2021 Leroy P. Steele Prizes - American Mathematical Society
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FROM THE AMS SECRETARY
2021 Leroy P. Steele Prizes
The 2021 Leroy P. Steele Prizes were presented at the Annual Meeting of the AMS, held virtually January 6–9, 2021.
Noga Alon and Joel Spencer received the Steele Prize for Mathematical Exposition. Murray Gerstenhaber was awarded
the Prize for Seminal Contribution to Research. Spencer Bloch was honored with the Prize for Lifetime Achievement.
Citation for Mathematical Biographical Sketch: Noga Alon
Exposition: Noga Alon Noga Alon is a Professor of Mathematics at Princeton
and Joel Spencer University and a Professor Emeritus of Mathematics and
The 2021 Steele Prize for Math- Computer Science at Tel Aviv University, Israel. He received
ematical Exposition is awarded his PhD in Mathematics at the Hebrew University of Jeru-
to Noga Alon and Joel Spencer salem in 1983 and had visiting and part-time positions in
for the book The Probabilistic various research institutes, including the Massachusetts
Method, published by Wiley Institute of Technology, Harvard University, the Institute
and Sons, Inc., in 1992. for Advanced Study in Princeton, IBM Almaden Research
Now in its fourth edition, Center, Bell Laboratories, Bellcore, and Microsoft Research
The Probabilistic Method is an (Redmond and Israel). He joined Tel Aviv University in
invaluable toolbox for both 1985, served as the head of the School of Mathematical
Noga Alon the beginner and the experi- Sciences in 1999–2001, and moved to Princeton in 2018.
enced researcher in discrete He supervised more than twenty PhD students. He serves
probability. It brings together on the editorial boards of more than a dozen international
through one unifying perspec- technical journals and has given invited lectures in numer-
tive a head-spinning variety of ous conferences, including plenary addresses in the 1996
results and methods, linked to European Congress of Mathematics and in the 2002 Inter-
applications in graph theory, national Congress of Mathematicians. He has published
combinatorics, number theory, one book and more than 500 research papers.
and geometry. His research interests are mainly in combinatorics, graph
This enduring book has been theory, and their applications in theoretical computer
used around the world. Much science. His main contributions include the study of ex-
cited by important papers in pander graphs and their applications, the investigation of
leading journals, it functions derandomization techniques, the foundation of streaming
as both on-ramp and toolbox. algorithms, the development and applications of algebraic
Joel Spencer The Probabilistic Method has its and probabilistic methods in discrete mathematics, and the
roots in the work of Paul Erdős, study of problems in information theory, combinatorial
and this volume brought together a dizzying array of ideas, geometry, and combinatorial number theory.
methods, and applications, using them to prove deter- Alon is a Fellow of the AMS and of the Association for
ministic properties of combinatorial systems and typical Computing Machinery (ACM). He is a member of the Israel
properties of random discrete systems. Academy of Sciences and Humanities, of the Academia
Applications presented are frequently state-of-the-art Europaea, and of the Hungarian Academy of Sciences. His
bounds, and Alon and Spencer have synthesized their honors include the Erdős Prize, the Feher Prize, the Polya
various proofs and at times offer alternative proofs of their Prize, the Bruno Memorial Award, the Landau Prize, the
own. As is often the case for books that help to seed a field, Gödel Prize, the Israel Prize, the EMET Prize, the Dijkstra
the material brought together here into a unified fabric was Prize, the Nerode Prize, and the Kanellakis Prize. He holds
previously unavailable in one place. The style of the book honorary doctorates from ETH Zürich and the University
is both crystalline and engaging. of Waterloo.
