2020 Leroy P. Steele Prizes - American Mathematical Society
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FROM THE AMS SECRETARY
2020
Leroy P. Steele Prizes
The 2020 Leroy P. Steele Prizes were presented at the 126th Annual Meeting of the AMS in Denver, Colorado, in Jan-
uary 2020. The Steele Prize for Mathematical Exposition was awarded to Martin R. Bridson and André Haefliger; the
Prize for Seminal Contribution to Research in Analysis/Probability was awarded to Craig Tracy and Harold Widom;
and the Prize for Lifetime Achievement was awarded to Karen Uhlenbeck.
Citation for Riemannian geometry and group theory, that the field of
Mathematical Exposition: geometric group theory came into being. Much of the 1990s
Martin R. Bridson was spent finding rigorous proofs of Gromov’s insights
and André Haefliger and expanding upon them. Metric Spaces of Non-Positive
The 2020 Steele Prize for Math- Curvature is the outcome of that decade of work, and has
ematical Exposition is awarded been the standard textbook and reference work throughout
to Martin R. Bridson and André the field in the two decades of dramatic progress since its
Haefliger for the book Metric publication in 1999.
Spaces of Non-Positive Curvature, A metric space of non-positive curvature is a geodesic
published by Springer-Verlag metric space satisfying (local) CAT(0) condition, that every
in 1999. pair of points on a geodesic triangle should be no further
Metric Spaces of Non-Positive apart than the corresponding points on the “comparison
Martin R. Bridson Curvature is the authoritative triangle” in the Euclidean plane. Examples of such spaces
reference for a huge swath of include non-positively curved Riemannian manifolds,
modern geometric group the- Bruhat–Tits buildings, and a wide range of polyhedral
ory. It realizes Gromov’s vision complexes.
of group theory studied via This book is the definitive text on these spaces and the
geometry, has been the fun - groups associated with them. The theory is developed
damental textbook for many carefully, in great generality. All the foundational theorems
graduate students learning the are proved, and the important examples are covered. The
subject, and paved the way for proofs are clear and comprehensive. The necessary density
the developments of the subse- of such a work is offset by the inclusion of a large number
quent decades. of exercises, making it invaluable both as a graduate text
At the turn of the 20th cen- and as a reference for active researchers.
tury, Max Dehn was interested
in topological problems about Biographical Note: Martin R. Bridson
André Haefliger closed surfaces. He translated Martin R. Bridson was born in the Isle of Man in 1964.
these problems into algebraic He was an undergraduate at Hertford College Oxford and
questions about the fundamental group and then solved received his PhD from Cornell University in 1991, advised
them using the geometry of the action of the fundamental by Karen Vogtmann. He was an assistant professor at Princ-
group on the universal cover. Subsequently, Dehn and eton until 1996, with extended leaves spent in Geneva and
others used combinatorial properties of group presenta- Oxford. He was a tutorial fellow and professor of topology
tions in place of geometric properties of spaces to develop at Oxford (Pembroke College), then professor of pure
combinatorial group theory. It was only in the 1980s, mathematics at Imperial College London. Since 2007 he
with Gromov’s seminal papers drawing parallels between has been the Whitehead Professor of Pure Mathematics at
April 2020 Notices of the American Mathematical Society 563
FROM THE AMS SECRETARY
the University of Oxford, where he served as head of the gave us an inspiring vision that melded the metric geometry
Mathematical Institute 2015–18. He is now president of of the Russian school with numerous novel ideas that drew
the Clay Mathematics Institute. on his unique insights into differential geometry, topol-
Bridson’s research interests revolve around the interac- ogy, and group theory. Finitely generated groups, viewed
tion of geometry, topology, and group theory. He has been as geometric objects, were at the heart of this vision, and
awarded the Whitehead Prize of the London Mathematical the interaction of groups and geometry is correspondingly
Society, the Forder Lectureship of the New Zealand Mathe- central to our book.
matical Society, and a Royal Society Wolfson Research Merit It was a desire to extend Serre’s theory of graphs-of-
Award. He gave an Invited Address to the Joint Mathematics groups to higher dimensions that led to our collaboration.
