receive a stipend of US $8,000, plus travel expenses to and from the AAAS in Washington, DC, and the media outlets where they will work as reporters, researchers, and production assistants. Applicants for the 2024 Mass Media Fellowship must a) be enrolled as upper-level undergraduate or graduate students; b) be postdoctoral trainees; or c) within one year of the completion of a) or b). Applicants must be in the life, physical, health, engineering, computer, or social sciences or mathematics and related fields. They must have outstanding written and oral communication skills and a strong interest in learning about the media.

DOI: https://doi.org/10.1090/noti2813

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For more information about the AMS Mass Media Fellowship, please visit https://www.ams.org/programs /ams-fellowships/media-fellow/massmediafellow. Apply at https://www.aaas.org/programs/mass -media-fellowship. —AMS Office of Government Relations

Applications Accepted for AMS Congressional Fellowship 2024–2025 The American Mathematical Society (AMS) will sponsor a Congressional Fellow from September 2024 through August 2025. The Fellow will spend the year working on the staff of either a member of Congress or a congressional committee, working in legislative and policy areas requiring scientific and technical input. The Fellow brings technical background and external perspective to the decision-making process in Congress. Prospective Fellows must be cognizant of and demonstrate sensitivity toward political and social issues and have a strong interest in applying personal knowledge toward solutions to societal problems. Now in its 19th year, the AMS Congressional Fellowship provides a unique public policy learning experience and demonstrates the value of science-government interaction. The program includes an orientation on congressional and executive branch operations and a yearlong seminar series on issues involving science, technology, and public policy. Applicants must have a PhD or an equivalent doctorallevel degree in the mathematical sciences by the application deadline (February 1, 2024). Applicants must be US citizens. Federal employees are not eligible. The Fellowship stipend is US $100,479 for the fellowship period with additional allowances for relocation and professional travel, as well as a contribution toward health insurance.

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 70, NUMBER 10

AMS Updates

NEWS Applicants must submit a statement expressing interest and qualifications for the AMS Congressional Fellowship as well as a current curriculum vitae. Candidates must have three letters of recommendation sent to the AMS by the February 1, 2024 deadline. For information and to apply, go to https://www.ams.org/government/government/ams -congressional-fellowship. Contact [email protected]

with questions. Deadline for receipt of applications is February 1, 2024. At JMM 2024: Learn more about this and other Washington, DC fellowship opportunities at the session on AMS DC-based policy and communications opportunities, 4:30 p.m., January 5, 2024. —AMS Office of Government Relations

NOVEMBER 2023

Deaths of AMS Members Ludo C. M. C. Buyst, of Belgium, died on June 9, 2021. Born on July 23, 1931, he was a member of the Society for 52 years. William J. Clover Jr., of Agusta, Missouri, died on March 29, 2018. Born in 1944, he was a member of the Society for 47 years. D. G. James, of State College, Pennsylvania, died on December 21, 2020. Born on March 18, 1938, he was a member of the Society for 56 years. Padmini T. Joshi, of Muncie, Indiana, died on January 2, 2023. Born on October 17, 1927, she was a member of the Society for 72 years. Clement F. Kent, of Canada, died on August 17, 2020. Born on March 15, 1927, he was a member of the Society for 66 years. Roland R. Kneece Jr., of Springfield, Virginia, died on June 7, 2023. Born on October 15, 1939, he was a member of the Society for 59 years. Henry A. Krieger, of Claremont, California, died on June 29, 2022. Born on May 7, 1936, he was a member of the Society for 57 years. L. Carl Leinbach, of Orrtanna, Pennsylvania, died on March 16, 2023. Born on March 29, 1941, he was a member of the Society for 57 years. John A. Thorpe, of Pensacola, Florida, died on January 18, 2021. Born on February 29, 1936, he was a member of the Society for 62 years.

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NEWS

Mathematics People Stroppel Wins 2023 Leibniz Prize Catharina Stroppel, University of Bonn, received the 2023 Gottfried Wilhelm Leibniz Prize in Pure Mathematics “in recognition of her excellent work in representation theory, especially on the topic of category theory,” according to a press release. The Leibniz prizes were awarded in Berlin, Germany, in March 2023 by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Germany’s central, self-governing, research funding organization. Of the ten Leibniz prizewinners, two each are from the humanities and social sciences, the natural sciences, and the engineering sciences, and four are from the life sciences. Each winner receives 2.5 million euros in prize money and is entitled to use these funds for their research work in any way they wish, without bureaucratic obstacles, for up to seven years. Stroppel studied mathematics and theology at the University of Freiburg, where she obtained her doctorate in mathematics. She has been professor of mathematics at the University of Bonn since 2008, deputy director of the Bonn International Graduate School since 2014, and a member of the University Senate since 2019. She has conducted research at Leicester, Aarhus, and Glasgow and has been a visiting professor at Chicago, Princeton, and elsewhere. Stroppel is involved in the Bonn Cluster of Excellence in Mathematics and delivered a plenary lecture at the International Congress of Mathematicians in 2022. —Leibniz Prize

IEEE Information Theory Society Names 2023 Awardees in Taipei The IEEE Information Theory Society announced its slate of 2023 awards at the 2023 IEEE International Symposium on Information Theory (ISIT ’23), held in June 2023 in Taipei, Taiwan. ISIT describes itself as “the flagship international conference dedicated to the advancement of information theory and related areas.” Andrew Barron of Yale University was named the recipient of the 2023 Claude E. Shannon Award of the IT Society for consistent and profound contributions to the field of information theory. Starting with the 2009 award, each Shannon Award winner has presented a Shannon Lecture at the following IEEE symposium. Barron will deliver his Shannon Lecture at ISIT 2024 in Athens, Greece. The Information Theory Society Paper Award is given annually for an outstanding publication in the fields of interest to the Society appearing anywhere during the preceding four calendar years. The 2023 award winner is “A unified framework for one-shot achievability via the Poisson matching lemma” by Cheuk Ting Li of the Chinese University of Hong Kong and Venkatachalam Anantharam of the University of California, Berkeley, which appeared in the IEEE Trans. Inf. Theory in February 2021. According to the abstract, “This paper extends the work of Li and El Gamal on Poisson functional representation for variablelength source coding settings, showing that the Poisson functional representation is a viable alternative to typicality for most problems in network information theory.” The 2023 ITSoc Paper Award will be presented formally at ISIT 2024. —IEEE Information Theory Society

DOI: https://doi.org/10.1090/noti2812

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NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 70, NUMBER 10

MATH VARIETY SHOW

Eight Math Scientists Receive Guggenheims In April 2023, the Board of Trustees of the John Simon Guggenheim Memorial Foundation awarded Guggenheim Fellowships to 171 individuals, including eight in the mathematical sciences, from an application pool of nearly 2,500 people. “These successful applicants were appointed on the basis of prior achievement and exceptional promise,” according to a press release. The full list of new fellows is posted at http://www.gf.org/. Mathematics: Lillian B. Pierce, Duke University Applied Mathematics: Erkki Somersalo, Case Western Reserve University Computer Science: Anima Anandkumar, California Institute of Technology; Venkatesan Guruswami, University of California, Berkeley; Ronitt Rubinfeld, Massachusetts Institute of Technology Physics: Prineha Narang, University of California, Los Angeles; Hirosi Ooguri, CalTech; Christopher Walter, Duke. Guruswami, Ooguri, and Pierce are fellows of and members of the American Mathematical Society.

Friday, January 5, 2024 JMM, San Francisco

Seeking Performers! • • • •

magic a cappella mime slam poetry

• juggling • dance • comedy

• and YOU!

—John Simon Guggenheim Memorial Foundation

AIM Changes Locations On July 10, 2023, the American Institute of Mathematics (AIM) began operating in its new home at the California Institute of Technology. AIM is now located on the eighth floor of Caltech Hall on the Caltech campus in Pasadena, CA. The newly renovated space has a lecture area and seminar rooms designed for AIM-style workshops and collaborative research. Established in 1994, AIM “strives to broaden participation in the mathematical sciences at every level, from supporting the research of professional mathematicians working on the most important mathematical problems of our day to encouraging young students to get excited about math and become the STEM professionals of the future,” according to the institute. Since 2002, AIM has received National Science Foundation funding to hold weeklong, focused workshops in all areas of the mathematical sciences. Each year, AIM hosts 20 workshops and more than 30 small research groups.

APPLY NOW

In association with

—American Institute of Mathematics

NOVEMBER 2023

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Classified Advertising

Employment Opportunities

MASSACHUSETTS Assistant Professor of Mathematics at Williams College

The Williams College Department of Mathematics and Statistics invites applications for a tenure-track position in applied mathematics, beginning fall 2024, at the rank of assistant professor (a more senior appointment is possible under special circumstances). The candidate should have a PhD in Applied Mathematics, Mathematics, or a closely related field by the time of appointment. We are seeking candidates who are committed to inclusive undergraduate education and show evidence and/or promise of excellence in teaching students from diverse backgrounds as well as a strong research program in applied mathematics that can engage undergraduate students. The candidate will join a department that actively supports interdisciplinary research and is expanding its existing applied curriculum. We welcome applications from all applied areas. Our department offers a vibrant undergraduate program with majors in mathematics (including an applied mathematics emphasis) and statistics; for more information, see https://math.williams.edu. The multidisciplinary environment is a rich and collegial setting for student education and faculty research. Williams College provides the opportunity to apply for student research assistant support, an annual allocation of funds to support travel and research, a shared computer cluster for parallel computation, a grants office, and several internal research funding opportunities. In addition, Williams College offers faculty participation

in the college’s professional development program First3 and in the NCFDD Faculty Success Program, and support through the newly established Rice Center for Teaching. Please submit your application via MathJobs: https:// www.mathjobs.org/jobs/list/22682. In your application materials, we ask you to address how your teaching, scholarship, mentorship and/or community service might support our commitment to diversity and inclusion. Your application should include the following components: 1. A cover letter. This should provide a brief summary of your professional experience and future goals, and should address your interest in working at Williams College in particular. 2. A current CV. 3. A research statement. 4. A teaching statement. This should address your teaching philosophy and experience, ways in which you foster an inclusive learning environment, and other reflections or relevant information you would like to share. 5. Three recommendation letters, at least one of which addresses your teaching experience. We encourage applications from members of underrepresented groups with respect to gender, race and ethnicity, religion, sexual orientation, disability status, socioeconomic background, and other axes of diversity. If you have questions about this position, contact the Chair of the Hiring Committee, Julie Blackwood (jcb5@williams .edu). Applications will be accepted until the position is filled, but all applications received by November 15, 2023 will be guaranteed full consideration. All offers of

The Notices Classified Advertising section is devoted to listings of current employment opportunities. The publisher reserves the right to reject any listing not in keeping with the Society’s standards. Acceptance shall not be construed as approval of the accuracy or the legality of any information therein. Advertisers are neither screened nor recommended by the publisher. The publisher is not responsible for agreements or transactions executed in part or in full based on classified advertisements. The 2023 rate is $3.65 per word. Advertisements will be set with a minimum one-line headline, consisting of the institution name above body copy, unless additional headline copy is specified by the advertiser. Headlines will be centered in boldface at no extra charge. Ads will appear in the language in which they are submitted. There are no member discounts for classified ads. Dictation over the telephone will not be accepted for classified ads. Upcoming deadlines for classified advertising are as follows: January 2024—October 27, 2023; February 2024—November 22, 2023; March 2024—December 29, 2023; April 2024—January 26, 2024; May 2024—February 23, 2024; June/July 2024—April 26, 2024; August 2024—May 24, 2024; September 2024—June 28, 2024; October 2024—July 26, 2024; November 2024—August 23, 2024; December 2024—September 27, 2024. US laws prohibit discrimination in employment on the basis of color, age, sex, race, religion, or national origin. Advertisements from institutions outside the US cannot be published unless they are accompanied by a statement that the institution does not discriminate on these grounds whether or not it is subject to US laws. Submission: Send email to [email protected].

