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 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 , . 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 , national Congress of Mathematicians. He has published , , 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 Erdo˝s, 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 Erdo˝s 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.

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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 Erdo˝s 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 Erdo˝s, 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 Erdo˝s 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.” Erdo˝s’s style was to prove specific individual In these two seminal and much-cited papers, Gersten- results. A mathematician once said “Erdo˝s 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 Biographical Sketch member of an advisory committee of the National Science 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.

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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 , 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 K and the 2 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 . 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 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.

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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. of US$10,000. Photo of Joel Spencer is courtesy of Maryann Spencer. 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

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