DOCSLIB.ORG

With Andrei Okounkov Interviewer: Hiraku Nakajima

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
With Andrei Okounkov Interviewer: Hiraku Nakajima Interview with Andrei Okounkov Interviewer: Hiraku Nakajima Studied Real Math in Special physics only in the 1.5 years Courses in the Evening at of the undergraduate course. Moscow State University I heard only basic things Nakajima: Thank you very (including experiments, much for making time for which I did not like at all). our conversation today. Later I learned physics when This is a nice opportunity Witten’s paper on Chern- for me to ask you some Simons appeared, but not questions. I wanted to ask you from physics colleagues, but them during your stay at our from Tsuchiya (who worked RIMS. on CFT) and also the notes of Okounkov: It’s my pleasure. an Oxford seminar on Jones- I also have some questions I Witten theory. Then I heard want to ask you later. physicists talks on Seiberg- Nakajima: Okay, let me start Witten theory after 1994. It is with hearing about your not systematic, hence I cannot academic background. What give advice to younger readers did you study at Moscow on my path. University, especially in Okounkov: I studied mathematics and physics? mathematics at Moscow State Your supervisors were Kirillov University from 1989 to 1993, and Olshanski, so I guess you so missed the golden age of studied representation theory mathematics there and met originally, but your current many of its heroes only in the works are linked with many West. When I was a student, elds: algebraic geometry, there were two very different probability, and also physics, layers to our education. The gauge theory, string theory, regular curriculum was, I think, integrable systems. Why do very much oriented towards you have such a wide range employment in the space of knowledge? or defense industries, with At the University of Tokyo, a lot of numerical methods, which I graduated from mechanics, and core physics. many years ago, I studied I enjoyed many aspects of it, and later really liked teaching Andrei Okounkov is Professor of Mathematics at Columbia numerical methods myself, University (since 2010). In 2006, but for me the real math he was awarded the Fields Medal, was in the special courses the world’s highest honor in mathematics, for his contribution to and the seminars that were bridging probability, representation happening in the evenings. theory, and algebraic geometry. He I believe the only regular received his Ph.D. in mathematics from Moscow State University in course that contained Lie 1995. groups was M. Zelikin’s course 24 Kavli IPMU News No. 40 December 2017 on optimal control. But in the really do mathematical Kirillov seminar (often led by physics, it was completely Olshanski when Kirillov was different, and very intuitive away), the Gelfand seminar in its unpolished form. I think (led by Rudakov after Gelfand I’ve been very fortunate to be left), as well as in the courses able to learn a lot of things by Beilinson and Feigin, practically, while working on representation theory was projects and papers, including certainly very much at the learning so much directly center of things. from my collaborators. My Ph.D. project, inspired Nakajima: I have heard of by the work of Olshanski, was the“ two layer” system in on representation theory of Moscow several times. It does the innite symmetric group. not exist in Japan and other Other projects we worked on countries. Could you explain with Olshanski while I was it to readers? Does it still exist a graduate student could now? perhaps be described as Okounkov: I am surprised classical and combinatorial to hear this, in Russia there representation theory inspired has been a strong tradition by the innite-dimensional of what we call "circles" and asymptotic points of view in schools after regular (that may be traced to the classes, continuing with Gelfand school, and to the various“ schools,” etc. where ideas of Vershik and Kerov, university students and faculty respectively). So, by training, teach high-school students in this is my mathematical home. the evenings, and then on to I learned very early from special courses and seminars Olshanski, and I now repeat outside of the regular to my students, that there is curriculum at the university. a huge difference between I don’t know the history, but learning a subject abstractly, maybe this enthusiasm for in libro, and learning to work teaching has its roots in the with it in vivo. At Moscow effort of the intelligentsia to Interview state, we had a physics course educate the masses in the based on Landau and Lifshitz, 19th century? It is charming and also courses in probability to see these traditions now and stochastic processes, blossoming on foreign soil, which were certainly very e.g., in some parts of the US. solid and rigorous courses, HIraku Nakajima is Professor at the but... When, by a very lucky Research Institute for Mathematical chance, I got into Dobrushin s Sciences (RIMS), Kyoto University. ’ He will join the Kavli IPMU in April, laboratory, and saw people 2018. 