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Classified Advertisements employment are contingent upon completion of a background check. Further information is available at https:// faculty.williams.edu/prospective-faculty /background-check-policy.

Williams College is a liberal arts institution located in the Berkshire Hills of western Massachusetts. The college has built its reputation on outstanding teaching and scholarship and on the academic excellence of its approximately 2,000 students. Please visit the Williams College website (http://www.williams.edu). Beyond meeting fully its legal obligations for non-discrimination, Williams College is committed to building a diverse and inclusive community where members from all backgrounds can live, learn, and thrive. 15

UTAH University of Utah, Department of Mathematics Current Positions (2023–2024)

The Department of Mathematics at the University of Utah invites applications for the following positions: 1. A tenure-track or tenured faculty position in Pure Mathematics at the level of Assistant Professor, Associate Professor, or Professor. 2. One or more Career-line faculty positions at the anticipated rank of Assistant Professor (Lecturer). 3. Several 3-year, termed research postdoctoral positions that involve research and teaching. Successful applicants might also receive additional opportunities through the Cecil Edmund Burgess and C. R. Wiley endowed Instructorships, as well as individual faculty research. 4. Several 3-year, termed postdoctoral positions in Applied Mathematics that involve research and teaching. This is in collaboration with the NSF Research Training Grant (RTG) “Optimization and Inversion for the 21st Century Workforce.” For more information on this grant see www.math.utah.edu/amrtg/. Applicants to this RTG program must have completed a PhD in Mathematics or a closely related field at the time of appointment and must be US citizens, nationals, or permanent residents. Outstanding candidates in areas of optimization, inverse problems, mathematics of materials and fluids, and mathematical climate science, as well as related areas in data science, machine learning and statistics will be considered. Applications from other fields of applied mathematics may be considered as long as they fit in the overarching goals of the RTG. The anticipated start date for these positions is July 1, 2024. All applications must be completed through MathJobs at https://www.mathjobs.org/jobs/UofUtah. Please refer

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to the individual job posts on MathJobs for a list of required application materials and search timeline for each position. The Carnegie Foundation has placed the University of Utah in their “highest research activity” category, and the University of Utah is the flagship institution of the Utah System of Higher Education. The University is located in Salt Lake City at the foot of the spectacular Wasatch Mountains. This location offers unparalleled opportunities for outdoor recreation, with nine world-class ski resorts within an hour of Salt Lake City, and five national parks only a few hours away. Salt Lake City is the center of the Wasatch Front metropolitan area, with a population of approximately 2.6 million residents. The city has extensive arts and cultural activities, and a major, newly renovated international airport. The area has received international recognition for its new light rail system, foodie culture, downtown renewal, increasing diversity, and welcoming culture. The University of Utah is an Affirmative Action/Equal Opportunity employer and does not discriminate based upon race, national origin, color, religion, sex, age, sexual orientation, gender identity/expression, status as a person with a disability, genetic information, or Protected Veteran status. Individuals from historically underrepresented groups, such as minorities, women, qualified persons with disabilities and protected veterans are encouraged to apply. Veterans’ preference is extended to qualified applicants, upon request and consistent with University policy and Utah state law. Upon request, reasonable accommodations in the application process will be provided to individuals with disabilities. To inquire about the University’s nondiscrimination or affirmative action policies or to request disability accommodation, please contact: Director, Office of Equal Opportunity and Affirmative Action, 201 S. Presidents Circle, Rm 135, (801)581-8365. Additional information can be found at http://www.utah.edu /nondiscrimination/. 17

Notices of the American Mathematical Society Volume 70, Number 10

Classified Advertisements

SOUTH KOREA Korea Institute for Advanced Study (KIAS) Call for Applications: Positions in Pure and Applied Mathematics

Founded in 1996, KIAS is committed to the excellence of research in basic sciences, namely mathematics, theoretical physics, and computational sciences, through high-quality research programs and a strong faculty body consisting of distinguished scientists and visiting scholars. The School of Mathematics and June E Huh Center for Mathematical Challenges boast internationally-renowned

November 2023

faculty members whose research bring prestigious visitors from diverse research areas, nurturing a research environment that encourages interaction and collaboration not only on-campus but beyond. Qualified, outstanding candidates in the field are encouraged to frequently check Mathjobs.org and KIAS Jobs website (https://jobs.kias.re.kr), where detailed information is updated when faculty and postdoctoral research fellow positions become available.

Notices of the American Mathematical Society

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NEW BOOKS

New Books Offered by the AMS Analysis Teaching and Learning with Primary Source Projects Real Analysis,Topology, and Complex Variables Janet Heine Barnett, Colorado State University Pueblo, David K. Ruch, Metropolitan State University of Denver, CO, and Nicholas A. Scoville, Ursinus College, Collegeville, PA, Editors “It appears to me that if one wants to make progress in mathematics one should study the masters and not the pupils.” —Niels Henrik Abel Recent pedagogical research has supported Abel’s claim of the effectiveness of reading the masters. Students exposed to historically based pedagogy see mathematics not as a monolithic assemblage of facts but as a collection of mental processes and an evolving cultural construct built to solve actual problems. Exposure to the immediacy of the original investigations can inspire an inquiry mindset in students and lead to an appreciation of mathematics as a living intellectual activity. TRIUMPHS (Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources) is an NSF-funded initiative to design materials that effectively harness the power of reading primary historical documents in undergraduate mathematics instruction. Teaching and Learning with Primary Source Projects is a collection of 24 classroom modules (PSPs) produced by TRIUMPHS that incorporate the reading of primary source excerpts to teach core mathematical topics. The selected excerpts are intertwined with thoughtfully designed student tasks that prompt students to actively engage with and explore the source material. Rigorously classroom tested and scrupulously edited to comply with the standards developed by the TRIUMPHS project, each of the PSPs in this volume can be inserted directly into a course in real analysis, 1736

complex variables, or topology and used to replace a standard textbook treatment of core course content. The volume also contains a comprehensive historical overview of the sociocultural and mathematical contexts within which the three subjects developed, along with extensive implementation guidance. Students and faculty alike are afforded a deeper classroom experience as they heed Abel’s advice by studying today’s mathematics through the words of the masters who brought that mathematics to life. This item will also be of interest to those working in geometry and topology. Classroom Resource Materials, Volume 71 December 2023, 456 pages, Softcover, ISBN: 978-1-47046989-4, LC 2023026255, 2020 Mathematics Subject Classification: 01A55, 01A60, 26–01, 26–03, 30–00, 30–03, 54–01, 54–03, 30–01, List US$65, AMS Individual member US$48.75, AMS Institutional member US$52, MAA members US$48.75, Order code CLRM/71 bookstore.ams.org/clrm-71

Geometry and Topology Multidimensional Residue Theory and Applications

Mathematical Surveys and Monographs Volume 275

Multidimensional Residue Theory and Applications

Alekos Vidras, University of Cyprus, Nicosia, and Alain Yger, University of Bordeaux, Talence, France

Alekos Vidras Alain Yger

Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundle-valued current maps. This point of view offers versatility and flexibility to the tools

Notices of the American Mathematical Society

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NEW BOOKS and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briançon–Skoda theorem and Hilbert’s Nullstellensatz, to arithmetic intersection theory and to tropical geometry. This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the one-dimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra.

un lien concret entre des objets de nature topologique et algébrique. This book provides a detailed exposition of the material presented in a series of lectures given in 2020 by Prof. Nicolas Bergeron while he held the Aisenstadt Chair at the CRM in Montréal. The topic is a broad generalization of certain classical identities such as the addition formulas for the cotangent function and for Eisenstein series. The book relates these identities to the cohomology of arithmetic subgroups of the general linear group. It shows that the relations can be made explicit using the theory of higher rank modular symbols, ultimately unveiling a concrete link between topological and algebraic objects.

Mathematical Surveys and Monographs, Volume 275 December 2023, approximately 546 pages, Softcover, ISBN: 978-1-4704-7112-5, 2020 Mathematics Subject Classification: 13Pxx, 14Q25, 32–02, 32A05, 32A10, 32A26, 32A27, 32Cxx, 32C05, 42Bxx, List US$129, AMS members US$103.20, MAA members US$116.10, Order code SURV/275

Titles in this series are co-published with the Centre de recherches mathématiques.

bookstore.ams.org/surv-275

Centre de recherches mathématiques

Cocycles de groupe pour GLn et arrangements d’hyperplans Nicolas Bergeron Pierre Charollois Luis E. García

CRM Monograph Series, Volume 39 November 2023, 127 pages, Hardcover, ISBN: 978-1-47047411-9, 2020 Mathematics Subject Classification: 11F41, 11F67, 11F66, 11F75, 11F25, 57R20, 14N20, 32S22, List US$130, AMS members US$104, MAA members US$117, Order code CRMM/39 bookstore.ams.org/crmm-39

Number Theory Volume 39

This item will also be of interest to those working in geometry and topology.