Meetings in 2001 and was an Invited Speaker at the ICM André, who was developing a theory of complexes of
in Madrid in 2006. He is a Fellow of the American Math- groups, visited John Stallings in Berkeley in 1989. Stall-
ematical Society and was elected a Fellow of the Royal ings, working with Gersten on triangles of groups, was
Society in 2016. developing similar ideas. Martin, struggling to understand
Gromov’s essay “Hyperbolic Groups” while a graduate
Biographical Note: André Haefliger student at Cornell, had resolved a challenge in the geo-
André Haefliger was born in Nyon, Switzerland, in 1929. metric foundations of polyhedral geometry that had been
He received his PhD from Paris-Sorbonne in 1958; his obstructing the work of both André and Gersten–Stallings.
thesis director was Charles Ehresmann, and the president When André learned of this from Stallings, he wrote to
of the jury was Henri Cartan. From 1961, he spent two Martin and subsequently arranged a position for him in
years at the Institute for Advanced Study in Princeton. With Geneva. It was there in 1992–93 that we decided to write
the help of George de Rham, he created the Department our book, naively assuming that we would finish during
of Mathematics at the University of Geneva, where he our stay at a chalet in the Swiss mountains in July 1993,
remained as Professeur until retiring in 1996. He traveled allowing time for long walks in the afternoons. Our sense
widely, visiting universities across Europe, the Americas, of what the book should contain expanded in the years that
the Soviet Union, Japan, China, and (many times) India. followed, but as the field expanded we had to accept that
He was honored by a Doctorat Honoris Causa from ETH there were many things we could not cover. We sent the
Zurich in 1992 and the University of Dijon in 1997. final manuscript to Springer on the first day of spring 1998.
His research interests have ranged widely, including: We were at opposite ends of our careers when we em-
diverse aspects of the theory of foliations; differentiable barked on this project, and we came with our own tastes,
maps—jet spaces, immersions and embeddings, knotting but it was a joy to explore the mathematics together and
for high-dimensional spheres; complex analytic structures; to argue until we agreed on how to present each idea. The
orbifolds and complexes of groups. For the past fifteen years structure of our profession does not reward the effort of
he has concentrated his efforts on the archives of Armand writing a monograph as readily as it rewards theorems
Borel (now in Geneva) and René Thom. He also initiated presented in discrete papers published promptly. There
the publication of the complete mathematical works of are good reasons for this, but the enduring value of a book
René Thom, with critical notes and significant unpublished that gives students and colleagues access to a coherent body
documents. of ideas is something to be treasured, and we applaud the
American Mathematical Society for recognising that value
Response from Martin R. Bridson through the Steele Prize.
and André Haefliger
We are honoured and delighted to receive the Steele Prize The Steele Prizes are awarded by the AMS Council acting
for Mathematical Exposition. We are particularly pleased on the recommendation of a selection committee. The
that the Prize Committee commented on the value that members of the Steele Prize Subcommittee for Mathemat-
students have found in our book; to see it used widely ical Exposition were:
as a textbook has been immensely gratifying. It has also •• Charles Fefferman •• Alice Guionnet
been rewarding to see it serve as a reference for the many •• Eric Friedlander •• Michael Jordan
colleagues who have advanced geometric group theory so (Chair) •• Dusa McDuff
spectacularly over the past twenty years. •• Mark Green •• Victor Reiner
We wrote, “The purpose of this book is to describe the •• Benedict Gross •• Thomas Scanlon
global properties of complete simply-connected spaces that
are non-positively curved in the sense of A. D. Alexandrov Citation for Seminal Contribution to Research:
and to examine the structure of groups that act on such Craig Tracy and Harold Widom
spaces by isometries.” Misha Gromov brought many ideas The 2020 Steele Prize for a Seminal Contribution to Re-
from the Alexandrov school to prominence in the West, and search in Analysis/Probability Theory is awarded to Craig
his contribution goes far beyond an act of transmission: he Tracy and Harold Widom for the paper “Level-Spacing
564 Notices of the American Mathematical Society Volume 67, Number 4
FROM THE AMS SECRETARY
Distributions and the Airy Biographical Sketch: Craig Tracy
Kernel,” published in 1994 in Craig Tracy was born in England on September 9, 1945, the
Communications in Mathemati- son of Eileen Arnold, a British subject, and Robert Tracy,
cal Physics. an American serving in the U.S. Army. After immigrating to
In this work, Tracy and the United States as an infant, Tracy grew up in Missouri,
Widom found the exact as - where he attended the University of Missouri at Colum-
ymptotics of the nth largest bia, graduating in 1967 as an O. M. Stewart Fellow with
a BS degree in physics. He began his graduate studies as a
eigenvalue (n = 1, 2, ...) of an
Woodrow Wilson Fellow at Stony Brook University, where
N × N random Hermitian ma- he wrote his doctoral dissertation under the supervision of
trix, suitably scaled, as N goes Barry McCoy. After postdoctoral positions at the University
to ∞. Tracy and Widom showed of Rochester (1973–75) and the C. N. Yang Institute for
Craig Tracy that the scaled eigenvalues Theoretical Physics (1975–78), Tracy was at Dartmouth
converge to random variables, College for six years before joining the University of Califor-
each with a specific distribu- nia, Davis, in 1984. He is currently Distinguished Professor
tion function F(n; s), which of Mathematics at UC Davis.