25 I think I learned more while looking for food to Rahul (Rahul Pandharipande) and Yang for the Gromov- through such channels than buy, cooking, and washing on quantum cohomology. Witten theory of P1, including through the regular ones, and ironing the home-made How did it start? You did a conjecture about Hurwitz and this is also how I met diapers, etc. We didn’t have know Hurwitz theory before, numbers. Hurwitz numbers, many dear friends, including disposable diapers, nor a but Gromov-Witten invariants of course, I knew since rst, my wife Inna, when we were washing machine, and the were new to you at that time. this is just a different name both at one such school iron was a combined wedding Did you have an outlook for the characters of the called EMSch for Economics gift from all of our friends. before you started the symmetric groups, and so + Mathematics + School. In In fact, on the day of my collaboration? I have several was something from the fact, when we rst met, I was thesis defense I mishandled collaborations (Yoshioka, world that Olshanski and I a student and she was already the boiling diapers, and so Goettsche, Braverman, spent a long time rethinking; a teacher. EMSch is still going came to the defense with Finkelberg, and others), but and second, they were very very strong, and just had its one arm bandaged. I don’t I needed several years of much on my mind because 50th anniversary. recall people on the defense mutual understanding of at the time I had just proved committee expressing any respective work before we the Baik-Deif-Johansson Happy Graduate Student concern; nothing was out of actually started collaboration. conjecture on increasing Days with Family and Plenty the ordinary in those times. For you and Rahul, did you subsequences in random of Free Time But all things considered, I know each other’s work well permutations precisely by a Nakajima: You mentioned think those were exceptionally before you started? And how geometric argument with to me that you had plenty of happy years, as a family is did it go after you started? Hurwitz numbers. So, I time in your graduate course certainly the greatest source Okounkov: By another thought it couldn’t be such (and have two daughters). of happiness in life, and the lucky chance, Rahul was my a hard conjecture to prove You also told me that second largest source of next-door ofce neighbor in (which was indeed the case) you read textbooks in the happiness is to understand Chicago, and I was coming and this was the rst point of Moscow subway. How was something new, of which to a very lively seminar actual mathematical contact. that possible? It is different there was also plenty. that Fulton, Rahul, and In the work on BDJ, I from me, and probably many Compared to the stress of others were running at observed a connection with others. People study hard in being a junior faculty member the University of Chicago Kontsevich’s combinatorial graduate course in order to on a short-term contract, I still on quantum cohomology. formula for Witten’s arrive at the cutting edge of think that graduate school Amusingly, in one talk by intersection numbers on the the eld. I was fortunate to is a local maximum of free Rahul, Spencer Bloch pointed moduli spaces of curves, a get a permanent position time for a young researcher in out that Bernoulli numbers very fashionable topic at the after graduating from my mathematics and one should are appearing in Faber- time. One could see how master’s course (that was really make the best possible Pandharipande computations through Hurwitz theory and popular in my day in Japan), use of it. This means, of of Hodge integrals in the ELSV formula (which, at but young people nowadays course, that one should study exactly the same form as in the time, was fresh from the have only temporary postdoc hard, but also take time to Spencer’s and my work [B. oven) one could reach an positions for many years. They simply ponder or to consider Spencer and A. Okounkov, independent and, all things must study even harder than examples, as well as to be The character of the innite considered, transparent proof we did, I think. curious about mathematics wedge representation, Adv. of that combinatorial formula. Okounkov: Maybe my case and science in general. It is Math. 149 (2000), 1] on I was very curious to learn is not so representative, true that many books that the character of the innite more algebraic geometry, because when I started were formative for me I read wedge (with hindsight, this and I hope Rahul was graduate school in 1993, on the subway rides to and is the degree 0 term in the equally happy to listen to my I already had a family, the from the University, and to eventual theory of completed explanations of other relevant country’s economy was in this day I try to always have a cycles of Rahul, Eskin, and I).
Load more

Recommended publications
  • Interview with Andrei Okounkov “I Try Hard to Learn to See the World The
    Interview with Andrei Okounkov “I try hard to learn to see the world the way physicists do” Born in Moscow in 1969, he obtained his doctorate in his native city. He is currently professor of Representation Theory at Princeton University (New Jersey). His work has led to applications in a variety of fields in both mathematics and physics. Thanks to this versatility, Okounkov was chosen as a researcher for the Sloan (2000) and Packard (2001) Foundations, and in 2004 he was awarded the prestigious European Mathematical Society Prize. This might be a very standard question, but making it is mandatory for us: How do you feel, being the winner of a Fields Medal? How were you told and what did you do at that moment? Out of a whole spectrum of thoughts that I had in the time since receiving the phone call from the President of the IMU, two are especially recurrent. First, this is a great honour and it means a great responsibility. At times, I feel overwhelmed by both. Second, I can't wait to share this recognition with my friends and collaborators. Mathematics is both a personal and collective endeavour: while ideas are born in individual heads, the exchange of ideas is just as important for progress. I was very fortunate to work with many brilliant mathematician who also became my close personal friends. This is our joint success. Your work, as far as I understand, connects several areas of mathematics. Why is this important? When this connections arise, are they a surprise, or you suspect them somehow? Any mathematical proof should involve a new ingredient, something which was not already present in the statement of the problem.