Cocycles de groupe pour GLn et arrangements d’hyperplans Nicolas Bergeron, Ecole Normale Supérieure et Sorbonne Université, Paris, France, Pierre Charollois, Sorbonne Université, Paris, France, and Luis E. García, University College London, United Kingdom

New in Contemporary Mathematics Geometry and Topology C ONTEMPORARY M ATHEMATICS 790

Ce livre constitue un exposé détaillé de la série de cours donnés en 2020 par le Prof. Nicolas Bergeron, titulaire de la Chaire Aisenstadt au CRM de Montréal. L’objet de ce texte est une ample généralisation d’une famille d'identités classiques, notamment la formule d’addition de la fonction cotangente ou celle des séries d'Eisenstein. Le livre relie ces identités à la cohomologie de certains sous-groupes arithmétiques du groupe linéaire général. Il rend explicite ces relations au moyen de la théorie des symboles modulaires de rang supérieur, dévoilant finalement November 2023

Compactifications, Configurations, and Cohomology

Compactifications, Configurations, and Cohomology Peter Crooks, Utah State University, Logan, and Alexandru I. Suciu, Northeastern University, Boston, MA, Editors

This volume contains the proceedings of the Conference on Compactifications, Configurations, and Cohomology, held from October 22–24, 2021, at Northeastern University, Boston, MA. Some of the most active and fruitful mathematical research occurs at the interface of algebraic geometry, Peter Crooks Alexandru I. Suciu Editors

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NEW BOOKS representation theory, and topology. Noteworthy examples include the study of compactifications in three specific settings—algebraic group actions, configuration spaces, and hyperplane arrangements. These three types of compactifications enjoy common structural features, including relations to root systems, combinatorial descriptions of cohomology rings, the appearance of iterated blow-ups, the geometry of normal crossing divisors, and connections to mirror symmetry in physics. On the other hand, these compactifications are often studied independently of one another. The articles focus on new and existing connections between the aforementioned three types of compactifications, thereby setting the stage for further research. It draws on the discipline-specific expertise of all contributors, and at the same time gives a unified, self-contained reference for compactifications and related constructions in different contexts. This item will also be of interest to those working in algebra and algebraic geometry. Contemporary Mathematics, Volume 790 December 2023, 157 pages, Softcover, ISBN: 978-1-47046992-4, LC 2023014589, 2020 Mathematics Subject Classification: 14J42, 14L17, 14L30, 14M15, 14N20, 17B63, 32S22, 55N10, 55R80, 55U05, List US$130, AMS members US$104, MAA members US$117, Order code CONM/790 bookstore.ams.org/conm-790

New in Memoirs of the AMS

14F40, 14H10, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/288/1430 bookstore.ams.org/memo-288-1430

On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type David A. Craven, University of Birmingham, United Kingdom Memoirs of the American Mathematical Society, Volume 288, Number 1434 September 2023, 213 pages, Softcover, ISBN: 978-1-47046702-9, 2020 Mathematics Subject Classification: 20D06, 20E28, 20G41, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/288/1434 bookstore.ams.org/memo-288-1434

Analysis Euclidean Structures and Operator Theory in Banach Spaces Nigel J. Kalton, University of Missouri, Colombia, Emiel Lorist, Delft University of Technology, The Netherlands, and Lutz Weis, Karlsruhe Institute for Technology, Germany Memoirs of the American Mathematical Society, Volume 288, Number 1433 September 2023, 156 pages, Softcover, ISBN: 978-1-47046703-6, 2020 Mathematics Subject Classification: 47A60; 47A68, 42B25, 47A56, 46E30, 46B20, 46B70, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/288/1433

Algebra and Algebraic Geometry

bookstore.ams.org/memo-288-1433

Comparison of Relatively Unipotent Log de Rham Fundamental Groups

The Slice Spectral Sequence of a C4-Equivariant Height-4 Lubin–TateTheory

Bruno Chiarellotto, Università di Padova, Italy, Valentina Di Proietto, Paris, France, and Atsushi Shiho, University of Tokyo, Japan

Michael A. Hill, University of California, Los Angeles, XiaoLin Danny Shi, University of Chicago, IL, Guozhen Wang, Fudan University, Shanghai, China, and Zhouli Xu, University of California, San Diego

Geometry and Topology

Memoirs of the American Mathematical Society, Volume 288, Number 1430 September 2023, 111 pages, Softcover, ISBN: 978-1-47046706-7, 2020 Mathematics Subject Classification: 14F35;

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Memoirs of the American Mathematical Society, Volume 288, Number 1429 September 2023, 119 pages, Softcover, ISBN: 978-14704-7468-3, 2020 Mathematics Subject Classification: 55P91, 55P92, 55Q40, 55Q10, 55Q91, List US$85, AMS

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NEW BOOKS members US$68, MAA members US$76.50, Order code MEMO/288/1429 bookstore.ams.org/memo-288-1429

Proper Equivariant Stable Homotopy Theory Dieter Degrijse, Keylane BV, Copenhagen, Denmark, Markus Hausmann, Univeristät Bonn, Germany, Wolfgang Lück, Universität Bonn, Germany, Irakli Patchkoria, University of Aberdeen, United Kingdom, and Stefan Schwede, Universität Bonn, Germany

New AMS-Distributed Publications Differential Equations An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows

Memoirs of the American Mathematical Society, Volume 288, Number 1432 September 2023, 142 pages, Softcover, ISBN: 978-1-47046704-3, 2020 Mathematics Subject Classification: 55P91, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/288/1432 bookstore.ams.org/memo-288-1432

Probability and Statistics Analyticity Results in Bernoulli Percolation Agelos Georgakopoulos, University of Warwick, Coventry, United Kingdom, and Christoforos Panagiotis, Université de Genève, Switzerland Memoirs of the American Mathematical Society, Volume 288, Number 1431 September 2023, 111 pages, Softcover, ISBN: 978-1-47046705-0, 2020 Mathematics Subject Classification: 60K35, 82B43, 05C30, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/288/1431 bookstore.ams.org/memo-288-1431

Second Edition Alessio Figalli, ETH Zürich, Switzerland, and Federico Glaudo, Institute for Advanced Study, Princeton, NJ This book provides a self-contained introduction to optimal transport, and it is intended as a starting point for any researcher who wants to enter into this beautiful subject. The presentation focuses on the essential topics of the theory: Kantorovich duality, existence and uniqueness of optimal transport maps, Wasserstein distances, the JKO scheme, Otto’s calculus, and Wasserstein gradient flows. At the end, a presentation of some selected applications of optimal transport is given. Suitable for a course at the graduate level, the book also includes an appendix with a series of exercises along with their solutions. The second edition contains a number of additions, such as a new section on the Brunn-Minkowski inequality, new exercises, and various corrections throughout the text. This item will also be of interest to those working in probability and statistics. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

EMS Textbooks in Mathematics, Volume 26 May 2023, 146 pages, Hardcover, ISBN: 978-3-98547-0501, 2020 Mathematics Subject Classification: 49Q22, 60B05, 28A33, 35A15, 35Q35, 49N15, 28A50, List US$45, AMS members US$36, Order code EMSTEXT/26 bookstore.ams.org/emstext-26

November 2023

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NEW BOOKS

General Interest

Hurwitz’s Lectures on the Number Theory of Quaternions

Essays in Geometry Dedicated to Norbert A’Campo Athanase Papadopoulos, Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, France, Editor This volume consists of a collection of essays dedicated to Norbert A’Campo on the occasion of his 80th birthday. The subject is geometry in the broadest sense. The topics include hyperbolic and super hyperbolic geometry, 3-manifolds, metric geometry, mapping class groups, linear groups, Riemann surfaces, Teichmüller spaces, high-dimensional complex geometry, differential topology, symplectic geometry, singularity theory, number theory, algebraic geometry, dynamics, mathematical physics and philosophy of mathematics. The book gives a fairly comprehensive overview of the wealth of current research in geometry. This item will also be of interest to those working in geometry and topology. A publication of the European Mathematical Society. Distributed within the Americas by the American Mathematical Society.

IRMA Lectures in Mathematics and Theoretical Physics, Volume 34 August 2023, 1028 pages, Hardcover, ISBN: 978-3-98547024-2, 2020 Mathematics Subject Classification: 01–02, 01–06, 01A70, 14B05, 34M35, 57K20, 57R45, 32S05, 32S55, 57K12, 57R17, 57K30, 57K45, 00A30, 57K10, 57K16, 58K30, 51M15, 53C70, 00B15, List US$139, AMS members US$111.20, Order code EMSILMTP/34 bookstore.ams.org/emsilmtp-34

Nicola Oswald, Bergische Universität Wuppertal, Germany, and Jörn Steuding, Julius-Maximilians Universität, Würzburg, Germany Quaternions are non-commutative generalizations of the complex numbers, invented by William Rowan Hamilton in 1843. Their number-theoretical aspects were first investigated by Rudolf Lipschitz in the 1880s, and, in a streamlined form, by Adolf Hurwitz in 1896. This book contains an English translation of Hurwitz’s 1919 textbook on this topic as well as his famous 1-2-3-4 theorem on composition algebras. In addition, the reader can find commentaries that shed historical light on the development of this number theory of quaternions, for example, the classical preparatory works of Fermat, Euler, Lagrange and Gauss, to name but a few, the different notions of quaternion integers in the works of Lipschitz and Hurwitz, analogies to the theory of algebraic numbers, and the further development (including Dickson’s work in particular). The authors have implemented parts of the book in stand-alone courses, and they believe that the present book can also complement a course on algebraic number theory (with respect to a noncommutative extension of the rational numbers). This item will also be of interest to those working in number theory. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Heritage of European Mathematics, Volume 13 May 2023, 293 pages, Hardcover, ISBN: 978-3-98547-0112, 2020 Mathematics Subject Classification: 01A55; 01A60, 01A75, 11E25, 11R52, 13G05, 16–03, 17A75, List US$85, AMS members US$68, Order code EMSHEM/13 bookstore.ams.org/emshem-13

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NEW BOOKS

Mathematical Physics

Number Theory

An Introduction to the Mathematical Fluid Dynamics of Oceanic and Atmospheric Flows

Purely Arithmetic PDEs Over a p-Adic Field: δ-Characters and δ-Modular Forms

Robin S. Johnson, Newcastle University, UK

Alexandru Buium, University of New Mexico, Albuquerque, NM, and Lance Edward Miller, University of Arkansas, Fayetteville, AR

The study of the movement of the atmosphere and the oceans is intriguing, challenging, and important, particularly in the context of current concerns about the climate. The familiar and tested approach to these problems is based on the construction of model equations, tailored to address specific flow scenarios. In this book, the author presents a single, overarching approach which uses the thin-shell approximation—and nothing more—applied to the general equations of fluid dynamics. This allows a range of classical problems, and some new applications, to be accessed from a single formulation which retains all the relevant physical attributes, as well as the essential characteristics of the spherical geometry. The approximations and assumptions are clear, and higher-order terms are readily accessible. The main aim is to present the material in a mathematically consistent and robust fashion—in the applied sense—emphasising the systematic, asymptotic methods usually employed in mathematical fluid dynamics. This is not a textbook that introduces the physical principles underpinning the study of the oceans and the atmosphere. Rather, it is intended to enhance the more usual modelling approach to these studies and, more significantly, to introduce those with mathematical interests, but no expertise in these particular applications, to these types of problems. The book is suitable for researchers and students in the oceanic and atmospheric sciences, and for mathematicians, researchers, and students with an interest in the application of fluid dynamics to more complicated flow scenarios. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory, solutions can be defined in algebraically closed p-adic fields. As an application, the authors show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly, the authors construct and study “genuinely PDE” differential modular forms. As further applications, the authors derive a theorem of the kernel and a reciprocity theorem for arithmetic PDE Manin maps and also a finiteness Diophantine result for modular parameterizations. The authors also prove structure results for the spaces of “PDE differential modular forms defined on the ordinary locus”. They also produce a system of differential equations satisfied by their PDE modular forms based on Serre and Euler operators. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Memoirs of the European Mathematical Society, Volume 6 August 2023, 116 pages, Softcover, ISBN: 978-3-98547-0570, 2020 Mathematics Subject Classification: 11F32; 11F85, 11G07, 11G18, List US$75, AMS members US$60, Order code EMSMEM/6 bookstore.ams.org/emsmem-6