they characterized in terms of In 2002 Tracy was awarded, jointly with Harold Widom,
the SIAM George Pólya Prize, and in 2007 the AMS-SIAM
a particular solution of a Pain-
Norbert Wiener Prize, also jointly with Widom. In 2006
levé II differential equation.
Tracy was elected a member of the American Academy of
The paper emphasizes the close Arts and Sciences.
connection between operator Tracy has two daughters, two stepdaughters, seven grand-
theory and random matrices children, and one stepgrandchild. He is married to Barbara
and proved to be an important Nelson, and they reside in Sonoma, California.
step in the development of “in-
tegrable probability.” Biographical Sketch: Harold Widom
Harold Widom From a statistician’s perspec- Harold Widom grew up in New York City, where he at-
tended Stuyvesant High School and the City College of
tive, the introduction of the
New York. He did his graduate work at the University of
Tracy–Widom distributions has been a breakthrough of
Chicago, receiving his PhD under the supervision of Irving
lasting importance. The eigenvalues of sample covariance Kaplansky. He is now Distinguished Professor Emeritus at
matrices are fundamental for the analysis of high-dimen- the University of California, Santa Cruz. His first academic
sional data. For reasons of dimension reduction or other- position was at Cornell University, where, inspired by Mark
wise, interest often focuses on the top sample eigenvalues. Kac, he turned his attention to the study of Toeplitz and
The Tracy–Widom distributions characterize the limiting Wiener–Hopf operators. This influenced much of his sub-
distribution of the top eigenvalue in the “null hypothesis” sequent research and led ultimately to his work (largely in
case of no structure, a challenge for statisticians since the collaboration with Craig Tracy) in integrable systems and
1950s. In particular, the distribution function F(2; s) gov- random matrix theory. He is an associate editor of Asymp-
totic Analysis, Journal of Integral Equations and Applications,
erns the complex-valued data of signal processing.
and Mathematical Physics, Analysis and Geometry. He is an
Further notable applications of these distribution func-
honorary editor of Integral Equations and Operator Theory.
tions include the celebrated distribution of the length of He is a member of the American Academy of Arts and Sci-
the longest increasing subsequence in a random permuta- ences. In 2002 he was awarded, jointly with Craig Tracy,
tion and the height-fluctuations random growth models the SIAM George Pólya Prize, and in 2007, also with Craig
in the KPZ (Kardar–Parisi–Zhang) universality class. The Tracy, the AMS-SIAM Norbert Wiener Prize. He has three
Airy kernel appearing in the title of the nominated paper children and four grandchildren.
has been generalized to the Airy process and then to the
Response from Craig Tracy and Harold Widom
Airy sheet, used recently in describing the KPZ fixed point.
We are honored to be named the recipients of the 2020
Craig Tracy and Harold Widom have collaborated on
Leroy P. Steele Prize for Seminal Contribution to Research
many other works that have had a significant impact. Their in Analysis/Probability Theory. We thank the members of
discovery of the Tracy–Widom distribution, however, is a the Selection Committee and the Executive Committee of
towering accomplishment exactly of the kind the Steele the AMS Council for their decision.
Prize for Seminal Contribution to Research is designed to We first express our appreciation of Estelle L. Basor, with
recognize. whom we wrote our first joint paper on random matrices.