    [Show full text]
  • The 68Th Geometry Symposium : Program (Aug
    The 68th Geometry Symposium : Program (Aug. 31st 〜 Sep. 3rd, 2021 : Zoom) Aug. 31st 9:30-10:30 Plenary Hiraku Nakajima(KAVLI IPMU, University of Tokyo) Intersection cohomology groups of instanton moduli spaces and cotangent bundles of affine flag varieties 10:50-11:50 Plenary Masahito Yamazaki (KAVLI IPMU, Univeristy of Tokyo) Quiver Yangians and Donaldson-Thomas invariants Session A： Session B： 13:30-14:10 Parallel Shintaro Akamine (Nihon University) Tomoya Akamatsu (Osaka University) Reflection principles for maximal surfaces A new transport distance and its associated Ricci curvature of hypergraphs 14:30-15:10 Parallel Kazuhiro Okumura (NIT, Asahikawa College) Hiroshi Tsuji (Osaka University) A certain tensor on real hypersurfaces in a nonflat Dilation type inequalities for strongly-convex sets in weighted complex space form Riemannian manifolds 15:30-16:10 Parallel Naomichi Nakajima (Hokkaido University) Daisuke Kazukawa (Osaka University) Local normal forms of e/m-wavefronts in flat affine Convergence of high-dimensional ellipsoids, spheres, and coordinates projective spaces Sep. 1st 9:30-10:30 Plenary Qing-Ming Cheng (Fukuoka University) Chern problems on minimal hypersufaces 10:50-11:50 Plenary Yasushi Homma (Waseda University) Higgs algebra in harmonic analysis on the Grassmannian of 2-planes Session A： Session B： 13:30-14:10 Parallel Shinichiro Kobayashi (Tohoku University) Yoshiki Kaneko (Waseda University) On the higher Laplace eigenvalues of vertex-transitive Local solutions of the tt*-Toda equations and quantum graphs cohomology of inuscule flag manifolds 14:30-15:10 Parallel Hiroki Nakajima (Tohoku University) Takuma Tomihisa (Waseda University) Convergence of group actions in metric measure Spinor analysis on spaces of constant curvature geometry 15:30-16:10 Parallel Ryuya Namba (Ritsumeikan University) Yuya Ikeda (Hiroshima University) Random walks on covering graphs of polynomial volume Designs on vector bundles growth via discrete geometric analysis Sep.
    [Show full text]
  • LECTURE Series
    University LECTURE Series Volume 18 Lectures on Hilbert Schemes of Points on Surfaces Hiraku Nakajima American Mathematical Society http://dx.doi.org/10.1090/ulect/018 Selected Titles in This Series 18 Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, 1999 17 Marcel Berger, Riemannian geometry during the second half of the twentieth century, 1999 16 Harish-Chandra, Admissible invariant distributions on reductive p-adic groups (with notes by Stephen DeBacker and Paul J. Sally, Jr.), 1999 15 Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, 1999 14 Lars Kadison, New examples of Frobenius extensions, 1999 13 Yakov M. Eliashberg and William P. Thurston, Confoliations, 1998 12 I. G. Macdonald, Symmetric functions and orthogonal polynomials, 1998 11 Lars G˚arding, Some points of analysis and their history, 1997 10 Victor Kac, Vertex algebras for beginners, Second Edition, 1998 9 Stephen Gelbart, Lectures on the Arthur-Selberg trace formula, 1996 8 Bernd Sturmfels, Gr¨obner bases and convex polytopes, 1996 7 Andy R. Magid, Lectures on differential Galois theory, 1994 6 Dusa McDuff and Dietmar Salamon, J-holomorphic curves and quantum cohomology, 1994 5 V. I. Arnold, Topological invariants of plane curves and caustics, 1994 4 David M. Goldschmidt, Group characters, symmetric functions, and the Hecke algebra, 1993 3 A. N. Varchenko and P. I. Etingof, Why the boundary of a round drop becomes a curve of order four, 1992 2 Fritz John, Nonlinear wave equations, formation of singularities, 1990 1 Michael H. Freedman and Feng Luo, Selected applications of geometry to low-dimensional topology, 1989 Lectures on Hilbert Schemes of Points on Surfaces University LECTURE Series Volume 18 Lectures on Hilbert Schemes of Points on Surfaces Hiraku Nakajima American Mathematical Society Providence, Rhode Island Editorial Board Jerry L.