ESI Lectures in Mathematics and Physics, Volume 11 July 2023, 176 pages, Softcover, ISBN: 978-3-98547-0297, 2020 Mathematics Subject Classification: 76–02, 86–02, 76D05, 76D33, 76M45, 76N06, 76N30, 76U60, 86A05, 86A10, List US$55, AMS Individual member US$44, Order code EMSESILEC/11 bookstore.ams.org/emsesilec-11

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Meetings & Conferences of the AMS November Table of Contents The Meetings and Conferences section of the Notices gives information on all AMS meetings and conferences approved by press time for this issue. Please refer to the page numbers cited on this page for more detailed information on each event. Paid meeting registration is required to submit an abstract to a sectional meeting. Invited Speakers and Special Sessions are listed as soon as they are approved by the cognizant program committee; the codes listed are needed for electronic abstract submission. For some meetings the list may be incomplete. Information in this issue may be dated. The most up-to-date meeting and conference information can be found online at www.ams.org/meetings. Important Information About AMS Meetings: Potential organizers, speakers, and hosts should refer to https:// www.ams.org/meetings/meetings-general for general information regarding participation in AMS meetings and conferences. Abstracts: Speakers should submit abstracts on the easy-to-use interactive Web form. No knowledge of LaTeX is necessary to submit an electronic form, although those who use LaTeX may submit abstracts with such coding, and all math displays and similarly coded material (such as accent marks in text) must be typeset in LaTeX. Visit www.ams .org/cgi-bin/abstracts/abstract.pl . Questions about abstracts may be sent to [email protected]. Close attention should be paid to specified deadlines in this issue. Unfortunately, late abstracts cannot be accommodated.

Meetings in this Issue 2023 October 13–15

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2024 January 3–6 March 23–24 April 6–7 April 20–21 May 4–5 July 23–26 September 14–15 October 5–6 October 19–20 October 26–27 December 9–13

San Francisco, California (JMM 2024) Tallahassee, Florida Washington, DC Milwaukee, Wisconsin San Francisco, California Palermo, Italy San Antonio, Texas Savannah, Georgia Albany, New York Riverside, California Auckland, New Zealand

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2025 January 8–11 April 5–6

Seattle, Washington (JMM 2025) Hartford, Connecticut

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2026

Associate Secretaries of the AMS Central Section: Betsy Stovall, University of Wisconsin– Madison, 480 Lincoln Drive, Madison, WI 53706; email: [email protected]; telephone: (608) 262-2933. Eastern Section: Steven H. Weintraub, Department of Mathematics, Lehigh University, Bethlehem, PA 180153174; email: [email protected]; telephone: (610) 758-3717. Southeastern Section: Brian D. Boe, Department of Mathematics, University of Georgia, 220 D W Brooks Drive, Athens, GA 30602-7403; email: [email protected]; telephone: (706) 542-2547. Western Section: Michelle Manes, University of Hawaii, Department of Mathematics, 2565 McCarthy Mall, Keller 401A, Honolulu, HI 96822; email: [email protected]; telephone: (808) 956-4679.

Mobile, Alabama

January 4–7

Washington, DC (JMM 2026)

p. 1756

The AMS strives to ensure that participants in its activities enjoy a welcoming environment. Please see our full Policy on a Welcoming Environment at https://www.ams .org/welcoming-environment-policy.

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MEETINGS & CONFERENCES

Meetings & Conferences of the AMS IMPORTANT information regarding meetings programs: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See https://www.ams.org/meetings. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL. New: Sectional Meetings Require Registration to Submit Abstracts. In an effort to spread the cost of the sectional meetings more equitably among all who attend and hence help keep registration fees low, starting with the 2020 fall sectional meetings, you must be registered for a sectional meeting in order to submit an abstract for that meeting. You will be prompted to register on the Abstracts Submission Page. In the event that your abstract is not accepted or you have to cancel your participation in the program due to unforeseen circumstances, your registration fee will be reimbursed.

Mobile, Alabama University of South Alabama October 13–15, 2023 Friday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: Volume 44, Issue 4

Meeting #1190

Deadlines

Southeastern Section Associate Secretary for the AMS: Brian D. Boe

For organizers: Expired For abstracts: Expired

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html.

Invited Addresses Theresa Anderson, Carnegie Mellon, Number theory and friends: a mathematical journey. Laura Ann Miller, University of Arizona, Flows around some soft corals. Cornelius Pillen, University of South Alabama, Lifting to tilting: modular representations of algebraic groups and their Frobenius kernels.

Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl. Advances in Extremal Combinatorics, Joseph Guy Briggs, Auburn University, and Chris Cox, Iowa State University. Analysis, Computation, and Applications of Stochastic Models, Yukun Li, University of Central Florida, Feng Bao, Florida State University, Xiaobing Feng, The University of Tennessee, Junshan Lin, Auburn University, and Liet Anh Vo, University of Illinois, Chicago. November 2023

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MEETINGS & CONFERENCES Categorical Representations, Quantum Algebra, and Related Topics, Arik Wilbert, University of South Alabama, Mee Seong Im, United States Naval Academy, Annapolis, and Bach Nguyen, Xavier University of Louisiana. Combinatorics and Geometry Related to Representation Theory, Markus Hunziker and William Erickson, Baylor University. Cyberinfrastructure for Mathematics Research & Instruction, Steven Craig Clontz, University of South Alabama, and Tien Chih, Oxford College of Emory University. Discrete Geometry and Geometric Optimization, Andras Bezdek, Auburn University, Auburn AL, Ferenc Fodor, University of Szeged, and Woden Kusner, University of Georgia, Athens GA. Dynamics and Equilibria of Energies, Ryan W Matzke, Technische Universität Graz, and Liudmyla Kryvonos, Vanderbilt University. Dynamics of Fluids, I. Kukavica, University of Southern California, Dallas Albritton, Princeton University, and Wojciech S. Ozanski, Florida State University. Ergodic Theory and Dynamical Systems, Mrinal Kanti Roychowdhury, School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, and Joanna Marie Furno, University of South Alabama. Experimental Mathematics in Number Theory and Combinatorics, Armin Straub, University of South Alabama, Brandt Kronholm, University of Texas Rio Grande Valley, and Luis A. Medina, University of Puerto Rico. Extremal and Probabilistic Combinatorics, Sean English, University of North Carolina Wilmington, and Emily Heath, Iowa State University. New Directions in Noncommutative Algebras and Representation Theory, Jonas T. Hartwig, Iowa State University, and Erich C. Jauch, Westminster College. Number Theory and Friends, Robert James Lemke Oliver, Tufts University, Theresa Anderson, Carnegie Mellon, Ayla Gafni, University of Mississippi, and Edna Luo Jones, Duke University. Recent Advances in Low-dimensional and Quantum Topology, Christine Ruey Shan Lee, Texas State University, and Scott Carter, University of South Alabama. Recent Developments in Graph Theory, Andrei Bogdan Pavelescu, University of South Alabama, and Kenneth Roblee, Troy University. Recent Progress in Numerical Methods for PDEs, Muhammad Mohebujjaman, Texas A&M International University, Leo Rebholz, Clemson University, and Mengying Xiao, University of West Florida. Representation Theory of Finite and Algebraic Groups, Daniel K Nakano, University of Georgia, Pramod N. Achar, Louisiana State University, and Jonathan R. Kujawa, University of Oklahoma. Rings, Monoids, and Factorization, Jim Coykendall, Clemson University, and Scott Chapman, Sam Houston State University. Theory and Application of Parabolic PDEs, Wenxian Shen and Yuming Paul Zhang, Auburn University. Topics in Harmonic Analysis and Partial Differential Equations, Jiuyi Zhu and Phuc Cong Nguyen, Louisiana State University.

Contributed Paper Sessions AMS Contributed Paper Session, Brian D. Boe, University of Georgia.

San Francisco, California Moscone North/South, Moscone Center January 3–6, 2024

Issue of Abstracts: Volume 45, Issue 1

Wednesday – Saturday

Deadlines

Meeting #1192

For organizers: Expired For abstracts: Expired

Associate Secretary for the AMS: Michelle Ann Manes Program first available on AMS website: To be announced

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /national.html.

Joint Invited Addresses Maria Chudnovsky, Princeton University, What Makes a Problem Hard? (MAA-AMS-SIAM Gerald and Judith Porter Public Lecture). 1744

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MEETINGS & CONFERENCES Anne Schilling, University of California, Davis, The Ubiquity of Crystal Bases (AWM-AMS Noether Lecture). Peter M Winkler, Dartmouth College, Permutons (AAAS-AMS Invited Address). Kamuela E. Yong, University of Hawaii West Oahu, When Mathematicians Don’t Count (MAA-SIAM-AMS HrabowskiGates-Tapia-McBay Lecture).

AMS Invited Addresses Ruth Charney, Brandeis University, From Braid Groups to Artin Groups (AMS Retiring Presidential Address). Daniel Erman, University of Hawaii, From Hilbert to Mirror Symmetry. Suzanne Marie Lenhart, University of Tennessee, Knoxville, Natural System Management: A Mathematician’s Perspective (AMS Josiah Willard Gibbs Lecture). Ankur Moitra, Massachusetts Institute of Technology, Learning from Dynamics (von Neumann Lecture). Kimberly Sellers, North Carolina State University, Dispersed Methods for Handling Dispersed Count Data. Terence Tao, UCLA, Machine Assisted Proof (AMS Colloquium Lecture I - Terence Tao, University of California, Los Angeles). Terence Tao, UCLA, Translational tilings of Euclidean space (AMS Colloquium Lecture II - Terence Tao, University of California, Los Angeles). Terence Tao, UCLA, Correlations of multiplicative functions (AMS Colloquium Lecture III - Terence Tao, University of California, Los Angeles). John Urschel, MIT, From Moments to Matrices (AMS Erdo˝s Lecture for Students). Suzanne L Weekes, SIAM, To be announced (AMS Lecture on Education). Melanie Matchett Wood, Harvard University, An Application of Probability Theory for Groups to 3-Dimensional Manifolds (AMS Maryam Mirzakhani Lecture).

