2015 Steele Prizes

2015 Steele Prizes The 2015 Leroy P. Steele Prizes were presented at become of utmost importance in birational geom- the 121st Annual Meeting of the AMS in San Anto- etry; they were used in the proofs of landmark nio, Texas, in January 2015. The Steele Prizes were theorems such as the invariance of plurigenera, awarded to Robert Lazarsfeld for Mathematical the finite generation of the canonical ring, and Exposition; to Rostislav Grigorchuk for Seminal the existence of flips. His books are now used for Contribution to Research; and to Victor Kac for courses and for reading and research seminars all Lifetime Achievement. over the world. They can be used both as textbooks and as sources for current research problems, and Mathematical Exposition: Robert so have great value for both students and experts Lazarsfeld in the field. They are exceptionally well written, Citation with numerous examples, new proofs, and espe- The 2015 Leroy P. Steele Prize for Mathematical cially new results reflecting Lazarsfeld’s great taste Exposition is awarded to Robert Lazarsfeld for his and originality. books Positivity in Algebraic Geometry I (Ergebnisse Biographical Sketch der Mathematik und iher Grenzgebiete. 3. Folge. Robert Lazarsfeld was born in New York City in A Series of Modern Surveys in Mathematics, 48, 1953. He graduated from Harvard College in 1975 Springer-Verlag, Berlin, 2004) and Positivity in and received his PhD from Brown University in Algebraic Geometry II (Ergebnisse der Mathematik 1980 under the direction of William Fulton. After und iher Grenzgebiete. 3. Folge. A Series of Mod- a postdoctoral stint as an instructor at Harvard, ern Surveys in Mathematics, 49, Springer-Verlag, he went to the University of California Los Angeles Berlin, 2004). in 1983. He remained Lazarsfeld’s books Positivity in Algebraic Geom- there until moving etry I and II were instant classics. He has succeeded to the University of wonderfully in putting together under the same Michigan in 1997, heading most of the areas of classical and modern where in 2007 he be- complex algebraic geometry dedicated to, or influ- came the Raymond enced by, the study of positivity. The positivity of L. Wilder Collegiate the title encompasses a circle of ideas germinat- Professor. Since 2013, ing from the notion of an ample line bundle on a Lazarsfeld has been smooth projective variety. Roughly speaking, such on the faculty of the a line bundle induces an embedding of the variety mathematics depart- into a projective space. By means of this embed- ment at Stony Brook ding, the restriction of the classical Fubini-Study University. Robert Lazarsfeld metric gives rise to a positive, or Kähler, form on Lazarsfeld was the variety in the sense of differential geometry. a Sloan Foundation Fellow (1984), a Presiden- Fundamental work of Kodaira showed that, con- tial Young Investigator (1985–1990), an invited versely, positivity in the differential-geometric speaker at the International Congress of Mathema- sense implies positivity in the algebro-geometric ticians (1990), and a Guggenheim Fellow (1998). He one. This circle of ideas has been enlarged over the delivered the 2005 AMS Colloquium Lectures and years to include singular varieties, higher rank bun- was elected to the American Academy of Arts and dles, and eventually more general sheaves, singular Sciences in 2006. From 2002 to 2009, Lazarsfeld Kähler metrics, and fractional divisors, leading to served on the editorial board of the Journal of the the introduction of multiplier ideals in both ana- American Mathematical Society, and in 2012 he lytic and algebraic settings. Here, for the first time became a Fellow of the AMS. in textbook form, the resulting algebraic theory Response from Robert Lazarsfeld of multiplier ideals is fully developed. Further- I am delighted and honored to receive the 2015 more, Lazarsfeld’s books work out the theory of Leroy P. Steele Prize for Mathematical Exposition asymptotic multiplier ideals, tools that have since for my monograph on positivity in algebraic geometry. Writing it was a protracted labor of love—or, DOI: http://dx.doi.org/10.1090/noti1231 as it occasionally seemed, madness—so naturally 416 NOTICES OF THE AMS VOLUME 62, NUMBER 4 I’m very gratified that in the end the book appears Grigorchuk used the idea of Kolmogorov com- to have proved itself useful. I’ve always believed plexity to study the complexity of algorithmic that it’s worth striving to present mathematics problems in his groups. This work attracted the at- well, and I’m pleased to have these efforts ac- tention of computer scientists and cryptographers. knowledged here. The work of Grigorchuk has influenced several The book owes a great deal to many teachers, generations of researchers in group theory. It students, and colleagues, but I’d like to single out would be impossible to imagine modern group three to whom I have a particular debt. To begin theory without Grigorchuk’s work. with