    [Show full text]
  • Applications at the International Congress by Marty Golubitsky
    From SIAM News, Volume 39, Number 10, December 2006 Applications at the International Congress By Marty Golubitsky Grigori Perelman’s decision to decline the Fields Medal, coupled with the speculations surrounding this decision, propelled the 2006 Fields Medals to international prominence. Stories about the medals and the award ceremony at the International Congress of Mathematicians in Madrid this summer appeared in many influential news outlets (The New York Times, BBC, ABC, . .) and even in popular magazines (The New Yorker). In Madrid, the topologist John Morgan gave an excellent account of the history of the Poincaré conjecture and the ideas of Richard Hamilton and Perelman that led to the proof that the three-dimensional conjecture is correct. As Morgan pointed out, proofs of the Poincaré con- jecture and its direct generalizations have led to four Fields Medals: to Stephen Smale (1966), William Thurston (1982), Michael Freedman (1986), and now Grigori Perelman. The 2006 ICM was held in the Palacio Municipal de Congressos, a modern convention center on the outskirts of Madrid, which easily accommodated the 3600 or so participants. The interior of the convention center has a number of intriguing views—my favorite, shown below, is from the top of the three-floor-long descending escalator. Alfio Quarteroni’s plenary lecture on cardiovascular mathematics was among the many ses- The opening ceremony included a welcome sions of interest to applied mathematicians. from Juan Carlos, King of Spain, as well as the official announcement of the prize recipients—not only the four Fields Medals but also the Nevanlinna Prize and the (newly established) Gauss Prize.
    [Show full text]
  • Arxiv:2101.11672V2 [Math.AG] 3 Feb 2021 Ok Fwte [ Witten Interac of and Works Insights Geom of Symplectic Physics
    INTEGRABLE HIERARCHY FOR THE RESOLVED CONIFOLD MURAD ALIM AND ARPAN SAHA Abstract. We provide a direct proof of a conjecture of Brini relating the Gromov– Witten theory of the resolved conifold to the Ablowitz–Ladik integrable hierarchy at the level of primaries. We use a functional representation of the Ablowitz–Ladik hierarchy as well as a difference equation for the Gromov–Witten potential. We express the solution of the difference equation as an analytic function which is a specialization of a Tau function put forward by Bridgeland in the study of wall- crossing phenomena of Donaldson–Thomas invariants. 1. Introduction The study of integrable structures as the underlying organizing principle of Gromov– Witten (GW) theory has been a great source of insights and interactions between various areas of mathematics such as algebraic and symplectic geometry, the study of integrable systems, and mathematical physics. The KdV integrable hierarchy for GW theory of a point was put forward in the works of Witten [Wit91] and Kontsevich [Kon92], relating the intersection theory on the Deligne–Mumford compactification of the moduli space of curves to 2d topological gravity. For P1, the relation of GW theory to the Toda integrable hierarchy was studied in [EHY95, EY94, EYY95, EHX97, Pan00]. In [Pan00], a conjectured difference equation for the GW potential implied a corresponding difference equation for Hurwitz numbers which was proved in [Oko00]. A general construction of an integrable hierarchy for a given Gromov–Witten theory was given by Dubrovin and Zhang in [DZ01], see also [Dub14] for an overview, yet explicit examples remain sparse, see also [ADK+06] for a relation between topologi- cal string theory and integrable hierarchies.