AMS Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://jointmathematicsmeetings.org/meetings/abstracts/abstract.pl?type=jmm. Some sessions are cosponsored with other organizations. These are noted within the parenthesis at the end of each listing, where applicable. Advances in Analysis, PDE’s and Related Applications, Tepper L. Gill, Howard University, E. Kwessi, Trinity University, and Henok Mawi, Howard University (Washington, DC, US). Advances in Coding Theory, Emily McMillon, Rice University, Christine Ann Kelley and Tefjol Pllaha, University of Nebraska - Lincoln, and Mary Wootters, Stanford University. Algebraic Approaches to Mathematical Biology, Nicolette Meshkat, Santa Clara University, Cash Bortner, California State University, Stanislaus, and Anne Shiu, Texas A&M University. Algebraic Structures in Knot Theory, Sam Nelson, Claremont McKenna College, and Neslihan Gügümcu, Izmir Institute of Technology in Turkey. AMS-AWM Special Session for Women and Gender Minorities in Symplectic and Contact Geometry and Topology, Sarah Blackwell, Max Planck Institute for Mathematics, Luya Wang, University of California, Berkeley, and Nicole Magill, Cornell University (AMS-AWM). Analysis and Differential Equations at Undergraduate Institutions, Evan Randles, Colby College, and Lisa Naples, Macalester College, Saint Paul MN USA. Applications of Extremal Graph Theory to Network Design, Kelly Isham, Colgate University, and Laura Monroe, Los Alamos National Laboratory. Applications of Hypercomplex Analysis, Mihaela B. Vajiac, Chapman University, Orange, CA, Daniel Alpay, Chapman University, and Paula Cerejeiras, University of Aveiro, Portugal. Applied Topology Beyond Persistence Diagrams, Nikolas Schonsheck, University of Delaware, Lori Ziegelmeier, Macalester College, Gregory Henselman-Petrusek, University of Oxford, and Chad Giusti, Oregon State University. Applied Topology: Theory, Algorithms, and Applications, Woojin Kim, Duke University, Johnathan Bush, University of Florida, Alex McCleary, Ohio State University, Sarah Percival, Michigan State University, and Iris H. R. Yoon, University of Oxford. Arithmetic Geometry with a View toward Computation, David Lowry-Duda, ICERM & Brown University, Barinder Banwait, Boston University, Shiva Chidambaram, Massachusetts Institute of Technology, Juanita Duque-Rosero, Boston University, Brendan Hassett, ICERM/Brown University, and Ciaran Schembri, Dartmouth College. November 2023

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MEETINGS & CONFERENCES Bridging Applied and Quantitative Topology, Henry Adams, University of Florida, and Ling Zhou, Duke University. Coding Theory for Modern Applications, Rafael D’Oliveira, Clemson University, Hiram H. Lopez, Cleveland State University, and Allison Beemer, University of Wisconsin-Eau Claire. Combinatorial Insights into Algebraic Geometry, Javier Gonzalez Anaya, Harvey Mudd College. Combinatorial Perspectives on Algebraic Curves and their Moduli, Sam Payne, UT Austin, Melody Chan, Brown University, Hannah K. Larson, Harvard University and UC Berkeley, and Siddarth Kannan, Brown University. Combinatorics for Science, Stephen J Young, Bill Kay, and Sinan Aksoy, Pacific Northwest National Laboratory. Commutative Algebra and Algebraic Geometry (associated with Invited Address by Daniel Erman), Daniel Erman, University of Hawaii, and Aleksandra C Sobieska, University of Wisconsin - Madison. Complex Analysis, Operator Theory, and Real Algebraic Geometry, J. E. Pascoe, Drexel University, Kelly Bickel, Bucknell University, and Ryan K. Tully-Doyle, Cal Poly SLO. Complex Social Systems (a Mathematics Research Communities session) I, Ekaterina Landgren, University of Colorado, Boulder, Cara Sulyok, Lewis University, Casey Lynn Johnson, UCLA, Molly Lynch, Hollins University, and Rebecca Hardenbrook, Dartmouth College. Computable Mathematics: A Special Session Dedicated to Martin D. Davis, Valentina S Harizanov, George Washington University, Alexandra Shlapentokh, East Carolina University, and Wesley Calvert, Southern Illinois University. Computational Biomedicine: Methods - Models - Applications, Nektarios A. Valous, Center for Quantitative Analysis of Molecular and Cellular Biosystems (Bioquant), Heidelberg University, Im Neuenheimer Feld 267, 69120, Heidelberg, Germany, Anna Konstorum, Center for Computing Sciences, Institute for Defense Analyses, 17100 Science Drive, Bowie, MD, 20715, USA, Heiko Enderling, Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center & Research Institute, Tampa, FL, 33647, USA, and Dirk Jäger, Department of Medical Oncology, National Center for Tumor Diseases (NCT), University Hospital Heidelberg (UKHD), Im Neuenheimer Feld 460, 69120, Heidelberg, Germany. Computational Techniques to Study the Geometry of the Shape Space, Shira Faigenbaum-Golovin, Duke University, Shan Shan, University of Southern Denmark, and Ingrid Daubechies, Duke University. Covering Systems of the Integers and Their Applications, Joshua Harrington, Cedar Crest College, Tony Wing Hong Wong, Kutztown University of Pennsylvania, and Matthew Litman, University of California, Davis. Cryptography and Related Fields, Ryann Cartor, Clemson University, Angela Robinson, NIST, and Daniel Everett Martin, Clemson University. Derived Categories, Arithmetic, and Geometry (a Mathematics Research Communities session) I, Anirban Bhaduri, University of South Carolina, Gabriel Dorfsman-Hopkins, St. Lawrence University, Patrick Lank, University of South Carlina, and Peter McDonald, University of Utah. Developing Students’ Technical Communication Skills through Mathematics Courses, Michelle L. Ghrist, Gonzaga University, Timothy P Chartier, Davidson College, Maila B. Hallare, US Air Force Academy, USAFA CO USA, and Denise Taunton Reid, Valdosta State University. Diffusive Systems in the Natural Sciences, Francesca Bernardi, Worcester Polytechnic Institute, and Owen L Lewis, University of New Mexico. Discrete Homotopy Theory, Krzysztof R. Kapulkin, University of Western Ontario, Anton Dochtermann, Texas State University, and Antonio Rieser, CONACYT-CIMAT. Dynamical Systems Modeling for Biological and Social Systems, Daniel Brendan Cooney, University of Pennsylvania, Chadi M Saad-Roy, University of California, Berkeley, and Chris M. Heggerud, University of California, Davis. Dynamics and Management in Disease or Ecological Models (associated with Gibbs Lecture by Suzanne Lenhart), Suzanne Lenhart, University of Tennessee, Knoxville, Christina Edholm, Scripps College, and Wandi Ding, Middle Tennessee State University. Dynamics and Regularity of PDEs, Zongyuan Li, Rutgers University, Zhiyuan Zhang, Northeastern University, Xueying Yu, Oregon State University, and Weinan Wang, University of Oklahoma. Epistemologies of the South and the Mathematics of Indigenous Peoples, María Del Carmen Bonilla Tumialán, National University of Education Enrique Guzman y Valle, Wilfredo Vidal Alangui, University of the Philippines Baguio, and Domingo Yojcom Rocché, Center for Scientific and Cultural Research. Ergodic Theory, Symbolic Dynamics, and Related Topics, Andrew T. Dykstra, Hamilton College, and Shrey Sanadhya, Ben Gurion University of the Negev, Israel. Ethics in the Mathematics Classroom, Victor Piercey, Ferris State University, and Catherine Buell, Fitchburg State University. Explicit Computation with Stacks (a Mathematics Research Communities session) I, Santiago Arango, Emory University, Jonathan Richard Love, CRM Montreal, and Sameera Vemulapalli, Princeton University.

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MEETINGS & CONFERENCES Exploring Spatial Ecology via Reaction Diffusion Models: New Insights and Solutions, Jerome Goddard II, Auburn University Montgomery, and Ratnasingham Shivaji, University of North Carolina Greensboro. Extremal and Probabilistic Combinatorics, Sam Spiro, Rutgers University, and Corrine Yap, Georgia Institute of Technology. Geometric Analysis in Several Complex Variables, Ming Xiao, University of California, San Diego, Bernhard Lamel, Texas A&M University At Qatar, and Nordine Mir, Texas A&M University at Qatar. Geometric Group Theory (Associated with the AMS Retiring Presidential Address), Kasia Jankiewicz, University of California Santa Cruz, Edgar A. Bering, San José State University, Marion Campisi, San Jose State University, and Tim Hsu and Giang Le, San José State University. Geometry and Symmetry in Differential Equations, Control, and Applications, Taylor Joseph Klotz and George Wilkens, University of Hawai‘i. Geometry and Topology of High-Dimensional Biomedical Data, Smita Krishnaswamy, Yale, Dhananjay Bhaskar, Yale University, Bastian Rieck, Technical University of Munich, and Guy Wolf, Université de Montréal. Group Actions in Commutative Algebra, Alessandra Costantini, Oklahoma State University, Alexandra Seceleanu, University of Nebraska-Lincoln, and Andras Cristian Lorincz, University of Oklahoma. Hamiltonian Systems and Celestial Mechanics, Zhifu Xie, The University of Southern Mississippi, and Ernesto Perez-Chavela, ITAM. Harmonic Analysis, Geometry Measure Theory, and Fractals, Kyle Hambrook, San Jose State University, Chun-Kit Lai, San Francisco State University, and Caleb Z Marshall, University of British Columbia. History of Mathematics, Adrian Rice, Randolph-Macon College, Sloan Evans Despeaux, Western Carolina University, Deborah Kent, University of St. Andrews, and Jemma Lorenat, Pitzer College. Homological Techniques in Noncommutative Algebra, Robert Won, George Washington University, Ellen Kirkman, Wake Forest University, and James J. Zhang, University of Washington. Homotopy Theory, Krzysztof R. Kapulkin, University of Western Ontario, Daniel K. Dugger, University of Oregon, Jonathan Beardsley, University of Nevada, Reno, and Thomas Brazelton, University of Pennsylvania. Ideal and Factorization Theory in Rings and Semigroups, Scott Chapman, Sam Houston State University, and Alfred Geroldinger, University of Graz. Informal Learning, Identity, and Attitudes in Mathematics, Sergey Grigorian, Mayra Ortiz, Xiaohui Wang, and Aaron T Wilson, University of Texas Rio Grande Valley. Integer Partitions, Arc Spaces and Vertex Operators, Hussein Mourtada, Université Paris Cité, and Andrew R. Linshaw, University of Denver. Interplay Between Matrix Theory and Markov Systems: Applications to Queueing Systems and of Duality Theory, Alan Krinik and Randall J. Swift, California State Polytechnic University, Pomona. Issues, Challenges and Innovations in Instruction of Linear Algebra, Feroz Siddique, University of Wisconsin-Eau Claire, and Ashish K. Srivastava, Saint Louis University. Knots, Skein Modules, and Categorification, Rhea Palak Bakshi, ETH Institute for Theoretical Studies, Zurich, Sujoy Mukherjee, University of Denver, and Jozef Henryk Przytycki, George Washington University. Large Random Permutations (affiliated with AAAS-AMS Invited Address by Peter Winkler), Peter M Winkler, Dartmouth College, and Jacopo Borga, Stanford University. Loeb Measure after 50 Years, Yeneng Sun, National University of Singapore, Robert M Anderson, UC Berkeley, and Matt Insall, Missouri University of Science and Technology. Looking Forward and Back: Common Core State Standards in Mathematics (CCSSM), 12 Years Later, Younhee Lee, Southern Connecticut State University, James Alvarez, University of Texas Arlington, Ekaterina Fuchs, City College of San Francisco, Tyler Kloefkorn, American Mathematical Society, Yvonne Lai, University of Nebraska-Lincoln, and Carl Olimb, Augustana University. Mathematical Modeling and Simulation of Biomolecular Systems, Zhen Chao, Western Washington University, and Jiahui Chen, University of Arkansas. Mathematical Modeling of Nucleic Acid Structures, Pengyu Liu, University of California, Davis, Van Pham, University of South Florida, and Svetlana Poznanovic, Clemson University. Mathematical Physics and Future Directions, Shanna Dobson, University of California, Riverside, Tepper L. Gill, Howard University, Michael Anthony Maroun, University of California, Riverside, CA, and Lance Nielsen, Creighton University. Mathematics and Philosophy, Tom Morley, Georgia Tech, and Bonnie Gold, Monmouth University. Mathematics and Quantum, Kaifeng Bu and Arthur M. Jaffe, Harvard, Sui Tang, UCSB, and Jonathan Weitsman, Northeastern University.