April 2020 Notices of the American Mathematical Society 565
FROM THE AMS SECRETARY
She was a PhD student of Widom’s and had written a field of research on harmonic
joint paper with Tracy on tau-function asymptotics. It was maps and minimal surfaces
she who suggested that Widom be asked to join them in which allowed for their many
working on a random matrix problem of Freeman Dyson. applications in geometry and
This we did, and it was the beginning of our further col- topology, and it was a direct
laboration. inspiration for M. Gromov’s
Two who most influenced our early work were Madan work on pseudo-holomorphic
L. Mehta and Freeman Dyson. Mehta alerted us to an al- curves in symplectic geometry.
ternative derivation he devised for the equations of Jimbo, Karen Uhlenbeck was also
Miwa, Môri, and Sato governing eigenvalue spacings in the founder of analytic studies
the bulk of the spectrum. His derivation used Fredholm of gauge-theoretic equations in
expansions extensively. We saw that by using some operator higher dimensions. By showing
theory we could simplify his argument; then we were able Karen Uhlenbeck how to analyze the behavior
to extend the method to obtain equations for other oper- of non-convergent sequences
ator ensembles. With Freeman Dyson, we had extensive of solutions to these natural,
discussions and written correspondence in all aspects of semi-linear partial differential equations, she provided
random matrices (especially with regard to our work on precisely what is needed to apply these equations effectively
orthogonal and symplectic ensembles). to fundamental problems in topology and geometric to-
In our subsequent work on random matrices, we had pology. These ideas were elaborated in her work with S. T.
valuable interactions with, among others, Mark Adler, Yau on the existence of Yang–Mills connections, as well as
Pierre van Moerbeke, John Harnad, and Alexander Its. in the groundbreaking work by S. Donaldson and later C.
Some years later, we were able to use ideas from Bethe Taubes that applied gauge theory to analyze the structure
Ansatz to show that a largest eigenvalue distribution func- of four-dimensional manifolds.
tion arising in random matrices arises also in a scaling limit Throughout her career, Karen Uhlenbeck has worked to
of the asymmetric simple exclusion process (a model lying support young mathematicians and strengthen the mathe-
outside the class of determinantal processes). matical community. In particular, she was a cofounder both
We also thank the diverse group of researchers in ran- of the Park City Mathematical Institute and of the Women
dom matrix theory and integrable systems for making this and Mathematics Program at the Institute for Advanced
an exciting field in which to work. Study, codirecting the latter with C.-L. Terng for over twenty
years. As only the second woman to give a plenary ICM
The Steele Prizes are awarded by the AMS Council acting address and the first to be awarded the Abel Prize, she has
on the recommendation of a selection committee. The expanded the reach and visibility of women in mathematics
members of the Steele Prize Subcommittee for Seminal and is an inspiration to all mathematicians.
Contribution to Research were:
•• Eric Friedlander •• Michael Jordan Biographical Sketch: Karen Uhlenbeck
(Chair) •• Dusa McDuff Karen Uhlenbeck was born in 1942, the first of four chil-
•• Mark Green dren. Her mother was an artist and her father an engineer.
•• Benedict Gross •• Victor Reiner She grew up in rural northern New Jersey and graduated
•• Alice Guionnet •• Thomas Scanlon from Bernards High School in 1960. She received her BS in
mathematics from the University of Michigan in 1964 and
Citation for Lifetime Achievement: her PhD from Brandeis University under the direction of
Karen Uhlenbeck Richard Palais in 1968. After postdoctoral positions at the
The 2020 Steele Prize for Lifetime Achievement is awarded Massachusetts Institute of Technology and the University
to Karen Uhlenbeck for her long-lasting influence in geo- of California, Berkeley, she became an assistant professor
metric topology and analysis and for her mentorship of at the University of Illinois in Urbana in 1972, followed by
young people and women in mathematics. Karen Uhlen- associate and full professor at the University of Illinois in
beck’s mathematics has laid the foundation for a tremen- Chicago (1977–83). In 1983 she moved to the University
dous range of research in differential geometry and geo- of Chicago. After four years at the University of Chicago,
metric analysis over the past four decades. Her early work she moved to the University of Texas at Austin in 1987, and
with J. Sacks and R. Schoen analyzed the limiting behavior spent most of her career there as a Sid Richardson Professor
of sequences of minimizers of geometric functionals (such of Mathematics, retiring in 2014. Since this time she has
as the energy functional) and showed that, even though had a visiting position at the Institute for Advanced Study
there may be no limit in the conventional sense, such in Princeton.
sequences often have geometrically understandable limits Karen Uhlenbeck has been on the council of the AMS,
that appear as “bubbles.” This insight gave birth to a whole a vice president of the AMS, and was elected a Fellow of
566 Notices of the American Mathematical Society Volume 67, Number 4
FROM THE AMS SECRETARY
the AMS. Among her honors are a MacArthur Fellowship, up the profession to include women. And where would we
election to the National Academy of Sciences, the National all be without the women and men who continue on with
Medal of Science, the Steele Prize for Seminal Contribution new contributions to our subject?
to Research, election to the American Philosophical Society, Many thanks to all of the above.
and honorary degrees from seven colleges and universities.