I’m grateful to János Kollár for suggesting Biographical Sketch in the first place that I might write a book for the Rostislav Grigorchuk is Distinguished Professor Ergebnisse series. I am also indebted to Bill Fulton, at Texas A&M University. He received his PhD who not only introduced me as a graduate student from Lomonosov to the circle of questions around positivity but also Moscow State Uni- taught me the importance of good mathematical versity in 1978, writing. Finally, quite a bit of the material that ap- advised by Dmi- pears in the book I either learned from Lawrence try Anosov and Ein or worked out together with him, and I’d like to Anatoly Stepin. take this opportunity to express my thanks to him. He taught from 1978 to 1995 at Seminal Contribution to Research: the Moscow State Rostislav Grigorchuk University of Rail- Citation way Engineering, Rostislav Grigorchuk where he also The 2015 Leroy P. Steele Prize for Seminal Con- eventually occupied the position of head of the tribution to Research is awarded to Rostislav Department of Mathematics, and from 2000 to Grigorchuk for his article “Degrees of growth of 2002 at Lomonosov Moscow State University. He finitely generated groups and the theory of invari- was invited to the Steklov Institute of Mathematics ant means”, Izvestiya Akademii Nauk SSSR. Seriya (Moscow) in 1995 and worked there as a leading Matematicheskaya, 48 (1984), no. 5, 939–985. researcher from 1995 to 2002. In 2002 he moved In this paper Grigorchuk constructed the first to College Station, Texas, in the United States. examples of a finitely generated group whose Grigorchuk was born in 1953 in the small village growth was faster than polynomial but slower of Mukhavets in the Vishnivci district of the Ter- than exponential, answering a question raised nopilska region in Ukraine. He received the Prize by John Milnor in 1968. These groups had other of the Moscow Mathematical Society in 1979, was amazing properties: they are torsion, they give a an invited speaker at the International Congress of new counterexample to the Burnside problem, and Mathematicians in Kyoto (1990), was a Fulbright they are amenable, but not elementary amenable, Foundation Senior Fellow affiliated with Columbia thus answering another old question of Mahlon University in 1991 (by invitation of Hyman Bass), Day. Moreover, Grigorchuk’s groups turned out to twice received the Award for Best Scientific Results be just infinite and yet have finite nilpotent width. from the Russian Academy of Science (1999 and Grigorchuk showed that the class of groups of 2001), received the Distinguished Achievement intermediate growth is extremely rich: it contains Award from the Association of Former Students a continuum of 2-generator pairwise distinct of Texas A&M University (2009), and was honored groups with respect to quasi-isometry, a relation as a Fellow of the AMS in 2013. much weaker than isomorphism of groups. This His research interests are in group theory, dy- breakthrough greatly sharpened results of B. H. namical systems, random walks, spectral theory Neumann, who had proved the existence of a of groups and graphs, operator algebras, theory of continuum of nonisomorphic 2-generator groups fractals, theory of automata, and formal languages. in 1937. Response from Rostislav Grigorchuk It is important that Grigorchuk’s work not only It is a great honor to receive this award. I credit gave solutions to old standing problems but also my father, Ivan Grygorchuk, and my high school discovered new exciting classes of groups which math teachers, Vira Habryd in Ivanovo-Frankivsk became known as “branch groups” and “self- and Volodymyr Axelrod in Chernivtci (Ukraine), similar groups”. These groups provided examples for stimulating my interest in mathematics and with the previously unknown asymptotic behav- teaching me how to achieve my goals. From my ior of random walks and the spectral densities thesis adviser Anatoly Stepin at Moscow University of discrete Laplacians. They found applications and my further mentors Dmitri Anosov and Yakov in the theory of fractals, holomorphic dynamics, Sinai, I learned many things, in particular how to spectral theory of groups and graphs, and theory mix ergodic theory with algebra and functional of finite automata. analysis and not to be afraid to attack difficult APRIL 2015 NOTICES OF THE AMS 417 problems. It was a pleasure listening to a talk in They now are seen to form the backbone of many one of their seminars for three to four hours (for aspects of combinatorics, integrable systems, instance, a talk of G. Kasparov or M. Lyubich) and modular forms, enumerative algebraic geometry, then to play soccer the next day against the team and the Langlands program. They also play an of another seminar. I furthermore learned a lot important role in quantum field theory and sta- from Sergei Adian and Vladimir Arnold, to whom tistical mechanics.

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