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MEETINGS & CONFERENCES Mathematics and the Arts, Karl M Kattchee, University of Wisconsin-La Crosse, Doug Norton, Villanova University, and Anil Venkatesh, Adelphi University. Mathematics of Computer Vision, Timothy Duff and Max Lieblich, University of Washington. Mathematics of DNA and RNA, Marek Kimmel, Rice University, Chris McCarthy, BMCC, City University of New York, and Johannes Familton, Borough of Manhattan Community College, CUNY. Metric Dimension of Graphs and Related Topics, Briana Foster-Greenwood, Cal Poly Pomona, and Christine Uhl, St. Bonaventure University. Metric Geometry and Topology, Christine M. Escher, Oregon State University, and Catherine Searle, Wichita State University. Mock Modular forms, Physics, and Applications, Amanda Folsom, Amherst College, Terry Gannon, University of Alberta, and Larry Rolen, Vanderbilt University. Modeling Complex Adaptive Systems in Life and Social Sciences, Yun Kang and Theophilus Kwofie, Arizona State University, and Sabrina H Streipert, University of Pittsburgh. Modeling to Motivate the Teaching of the Mathematics of Differential Equations, Brian Winkel, SIMIODE, Chardon NY USA, Kyle T Allaire, Worcester State University, Worcester MA USA, Maila B. Hallare, US Air Force Academy, USAFA CO USA, Yanping Ma, Loyola Marymount University, Los Angeles CA USA, and Lisa Naples, Macalester College, Saint Paul MN USA. Modelling with Copulas: Discrete vs Continuous Dependent Data, Martial Longla, University of Mississippi, and Isidore Seraphin Ngongo, University of Yaounde I. Modern Developments in the Theory of Configuration Spaces, Christin Bibby, Louisiana State University, and Nir Gadish, University of Michigan. Modular Tensor Categories and TQFTs beyond the Finite and Semisimple, Colleen Delaney, UC Berkeley, and Nathan Geer, Utah State University. Navigating the Benefits and Challenges of Mentoring Students in Data-Driven Undergraduate Research Projects, Vinodh Kumar Chellamuthu, Utah Tech University, and Xiaoxia Xie, Idaho State University. New Faces in Operator Theory and Function Theory, Michael R Pilla, Ball State University, and William Thomas Ross, University of Richmond. Nonlinear Dynamics in Human Systems: Insights from Social and Biological Perspectives, Armando Roldan, University of Central Florida, and Thomas Dombrowski, Moffitt Cancer Center. Number Theory in Memory of Kevin James, Jim L. Brown, Occidental College, and Felice Manganiello, Clemson University. Numerical Analysis, Spectral Graph Theory, Orthogonal Polynomials, and Quantum Algorithms, Anastasiia Minenkova, University of Hartford, and Gamal Mograby, University of Cincinnati. Partition Theory and q-Series, William Jonathan Keith, Michigan Technological University, Brandt Kronholm, University of Texas Rio Grande Valley, and Dennis Eichhorn, University of California, Irvine. Polymath Jr REU Student Research, Steven Joel Miller, Williams College, and Alexandra Seceleanu, University of Nebraska-Lincoln. Principles, Spatial Reasoning, and Science in First-Year Calculus, Yat Sun Poon and Catherine Lussier, University of California, Riverside, and Bryan Carrillo, Saddleback College. Quantitative Justice, Ron Buckmire, Occidental College, Omayra Ortega, Sonoma State University, and Robin Wilson, California State Polytechnic University, Pomona (NAM-SIAM-AMS). Quaternions, Chris McCarthy, BMCC, City University of New York, Johannes Familton, Borough of Manhattan Community College, CUNY, and Terrence Richard Blackman, Medgar Evers Community College, CUNY. Recent Advances in Mathematical Models of Diseases: Analysis and Computation, Najat Ziyadi and Jemal S Mohammed-Awel, Department of Mathematics, Morgan State University. Recent Advances in Stochastic Differential Equation Theory and its Applications in Modeling Biological Systems, Tuan A. Phan, IMCI, University of Idaho, Nhu N. Nguyen, University of Rhode Island, and Jianjun P. Tian, New Mexico State University. Recent Developments in Commutative Algebra, Austyn Simpson and Alapan Mukhopadhyay, University of Michigan, and Thomas Marion Polstra, University of Virginia. Recent Developments in Numerical Methods for PDEs and Applications, Chunmei Wang, University of Florida, Long Chen, UC Irvine, Shuhao Cao, University of Missouri-Kansas City, and Haizhao Yang, University of Maryland College Park. Recent Developments on Markoff Triples, Elena Fuchs, UC Davis, and Daniel Everett Martin, Clemson University. Recent Progress in Inference and Sampling (Associated with AMS Invited Address by Ankur Moitra), Ankur Moitra, Massachusetts Institute of Technology, and Sitan Chen, Harvard University. 1748

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MEETINGS & CONFERENCES Research in Mathematics by Undergraduates and Students in Post-Baccalaureate Programs, Darren A. Narayan, Rochester Institute of Technology, John C. Wierman, Johns Hopkins University, Mark Daniel Ward, Purdue University, Khang Duc Tran, California State University, Fresno, and Christopher O’Neill, San Diego State University. Research Presentations by Math Alliance Scholar Doctorates, Theresa Martines, University of Texas, Austin, and David Goldberg, Math Alliance/Purdue University. Ricci Curvatures of Graphs and Applications to Data Science (a Mathematics Research Communities session) I, Aleyah Dawkins, George Mason University, Xavier Ramos Olive, Smith College, Zhaiming Shen, University of Georgia, David Harry Richman, University of Washington, and Michael G Rawson, PNNL. Roots of Unity - Mathematics from Graduate Students in the Roots of Unity Program, Allechar López, Montana State University, and Patricia Klein, University of Minnesota. Serious Recreational Mathematics, Erik Demaine, Massachusetts Institute of Technology, Robert A. Hearn, H3 Labs, and Tomas Rokicki, California. Solvable Lattice Models and their Applications Associated with the Noether Lecture, Anne Schilling, University of California, Davis, Amol Aggarwal, Columbia, Benjamin Brubaker, University of Minnesota - Twin Cities, Daniel Bump, Stanford, Andrew Hardt, Stanford University, Slava Naprienko, Stanford and University of North Carolina, Leonid Petrov, University of Virginia, and Anne Schilling, University of California, Davis. Spectral Methods in Quantum Systems, Matthew Powell, Georgia Institute of Technology, and Wencai Liu, Texas A&M University. Structure-preserving Algorithms, Analysis and Simulations for Differential Equations, Brian E Moore, University of Central Florida, and Qin Sheng, Baylor University. The EDGE (Enhancing Diversity in Graduate Education) Program: Pure and Applied Talks by Women Math Warriors, Quiyana Murphy, Virginia Tech, Sofia Rose Rose Martinez Alberga, Purdue University, Kelly Buch, Austin Peay State University, and Alexis Hardesty, Texas Tech University. The Mathematics of Decisions, Elections, and Games, David McCune, William Jewell College, Michael A. Jones, Mathematical Reviews | AMS, and Jennifer M. Wilson, Eugene Lang College, The New School. Theoretical and Numerical Aspects of Nonlocal Models, Nicole Buczkowski, Worcester Polytechnic Institute, Christian Alexander Glusa, Sandia National Laboratories, and Animesh Biswas, University of Nebraska Lincoln. Theta Correspondence, Edmund Karasiewicz and Petar Bakic, University of Utah. The Teaching and Learning of Undergraduate Ordinary Differential Equations, Viktoria Savatorova, Central Connecticut State University, Chris Goodrich, The University of New South Wales, Itai Seggev, Wolfram Research, Beverly H West, Cornell University, and Maila B. Hallare, US Air Force Academy, USAFA CO USA. Thresholds in Random Structures, Will Perkins, Georgia Tech. Topics in Combinatorics and Graph Theory, Cory Palmer and Anastasia Halfpap, University of Montana, and Neal Bushaw, Virginia Commonwealth University. Topics in Equivariant Algebra, Ben Spitz, University of California Los Angeles, and Christy Hazel and Michael A. Hill, UCLA. Topological and Algebraic Approaches for Optimization, Ali Mohammad Nezhad, Carnegie Mellon University. Undergraduate Research Activities in Mathematical and Computational Biology, Timothy D Comar, Benedictine University, and Anne E. Yust, University of Pittsburgh. Using 3D-Printed and Other Digitally-Fabricated Objects in the Mathematics Classroom, Shelby Stanhope, U.S. Air Force Academy, Paul E. Seeburger, Monroe Community College, and Stepan Paul, North Carolina State University. Water Waves, Anastassiya Semenova and Bernard Deconinck, University of Washington, John D Carter, Seattle University, and Eleanor Devin Byrnes, University of Washington.