Most recently, she received the Abel Prize in 2019. She is The Steele Prizes are awarded by the AMS Council acting
proud to have been one of the founders of the Park City on the recommendation of a selection committee. The
Mathematics Institute, of her work with Chuu-Lian Terng members of the Steele Prize Subcommittee for Lifetime
in starting the Women and Math Program at IAS, and of the Achievement were:
outreach projects at the University of Texas (which include •• Charles Fefferman •• Alice Guionnet
Saturday Morning Mathematics and the Distinguished •• Eric Friedlander •• Michael Jordan
Lectureships for Women). According to the Mathematics (Chair) •• Dusa McDuff
Genealogy Projects, she has had nineteen PhD students •• Mark Green •• Thomas Scanlon
(but she counts twenty). •• Benedict Gross
Response from Karen Uhlenbeck About the Prizes
It is an honor to be the second woman to receive the Steele The Steele Prizes were established in 1970 in honor of
Prize for Lifetime Achievement from the AMS. It has been George David Birkhoff, William Fogg Osgood, and William
an interesting and rewarding career. I have had ample Caspar Graustein. Osgood was president of the AMS during
opportunity in the past year to review my mathematical 1905–06, and Birkhoff served in that capacity during 1925–
career, and I continue to be amazed at how well it worked 26. The prizes are endowed under the terms of a bequest
out. I seem to have been at the right place at the right time, from Leroy P. Steele. Up to three prizes are awarded each
and to have received support from the right individuals year in the following categories: (1) Lifetime Achievement:
along the way. for the cumulative influence of the total mathematical
I would most like to talk about the mathematics; how work of the recipient, high level of research over a period
the subject of global analysis came about and developed of time, particular influence on the development of a field,
into geometric analysis, became entwined with theoretical and influence on mathematics through PhD students; (2)
physics, and continues to flourish. There is not space here Mathematical Exposition: for a book or substantial survey
for this story, but that I have been able to take part in this or expository research paper; (3) Seminal Contribution to
mathematics is due to the mathematical environment we Research: for a paper, whether recent or not, that has proved
live in. to be of fundamental or lasting importance in its field or a
I am indebted to the mathematics community for the model of important research. The Prize for Seminal Contri-
encouragement and support I received as an undergraduate, bution to Research is awarded on a six-year cycle of subject
a graduate student, a postdoc, and a developing researcher. areas. The 2020 prize was given in analysis/probability. The
My thesis advisor, Richard Palais, was a wonderful teacher 2021 prize will be given in algebra/number theory, the 2022
who pointed me in interesting mathematical directions prize in applied mathematics, the 2023 prize in geometry/
for many years. Dan Freed and I carried out many projects topology, and the 2024 prize in discrete mathematics/logic.
together, and my long-term collaboration with Chuu-Lian The Steele Prizes for Mathematical Exposition and
Terng took me in unexpected directions. My students have Seminal Contribution to Research carry a cash award of
more than paid me back over the years, and my many col- US$5,000; the Prize for Lifetime Achievement, a cash award
laborators allowed me to venture into many different types of US$10,000.
of mathematics. I have always credited S. T. Yau with finally The list of previous recipients of the Steele Prize may
establishing me in my own view as a real mathematician, be found on the AMS website at https://www.ams.org
not someone lurking on the edges of the subject. /steele-prize.
The years I spent at the University of Texas, with en-
dowment funds donated by Peter O’Donnell, gave me the Credits
opportunity to help build a major mathematics department Photo of Martin R. Bridson is courtesy of Martin R. Bridson.
Photo of André Haefliger is courtesy of Dr. Nikita Nikolayev.
and to become involved in many outreach projects. The
Photo of Craig Tracy is courtesy of Craig Tracy.
Institute for Advanced Study gave Chuu-Lian and me the Photo of Harold Widom is courtesy of Harold Widom.
opportunity to develop the Women and Math mentoring Photo of Karen Uhlenbeck is courtesy of Andrea Kane, Insti-
project and has taken me in many times, including in my tute for Advanced Study.
retirement. Finally, I would never have landed in this po-
sition without the women activists, those of the first and
second waves of feminism, as well as the mathematicians
who dedicated large chunks of their lives towards opening
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