April 2021 Notices of the American Mathematical Society 629
FROM THE AMS SECRETARY
Biographical Sketch: Joel Spencer to try to cover all the significant applications of the method,
Joel Spencer is a Silver Professor of Mathematics and and with all the beautiful results and techniques developed
Computer Science at the Courant Institute, New York since then, this is a totally impossible task now. The em-
University. He works in the fecund intersection of discrete phasis in the book is on methodology and ideas, with an
mathematics, probability, and logic. He is cofounder of the attempt to explain those in a precise and yet intuitive and
journal Random Structures and Algorithms. He has served on readable manner. As is often the case, the work often led us
the AMS Executive Committee, chaired the Meetings and to find new arguments and proofs, and this has been one
Conferences Committee, and, most proudly, helped found of the main satisfying aspects of the project.
and chaired the Epsilon Fund for High School Math Camps. The book, and the probabilistic method itself, would not
He is a Fellow of the AMS and the Society for Industrial and exist without the immense contributions of the superb re-
Applied Mathematics (SIAM). He has authored over 200 searchers working in the area, starting with the fundamental
research publications and seven books. His latest book, As- contributions of the giant founders and continuing with the
ymptopia, was published by the AMS. His Erdős number is 1. beautiful results of numerous others. We are indebted to
all of them, including our many colleagues, collaborators,
Response from Noga Alon and Joel Spencer and students. It is rewarding to see that the resulting text
We are delighted and honored to receive the Steele Prize is used extensively by researchers and students, and it is a
for Mathematical Exposition. Already, when we started great honor to thank the prize committee and the American
writing the first edition of the book, there had been a sub- Mathematical Society for the recognition.
stantial number of known impressive applications of the Finally, it is a special pleasure to thank our wives, Nurit
probabilistic method in the study of problems in discrete and MaryAnn. Their understanding and encouragement
mathematics, as well as in other areas, including infor- have been crucial in the successful writing enterprise.
mation theory, number theory, geometry, and theoretical
computer science. Three decades later, after four editions Citation for Seminal
of the book have been published, it is now clear that the Contribution to Research:
method is one of the most powerful and widely used tools Murray Gerstenhaber
in combinatorics and its applications. We believe and hope The 2021 Steele Prize for Sem-
that our book contributed to the success and popularity of inal Contribution to Research
the subject. is awarded to Murray Gersten-
I [Noga] personally first learned about the probabilistic haber for “The Cohomology
method when I was still in high school. I read a version of Structure of an Associative
one of the earliest results established using it: the proof of Ring,” Annals of Mathematics
the lower bound for Ramsey numbers discovered in 1947 78 (1963), 267–288, and “On
by Paul Erdős, the founder of the method. I still recall the the Deformation of Rings and
admiration I felt going through the concise and elegant Algebras,” Annals of Mathemat-
argument, an admiration that only increased when I kept Murray Gerstenhaber ics 79 (1964), 59–103. These
following the profound impact of other applications of two remarkable Annals of Math-
the method on the development of discrete mathematics. ematics papers established the
I [Joel] began working with Paul Erdős while still a foundations of algebraic deformation theory, developing
graduate student. Uncle Paul, as we all called him, was, is, a rich structure on the Hochschild cohomology. These
and forever will be the center of my professional life. He papers have had and continue to have a huge impact on
combined brilliance with a powerful personality—pushing many areas of mathematics and physics, including “higher
us always to new heights with his admonition “prove and algebra” and deformation quantization.