Invited Addresses of Other JMM Partners Henri Darmon, McGill University, Fourier Coefficients of Modular Forms (CRM-PIMS-AARMS Invited Address - Henri Darmon, McGill University). Ranthony A C Edmonds, Duke University, Hidden Figures Revealed (NAM Cox-Talbot Address). Katherine Ensor, Rice University, Celebrating Statistical Foundations Driving 21st -Century Innovation (ASA Invited Address- Kathy Ensor, Rice University). Stephan Ramon Garcia, Pomona College, Fast food for thought: what can chicken nuggets tell us about linear algebra? (ILAS Invited Address). November 2023

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MEETINGS & CONFERENCES Sylvester James Gates, Jr, Clark Leadership Chair in Science, University of Maryland; past president of American Physical Society, National Medal of Science, What Challenges Does Data Science Present to Mathematics Education? (TPSE Invited Address - Sylvester James Gates, Jr, Clark Leadership Chair in Science, University of Maryland). Matthew Harrison-Trainor, University of Illinois Chicago, The Complexity of Classifying Topological Spaces (ASL Invited Address). Åsa Hirvonen, University of Helsinki, Games for Measuring Distances Between Metric Structures (ASL Invited Address). Trachette Jackson, University of Michigan, Mobilizing Mathematics for the Fight Against Cancer (PME Invited Address). Shelly M Jones, Central Connecticut State University, Making the Case for Culturally Relevant Teaching in Math (NAM Claytor-Woodard Lecture). Francois Loeser, Institut Universitaire de France, Sorbonne, Model Theory and Non-Archimedean Geometry (ASL Invited Address). Toby Meadows, University of California, Irvine, A Modest Foundational Argument for the Generic Multiverse (ASL Invited Address). Dima Svetosla Sinapova, Rutgers University, Combinatorial Principles at Successors of Singular Cardinals (ASL Invited Address). Slawomir Solecki, Cornell University, Title to be announced (ASL Invited Address). Joni Teräväinen, University of Turku, Title to be announced (AIM Alexanderson Award Lecture - Joni Teräväinen). Mariel Vazquez, University of California Davis, Topological Considerations in Genome Biology (SIAM Invited Address). Mariana Vicaria, University of California, at Berkeley, Title to be announced (ASL Invited Address).

AIM Special Sessions AIM Special Session Associated with the Alexanderson Award and Lecture, Joni Teräväinen, University of Turku, Terence Tao, UCLA, Kasia Matomäki, University of Turku, Maksym Radziwill, Northwestern University, and Tamar Ziegler, Hebrew University. Equivariant Techniques in Stable Homotopy Theory, Michael A. Hill, UCLA, and Anna Marie Bohmann, Vanderbilt University. Graphs and Matrices, Mary Flagg, University of St. Thomas, and Bryan A Curtis, Iowa State University. Little School Dynamics: Cool Research by Researchers at PUIs, Kimberly Ayers, California State University, San Marcos, Ami Radunskaya, Pomona College, Andy Parrish, Eastern Illinois University, David M. McClendon, Ferris State University, and Han Li, Wesleyan University. Math Circle Activities as a Gateway Into Research, Jeffrey Musyt, Slippery Rock University, Lauren L Rose, Bard College, Tom G. Stojsavljevic, Beloit College, Nick Rauh, Julia Robinson Math Festivals, Edward Charles Keppelmann, University of Nevada Reno, Allison Henrich, Seattle University, Violeta Vasilevska, Utah Valley University, and Gabriella A. Pinter, University of Wisconsin, Milwaukee.

ASL Special Sessions Descriptive Methods in Dynamics, Combinatorics, and Large Scale Geometry, Jenna Zomback, University of Maryland, College Park, and Forte Shinko, UCLA.

AWM Special Sessions AWM Workshop: Women in Operator Theory, Catherine Anne Beneteau, University of South Florida, and Asuman Aksoy, Claremont McKenna College. EvenQuads Live and in person: The honorees and the games, sarah-marie belcastro, Mathematical Staircase, Inc., Sherli Koshy-Chenthittayil, Touro University Nevada, Oscar Vega, California State University, Fresno, Monica D. Morales-Hernandez, Adelphi University, Linda McGuire, Muhlenberg College, and Denise A. Rangel Tracy, Fairleigh Dickinson University. Mathematics in the Literary Arts and Pedagogy in Creative Settings, Shanna Dobson, University of California, Riverside, and Claudia Maria Schmidt, California State University. Recent Developments in Harmonic Analysis, Betsy Stovall, University of Wisconsin-Madison, and Sarah E Tammen, Massachusetts Institute of Technology. Women in Mathematical Biology, Christina Edholm, Scripps College, Lihong Zhao, University of California, Merced, and Lale Asik, University of the Incarnate Word. 1750

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MEETINGS & CONFERENCES COMAP Special Sessions Math Modeling Contests: What They Are, How They Benefit, What They Did – Discussions with the Students and Advisors, Kayla Blyman, Saint Martin’s University, and Kim A Kuda, American Mathematical Society.

ILAS Special Sessions Generalized Numerical Ranges and Related Topics, Tin-Yau Tam and Pan-Shun Lau, University of Nevada, Reno. Graphs and Matrices, Jane Breen, Ontario Tech University, and Stephen Kirkland, University of Manitoba. Innovative and Effective Ways to Teach Linear Algebra, David M. Strong, Pepperdine University, Sepideh Stewart, University of Oklahoma, Gil Strang, MIT, and Megan Wawro, Virginia Tech. Linear Algebra, Matrix theory, and its Applications, Stephan Ramon Garcia and Konrad Aguilar, Pomona College. Sign-pattern Matrices and Their Applications, Bryan L Shader, University of Wyoming, and Minerva Catral, Xavier University. Spectral and combinatorial problems for nonnegative matrices and their generalizations, Pietro Paparella, University of Washington Bothell, and Michael J. Tsatsomeros, Washington State University.

MSRI Special Sessions African Diaspora Joint Mathematics Working Groups (ADJOINT), Caleb Ashley, Boston College, and Anisah Nabilah Nu’Man, Spelman College. Summer Research in Mathematics (SRiM): Recent Trends in Nonlinear Boundary Value Problems, Maya Chhetri, UNC Greensboro, Elliott Zachary Hollifield, University of North Carolina at Pembroke, and Nsoki Mavinga, Swarthmore College. The MSRI Undergraduate Program (MSRI-UP), Maria Mercedes Franco, Queensborough Community College-CUNY.

PMA Special Sessions BSM Special Session: Mathematical Research in Budapest for Students and Faculty, Kristina Cole Garrett, St. Olaf College.

SIAM Minisymposium SIAM ED Session on Artificial Intelligence and its Uses in Mathematical Education, Research, and Automation in the Industry, Kathleen Kavanagh, Clarkson University, and Alvaro Alfredo Ortiz Lugo and Sergio Molina, University of Cincinnati. SIAM Minisymposium on Computational Mathematics and the Power Grid, Todd Munson, Argonne National Laboratory. SIAM Minisymposium on Current Advances in Modeling and Simulation to Uncover the Complexity of Disease Dynamics, Naveen K. Vaidya, San Diego State University, and Elissa Schwartz, Washington State University. SIAM Minisymposium on Mathematical Methods in Computer Vision and Image Analysis, Andreas Mang, University of Houston. SIAM Minisymposium on Mathematics of Bacterial Viruses: From Virus Discovery to Mathematical Principles, Javier Arsuaga, University of California,Davis, Carme Calderer, University of Minnesota, and Ami Bhatt, Stanford University. SIAM Minisymposium on Recent Developments in the Analysis and Control of Partial Differential Equations Arising in Fluid and Fluid-Structure Interactive Dynamics, George Avalos, University of Nebraska-Lincoln, and Pelin Guven Geredeli, Iowa State University. SIAM Minisymposium on Scientific Machine Learning to Advance Modeling and Decision Support, Erin Acquesta, Sandia National Laboratories, Timo Bremer, Lawrence Livermore National Laboratories, and Joseph Hart, Sandia National Laboratories. SIAM-USNCTAM Minisymposium on Mathematical Modeling of Complex Materials Systems, Maria G Emelianenko, George Mason University, and Dmitry Golovaty, The University of Akron.

SPECTRA Special Sessions Research by LGBTQ+ Mathematicians, Devavrat Dabke, Princeton University, Joseph Nakao, Swarthmore College, and Michael A. Hill, UCLA.

Invited Addresses of Other Organizations Arezoo Islami, San Francisco State University, The Unreasonable Effectiveness of Mathematics: Dissolving Wigner’s Applicability Problem (POMSIGMAA Guest Lecture and Discussion). Yvonne Lai, University of Nebraska-Lincoln, (Why) To Build Bridges in Mathematics Education (Project NExT Lecture on Teaching and Learning). November 2023

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MEETINGS & CONFERENCES Other Special Sessions Exploring Funding Opportunities in the Division of Mathematical Sciences, Elizabeth Wilmer, National Science Federation, and Junping Wang, National Science Foundation. Outcomes and Innovations from NSF Undergraduate Education Programs in the Mathematical Sciences I, Michael Ferrara, Division of Undergraduate Education, National Science Foundation.

AMS Contributed Paper Sessions AMS Contributed Paper Session, Michelle Ann Manes, University of Hawaii.

ASL Contributed Paper Sessions ASL Contributed Paper Session, David Reed Solomon, University of Connecticut.

COMAP Contributed Paper Sessions COMAP Contributed Paper Session: Integrating Modeling into Established Courses, Kayla Blyman, Saint Martin’s University.

NAM Contributed Paper Sessions NAM Haynes-Granville-Browne Session of Presentations by Recent Doctoral Recipients, Aris Winger, Georgia Gwinnett College, Torina D. Lewis, American Mathematical Society, and Omayra Ortega, Sonoma State University.

PME Contributed Paper Sessions PME Contributed Session on Research by Undergraduates, Thomas Philip Wakefield, Youngstown State University, and Jennifer Beineke, Western New England University.

TPSE Contributed Paper Sessions TPSE Contributed Paper Session on Using Institutional and National Data Sources to Recruit, Retain and Support a Diverse Population of Mathematics Students, Rick Cleary, Babson College, and Mitchel T. Keller, University of Wisconsin - Madison.

Other Events JMM Workshop on Building Conceptual Understanding of Multivariable Calculus using 3D Visualization in CalcPlot3D and 3D-Printed Surfaces, Rick Cleary, Babson College, and Mitchel T. Keller, University of Wisconsin - Madison. JMM Workshop on Teaching Student-Centered Mathematics: Active Learning & the Learning Assistant Model, Rick Cleary, Babson College, and Mitchel T. Keller, University of Wisconsin - Madison.

AMS Other Events AMS Current Events Bulletin, David Eisenbud, MSRI.

OTH Other Events AMS - PME Undergraduate Student Poster Session, Chad Awtrey and Frank Patane, Samford University. AWM Workshop Poster Presentations, Radmila Sazdanovic, NC State University.

Tallahassee, Florida Florida State University March 23–24, 2024 Saturday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: Volume 45, Issue 2

Meeting #1193

Deadlines

Southeastern Section Associate Secretary for the AMS:

For organizers: To be announced For abstracts: January 23, 2024

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html. 1752

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MEETINGS & CONFERENCES Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl. Advanced Numerical Methods for Partial Differential Equations and Their Applications (Code: SS 1A), Seonghee Jeong, Louisiana State University, Sanghyun Lee, Florida State University, and Seulip Lee, University of Georgia. Advances in Financial Mathematics (Code: SS 2A), Qi Feng, Alec N Kercheval, and Lingjiong Zhu, Florida State University. Advances in Shape and Topological Data Analysis (Code: SS 3A), Emmanuel L Hartman, Eric Klassen, and Ethan Semrad, Florida State University. Algebraic Groups and Local-Global Principles (Code: SS 4A), Suresh Venapally, Emory University, and Daniel Reuben Krashen, University of Pennsylvania. Bases and Frames in Hilbert spaces (Code: SS 5A), Laura De Carli, Florida International University, and Azita Mayeli, City University of New York. Combinatorics in Geometry of Polynomials (Code: SS 6A), Papri Dey, Georgia Institute of Technology. Control, Inverse Problems and Long Time Dynamics of Evolution Systems (Code: SS 7A), Shitao Liu, Clemson University, and Louis Tebou, Florida International University. Data Integration and Identifiability in Ecological and Epidemiological Models (Code: SS 8A), Omar Saucedo, Virginia Tech, and Olivia Prosper, University of Tennessee/Knoxville. Diversity in Mathematical Biology (Code: SS 9A), Daniel Alejandro Cruz and Skylar Grey, University of Florida. Fluids: Analysis, Applications, and Beyond (Code: SS 10A), Aseel Farhat and Anuj Kumar, Florida State University. Geometric Measure Theory and Partial Differential Equations (Code: SS 11A), Alexander B. Reznikov, John Hoffman, and Richard Oberlin, Florida State University. Geometry and Symmetry in Data Science (Code: SS 12A), Dustin G. Mixon, The Ohio State University, and Thomas Needham, Florida State University. Homotopy Theory and Category Theory in Interaction (Code: SS 13A), Ettore Aldrovandi and Brandon Doherty, Florida State University, and Philip John Hackney, University of Louisiana at Lafayette. Human Behavior and Infectious Disease Dynamics (Code: SS 14A), Bryce Morsky, Florida State University. Mathematical Advances in Scientific Machine Learning (Code: SS 15A), Wenjing Liao, Georgia Institute of Technology, and Feng Bao and Zecheng Zhang, Florida State University. Mathematical Modeling and Simulation in Fluid Dynamics (Code: SS 16A), Pejman Sanaei, Georgia State University. Mathematical Models for Population and Methods for Parameter Estimation in Epidemiology (Code: SS 17A), Yang LI, Georgia State University, and Guihong Fan, Columbus State University. Moduli Spaces in Algebraic Geometry (Code: SS 18A), Jeremy Usatine, Florida State University, Hulya Arguz and Pierrick Bousseau, University of Georgia, and Matthew Satriano, University of Waterloo. Nonlinear Evolution Partial Differential Equations in Physics and Geometry (Code: SS 19A), Jared Speck and Leonardo Abbrescia, Vanderbilt University. Numerical Methods and Deep Learning for PDEs (Code: SS 20A), Chunmei Wang, University of Florida, and Haizhao Yang, University of Maryland College Park. PDEs in Incompressible Fluid Mechanics (Code: SS 21A), Wojciech S. Ozanski, Florida State University, Stanley Palasek, UCLA, and Alexis F Vasseur, The University of Texas At Austin. Recent Advances in Geometry and Topology (Code: SS 22A), Thang Nguyen, Samuel Aaron Ballas, Philip L. Bowers, and Sergio Fenley, Florida State University. Recent Advances in Inverse Problems for Partial Differential Equations and Their Applications (Code: SS 23A), Anh-Khoa Vo, Florida A&M University, and Thuy T. Le, North Carolina State University. Recent Development in Deterministic and Stochastic PDEs (Code: SS 24A), Quyuan Lin, Clemson University, and Xin Liu, Texas A&M University. Recent Developments in Numerical Methods for Evolution Partial Differential Equations (Code: SS 25A), Thi-Thao-Phuong Hoang, Yanzhao Cao, and Hans-Werner Van Wyk, Auburn University. Regularity Theory and Free Boundary Problems (Code: SS 27A), Lei Zhang, University of Florida, and Eduardo V. Teixeira, University of Central Florida. Stochastic Analysis and Applications (Code: SS 28A), Hakima Bessaih, Florida International University, and Oussama Landoulsi, FLORIDA INTERNATIONAL UNIVERSITY. Stochastic Differential Equations: Modeling, Estimation, and Applications (Code: SS 29A), Sher B Chhetri, University of South Carolina Sumter, Hongwei Long, Florida Atlantic University, and Olusegun M. Otunuga, Augusta University. November 2023

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MEETINGS & CONFERENCES Theory of Nonlinear Waves (Code: SS 30A), Nicholas James Ossi and Ziad H Musslimani, Florida State University. Topics in Graph Theory (Code: SS 31A), Songling Shan, Auburn University, and Guantao Chen, Georgia State University. Topics in Stochastic Analysis/Rough Paths/SPDE and Applications in Machine Learning (Code: SS 32A), Cheng Ouyang, University of Illinois At Chicago, Fabrice Baudoin, University of Connecticut, and Qi Feng, Florida State University. Topological Algorithms for Complex Data and Biology (Code: SS 33A), Henry Adams, Johnathan Bush, and Hubert Wagner, University of Florida. Topological Interactions of Contact and Symplectic Manifolds (Code: SS 34A), Angela Wu, University College of London and Louisiana State University, and Austin Christian, Georgia Institute of Technology.

Washington, District of Columbia Howard University April 6–7, 2024 Saturday – Sunday

Program first available on AMS website: Not applicable Issue of Abstracts: Volume 45, Issue 2

Meeting #1194

Deadlines

Eastern Section Associate Secretary for the AMS: Steven H. Weintraub

For organizers: Expired For abstracts: February 13, 2024

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html.

Invited Addresses Ryan Charles Hynd, University of Pennsylvania, Title to be announced. Jinyoung Park, Institute for Advanced Study, Title to be announced. Jian Song, Rutgers, State University of New Jersey, Title to be announced. Talitha M Washington, Clark Atlanta University & Atlanta University Center, Title to be announced (Einstein Public Lecture in Mathematics).

Milwaukee, Wisconsin University of Wisconsin-Milwaukee April 20–21, 2024 Saturday – Sunday

Program first available on AMS website: Not applicable Issue of Abstracts: Volume 45, Issue 2

Meeting #1195

Deadlines

Central Section Associate Secretary for the AMS: Betsy Stovall

For organizers: Expired For abstracts: February 20, 2024

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html.

Invited Addresses Mihaela Ifrim, University of Wisconsin-Madison, Title To Be Announced. Lin Lin, University of California, Berkeley, Title To Be Announced. Kevin Schreve, LSU, Title To Be Announced.

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Notices of the American Mathematical Society Volume 70, Number 10

MEETINGS & CONFERENCES

San Francisco, California San Francisco State University May 4–5, 2024 Saturday – Sunday

Program first available on AMS website: Not applicable Issue of Abstracts: Volume 45, Issue 3

Meeting #1196

Deadlines

Western Section Associate Secretary for the AMS: Michelle Ann Manes

For organizers: October 4, 2023 For abstracts: March 12, 2024

Palermo, Italy July 23–26, 2024

Issue of Abstracts: To be announced

Tuesday – Friday Associate Secretary for the AMS: Brian D. Boe

Deadlines

Program first available on AMS website: To be announced

For organizers: To be announced For abstracts: To be announced

San Antonio, Texas University of Texas, San Antonio September 14–15, 2024 Saturday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: Volume 45, Issue 3

Meeting #1198

Deadlines

Central Section Associate Secretary for the AMS: Betsy Stovall

For organizers: February 13, 2024 For abstracts: July 23, 2024

Savannah, Georgia Georgia Southern University, Savannah October 5–6, 2024

Program first available on AMS website: To be announced

Saturday – Sunday

Issue of Abstracts: Volume 45, Issue 4

Meeting #1199 Southeastern Section Associate Secretary for the AMS: Brian D. Boe, University of Georgia

Deadlines For organizers: March 5, 2024 For abstracts: August 13, 2024

Albany, New York State University of New York at Albany October 19–20, 2024

Program first available on AMS website: To be announced

Saturday – Sunday

Issue of Abstracts: Volume 45, Issue 4

Meeting #1200 Eastern Section Associate Secretary for the AMS: Steven H. Weintraub, Lehigh University November 2023

Deadlines For organizers: March 19, 2024 For abstracts: August 27, 2024

Notices of the American Mathematical Society

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MEETINGS & CONFERENCES

Riverside, California University of California, Riverside October 26–27, 2024 Saturday – Sunday

Program first available on AMS website: Not applicable Issue of Abstracts: Volume 45, Issue 4

Meeting #1201

Deadlines

Western Section Associate Secretary for the AMS: Michelle Ann Manes

For organizers: March 26, 2024 For abstracts: September 3, 2024

Auckland, New Zealand December 9–13, 2024

Issue of Abstracts: To be announced

Monday – Friday Associate Secretary for the AMS: Steven H. Weintraub

Deadlines

Program first available on AMS website: To be announced

For organizers: To be announced For abstracts: To be announced

Seattle, Washington Washington State Convention Center and the Sheraton Seattle Hotel January 8–11, 2025

Issue of Abstracts: To be announced

Wednesday – Saturday Associate Secretary for the AMS: Steven H. Weintraub

Deadlines

Program first available on AMS website: To be announced

For organizers: To be announced For abstracts: To be announced

Hartford, Connecticut Hosted by University of Connecticut; taking place at the Connecticut Convention Center and Hartford Marriott Downtown April 5–6, 2025

Issue of Abstracts: To be announced

Saturday – Sunday Eastern Section Associate Secretary for the AMS: Steven H. Weintraub

Deadlines

Program first available on AMS website: To be announced

For organizers: To be announced For abstracts: To be announced

Washington, District of Columbia Walter E. Washington Convention Center and Marriott Marquis Washington DC January 4–7, 2026

Issue of Abstracts: To be announced

Sunday – Wednesday Associate Secretary for the AMS: Betsy Stovall

Deadlines

Program first available on AMS website: To be announced

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For organizers: To be announced For abstracts: To be announced

Notices of the American Mathematical Society Volume 70, Number 10

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The Mathematical Neighborhoods of School Mathematics Hyman Bass, University of Michigan, Ann Arbor, MI With the assistance of Jason Brasel The Mathematical Neighborhoods of School Mathematics visits regions beyond, but proximal to and accessible from, school mathematics. Its aim is to give readers a glimpse of not just the rich diversity and adaptability of mathematics, but, most importantly, its interconnections and overall coherence, a perspective not easily available from the school curriculum. The book is a valuable resource for professional development of mathematics teachers, and in mathematical enrichment programs, for both students and teachers. 2023; 339 pages; Softcover; ISBN: 978-1-4704-7247-4; List US$89; AMS members US$71.20; MAA members US$80.10; Order Code MBK/148

Visit bookstore.ams.org/mbk-148 Background image credit: Liudmila Chernetska / iStock / Getty Images Plus via Getty Images