conjecture.” Erdős’s style was to prove specific individual In these two seminal and much-cited papers, Gersten-
results. A mathematician once said “Erdős only gives us haber laid the foundation of algebraic deformation theory,
corollaries of the great metatheorems which remain unfor- discovering that formal deformations of an algebraic struc-
mulated in the back of his mind.” It was my hubris that the ture are governed by an appropriate cohomology theory
theorems could be made into a theory, that the methods and the existence and classification of such deformations by
could be made into a methodology. I was so fortunate a graded Lie algebra structure on the cohomology. The ideas
to find Noga, who shared my passion and surpassed my initiated in these papers have permeated a multiplicity of
abilities. To my great joy, the probabilistic method is now a subjects. Gerstenhaber’s theory of deformations of associa-
basic element of combinatorial and probabilistic thinking. tive algebras has been extended to commutative Lie, Hopf,
This award is surely icing on the cake. Poisson, Leibniz, and other classes of algebras. Algebras
Our book was never meant to be an encyclopedic treat- carrying a structure like the one introduced by Gerstenhaber
ment of the subject. Even in the late ’80s, it looked difficult are now called Gerstenhaber algebras, with examples the
630 Notices of the American Mathematical Society Volume 68, Number 4
FROM THE AMS SECRETARY
exterior algebra of a Lie algebra, the multivector fields on the Code of Ethics of the Society. As a Regional Secretary
a manifold using the Schouten–Nijenhuis bracket, and of the Society, Gerstenhaber reconnected the AMS with
differential forms on a Poisson manifold. the American Association for the Advancement of Science
Gerstenhaber’s homological approach to deformation (AAAS) by instituting a series of annual symposia on “Some
theory has been taken up by the physics community, be- Mathematical Questions in Biology,” the proceedings of
ginning with looking at special relativity as a deformation which were published by the Society until those symposia
of Newtonian mechanics. Work of Bayen, Flato, Fronsdal, were discontinued.
Lichnerowicz, and Sternheimer studied quantization in Gerstenhaber served as an editor of the Bulletin of the
terms of a deformation of the algebra of functions on a American Mathematical Society from 1966 to 1971 and as
Poisson manifold, initiating the subject of deformation managing editor from 1968 to 1971. As first chair of the
quantization. This led to applications to particle physics, Steele Prize committee, he nominated Solomon Lefschetz
string theory, and gauge theory, including remarkable work for the first award, given in 1970. Gerstenhaber is a Fellow
of Kontsevich that settled the deformation quantization of the AAAS and an Inaugural Fellow of the AMS. He was
of Poisson manifolds. The continued use of the methods also one of the founders of the Association of Members of
pioneered in these two papers is testimony to their endur- the Institute for Advanced Study (AMIAS), its alumni orga-
ing influence. nization, which he served for many years as treasurer. As a
member of an advisory committee of the National Science
Biographical Sketch
Foundation, Gerstenhaber moved to fund the Mathemati-
Murray Gerstenhaber was born in 1927 in Brooklyn to a
cal Sciences Research Institute at Berkeley and the Institute
family that lost much during the Great Depression, includ-
for Mathematical Analysis at Minnesota.
ing their brownstone home. They survived on the earnings
of his hardworking seamstress mother. Gerstenhaber Response from Murray Gerstenhaber
attended the Speyer School and the Bronx High School of Thank you for this honor. It has been a wonderful journey
Science. A scholarship allowed him to enter Yale Univer-
learning of Riemann surfaces from Weyl’s Die Idee der
sity in March 1944. There, his encounter with student and
Riemannschen Fläche, to Teichmüller’s attempt to define
institutional anti-Semitism was considerably mitigated by
their infinitesimal deformations, to the correct definition
Einar Hille and Deane Montgomery, both later presidents
by Frölicher and Nijenhuis of infinitesimal deformations
of the AMS. Gerstenhaber was drafted in May 1945. In Feb-
of complex manifolds of arbitrary dimension, to seeing
ruary 1946, he was sent to OMGUS, the Office of Military
Kodaira and Spencer develop the deformation theory of
Government, US, in Berlin for 10 months. There he was
complex manifolds. My own Annals of Mathematics papers
assigned first to the Transportation Division and then to a
of 1963 and 1964 creating algebraic deformation theory,
small Army university run for soldiers on leave.
for which you are now honoring me, were but the next step.
With the support of the GI Bill, Gerstenhaber returned
in March 1947 to a Yale transformed by war veterans, ma- Even more wonderful has been seeing later the work
tured by their experiences and driven to make up for time of Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer,
lost during the war. Gerstenhaber graduated in June 1948 who, with algebraic deformation theory, were able to
and then entered the University of Chicago. He received a deduce the spectrum of hydrogen without using either
PhD in 1951 with A. Adrian Albert; his mentor at Chicago Schrödinger or wave mechanics. They also recognized
was André Weil. With Frank B. Jewett Fellowships from Bell that Einstein’s special relativity can be viewed as a defor-
Laboratories, Gerstenhaber engaged in postdoctoral study mation of Newtonian mechanics, with the speed of light
at Harvard University in 1951–1952 and at the Institute for (more precisely, its inverse) as the deformation parameter.
Advanced Study in 1952–1953, where he was an assistant A polynomial algebra in two variables can deform, with
to Hermann Weyl. Planck’s constant as deformation parameter, to the first
In 1953, Gerstenhaber joined the faculty of the Uni- Weyl algebra, which expresses the quantum relationship
versity of Pennsylvania, from which he retired in 2011. At between position and momentum. Weyl algebras allow no
Penn, Gerstenhaber served as chair of the Mathematics further deformations; they are “rigid” or “stable,” which
Department and subsequently as chair of the Faculty Sen- may suggest that physical laws deform toward stability, but
ate. He earned a JD in 1973 at Penn’s Law School and was the laws themselves do not change; only our understanding
admitted to the Pennsylvania bar. At the Law School he of them continues to evolve.
taught a course on Statistics for Law, using Supreme Court Acknowledging with gratitude those before who brought
cases as illustrations—a first for this country. us to our present level of understanding, and with the firm
At the AMS, Gerstenhaber served on the Committee on belief that those who come after us will see much farther
Human Rights, the Committee on Academic Freedom and than we have, I gratefully accept this honor you have given
Tenure, and the Council of the Society. He helped draft me.
April 2021 Notices of the American Mathematical Society 631
FROM THE AMS SECRETARY
Citation for Lifetime Response from Spencer Bloch
Achievement: I am honored (humbled, actually) to have been awarded
Spencer Bloch the Steele Prize for Lifetime Achievement from the AMS.
The Steele Prize for Lifetime Surely, if the committee had asked me, I could have come
Achievement is awarded to up with any number of more suitable candidates. Some
Spencer Bloch for his seminal thoughts on 50+ years in mathematics:
contributions linking algebraic 1. As a math teacher, I am moved to cry out “Behold, the
geometry, algebraic K-theory, infinite variety of human intelligence!” For are we math
arithmetic, and Hodge theory. people not the best placed to observe? All the modern
Bloch’s ideas pervade modern emphasis on STEM. Surely it is for naught. What will
thinking on these subjects and happen to the LEAVES and FLOWERS and ROOTS and
laid the groundwork in both the other 22 letters, constituting acronyms we cannot
Spencer Bloch techniques and framework for even imagine? And STEM seems such a rigid thing. We
many of the most exciting de- must avoid rigidity at all costs. Science is expensive, and
velopments in these subjects. it changes rapidly. People find themselves stuck in an
A striking feature of Bloch’s work is its combination outdated technology. By good fortune, I was able to focus
of extraordinary results with its seminal nature. Starting on excellence. We should insure that many students
with his remarkable work linking algebraic K-theory and have that chance. I was also blessed with an extremely
algebraic cycles, leading to the Bloch–Quillen formula, a supportive intellectual atmosphere at the University of
body of visionary work emerged. Bloch’s conjecture on Chicago. If the current political environment is allowed
rational equivalence on surfaces, the Bloch–Beilinson to fester, we risk losing such support.
conjectures, the Bloch–Kato conjecture, the Bloch–Srinivas 2. How has math changed over the years? I am fascinated
by the jujitsu wrestling match currently playing out be-
theorem on decomposition of the diagonal, Bloch’s recast-
tween math and physics. Modern ideas like string theory
ing of the Birch–Swinnerton-Dyer conjecture as a volume
and mirror symmetry were introduced in physics. People
computation, Bloch’s work on motivic cohomology, his
thought they would lead to mathematical domination
development of higher Chow groups and the Bloch–Suslin
of physics. In fact, it is the reverse. Mathematicians
theorem, and his work on the link between K2 and the
are confronted with amazing conjectures, sometimes
dilogarithm function have energized entire fields and been
admitting mathematical proofs, but totally lacking in
remarkably productive of great mathematics.
mathematical motivation or intuition.
More recently, his work with a variety of collaborators on
3. Is it possible for a seventy-six-year-old to continue to do
“irregular differential equations” and on Feynman motives math? Yes but... It is difficult to “multi-task,” and one
and mathematical physics continues to reflect the innova- has to acknowledge there are ideas out there one will
tive nature of Bloch’s mathematical work. It is difficult to never grasp.
imagine algebraic geometry, algebraic K-theory, arithmetic,
and Hodge theory without Bloch’s contributions. About the Prizes
The Steele Prizes were established in 1970 in honor of
Biographical Sketch
George David Birkhoff, William Fogg Osgood, and William
Spencer Bloch was born in 1944 in New York City. He grew Caspar Graustein. Osgood was president of the AMS during
up in Ossining, New York, a suburb of New York City. He 1905–1906, and Birkhoff served in that capacity during
was educated at Scarborough School and Deerfield Acad- 1925–1926. The prizes are endowed under the terms of a
emy, graduating from high school in 1962. He attended bequest from Leroy P. Steele. Up to three prizes are awarded
Harvard University, graduating in 1966, and earned a each year in the following categories: (1) Lifetime Achieve-
PhD in mathematics from Columbia University under the ment: for the cumulative influence of the total mathemati-
direction of Steve Kleiman in 1971. He held nontenured cal work of the recipient, high level of research over a period
positions at Princeton University and the University of of time, particular influence on the development of a field,
Michigan, moving to a tenured post at the University of and influence on mathematics through PhD students; (2)
Chicago in 1976. Mathematical Exposition: for a book or substantial survey
Bloch was a speaker at the International Congress of or expository research paper; (3) Seminal Contribution to
Mathematicians (ICM) in Helsinki, Finland, in 1978 and Research: for a paper, whether recent or not, that has proved
an ICM plenary speaker in Kyoto, Japan, in 1990. He was to be of fundamental or lasting importance in its field or a
elected to the National Academy of Science in 1994. Over model of important research. The Prize for Seminal Contri-
the years, he has held temporary research positions in En- bution to Research is awarded on a six-year cycle of subject
gland, France, Germany, India, Japan, and China. areas. The 2021 prize was given in algebra/number theory.
632 Notices of the American Mathematical Society Volume 68, Number 4FROM THE AMS SECRETARY
The 2022 prize will be given in applied mathematics, the The list of previous recipients of the Steele Prize may
2023 prize in geometry/topology, and the 2024 prize in be found on the AMS website at https://www.ams.org
discrete mathematics/logic. /prizes-awards/palist.cgi.
The Steele Prizes for Mathematical Exposition and
Seminal Contribution to Research carry a cash award of Credits
US$5,000; the Prize for Lifetime Achievement, a cash award Photo of Noga Alon was taken by Nurit Alon.
Photo of Joel Spencer is courtesy of Maryann Spencer.
of US$10,000.
Photo of Murray Gerstenhaber is courtesy of Ruth P. Gersten-
The Steele Prizes are awarded by the AMS Council act- haber.
ing on the recommendation of a selection committee. The
members of the committee for the 2021 Steele Prizes were:
• Sun-Yung A. Chang
• Charles Fefferman
• Eric M. Friedlander
• Mark L. Green (Chair)
• Alice Guionnet
• Michael I. Jordan
• Dusa McDuff
• Sylvia Serfaty
• Marie-France Vigneras
April 2021 Notices of the American Mathematical Society 633You can also read