    [Show full text]
  • The Quaternion
    The Quaternion Volume 22: Number 1; Fall, 2007 The Newsletter of the Department of Mathematics and Statistics Just a Thought about Poincaré The R. Kent Nagle Lecture Series by Boris Shekhtman The Navier-Stokes (ordinary differential) equations are Andrei Okounkov, Grigori Perelman, Terence Tao, extensions of Newton’s laws of motion to flows of Wendelin Werner. Which of these names sounds liquids or gases. Developed by Claude-Louis Navier familiar? If you circled Perelman you are not alone. I do and George Gabriel Stokes (among others) during the not remember when so many people from different Nineteenth century, these equations are critical in walks of life asked me about a mathematician: modeling weather, air flow around planes and rockets, “Do you know the dude? What’s he done?” motions of stars in a galaxy, and many other physical “Not personally but he solved the Poincaré phenomena. Conjecture.” These equations are very difficult to solve, and the “What’s that?” Clay Institute has offered $ 1,000,000 for a tractable “He proved that a 3-dimensional, simply-connected solution as one of the seven “millennial problems” in manifold is homeomorphic to a sphere” modern mathematics. Then after a pause, “Yeah, whatever, dude. What do Fang-Hua Lin , Silver Professor of Mathematics at you think? Is he nuts?” the Courant Institute of Mathematical Sciences at NYU So who is Perelman and why is he crazy? and recipient of a Sloan fellowship, a Presidential Young Grigori “Grisha” Perelman, WJM, 40, Investigator Award, the Bochner Prize, and the Chern mathematician, lives as a recluse in St.
    [Show full text]
  • Andrei Okounkov [email protected]
    Andrei Okounkov [email protected] Born July 26, 1969, in Moscow. Citizen of the Russian Federation and of the United States of America. Current positions: | Samuel Eilenberg Professor of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027 USA | Full Professor, Center for Advanced Studies, Skolkovo Institute for Science and Technology, Nobel 3, Moscow Region, 143026 Russia | Academic Supervisor, International Laboratory of Representation Theory and Mathematical Physics, Higher School of Economics, Usacheva 6, Moscow, 119048 Russia Education: 1993 B.S. in Mathematics, Moscow State University, summa cum laude 1995 Ph.D. in Mathematics, Moscow State University Honors: Sloan Research Fellowship (2000), Packard Fellowship (2001), European Mathematical Society Prize (2004), Fields Medal (2006), Compositio Prize (2009). Member of the US National Academy of Science (2012) and of the American Academy of Arts and Sciences (2016). ICM Plenary talk (2018). Member of the Royal Swedish Academy of Sciences (2020). Service: I am a member of Executive Committee of the International Mathematics Union, the Executive Organizing Com- mittee of the 2022 International Congress of Mathematicians, and cochair of the Local Organizing Committee of the 2022 International Congress of Mathematicians. I am also member of both the Russian and the US National Com- mittees. I am a long-time supporter of the Mathematical Sciences Research Institute in Berkeley, first as a member and then chair of the Scientific Advisory Committee, and then as a member and vice-chair of the Board of Trustees. Other advisory and trustee board memberships include the Clay Mathematics Institute, Packard Foundation, Simons Center for Geometry and Physics, Hamilton Mathematics Institute, and others.
    [Show full text]
  • World Premier International Research Center Initiative (WPI) List of Working Group Leaders and Assigned Members
    World Premier International Research Center Initiative (WPI) List of Working Group Leaders and Assigned Members April 2013 WPI Advanced Institute for Materials Research (AIMR), Tohoku University Name Affliation/Title PO Yoshihito Osada Senior Visiting Scientist, RIKEN Advanced Science Institute Hideo Hosono Professor, Frontier Research Center, Tokyo Institute of Technology Yasuhiko Shirota Professor Emeritus, Osaka University Director, Nanosystem Research Institute, The National Institute of Advanced Industrial Japanese Tomohiko Yamaguchi Science and Technology Toyonobu Yoshida Fellow of National Institute for Materials Science POSCO Professor of Physical Metallurgy, Chair of the Faculty, Department of Materials Samuel M. Allen Science and Engineering, Massachusetts Institute of Technology, USA Professor and Director of Research, Department of Materials Science & Metallurgy, Colin Humphreys University of Cambridge, UK Overseas Samuel l. Stupp Director, Institute for BioNanotechnology in Medicine Northwestern University, USA Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), The University of Tokyo Name Affliation/Title PO Ichiro Sanda Professor, Department of Physics, Kanagawa University Yutaka Hosotani Professor, Department of Physics, Graduate School of Science, Osaka University Tetsuji Miwa Program-Specific Professor, Institute for Liberal Arts and Sciences, Kyoto University Japanese Hiraku Nakajima Professor, Research Institute for Mathematical Science, Kyoto University Julian Schwinger Distinguished Professor
    [Show full text]
  • Letters to the Editor
    LETTERS TO THE EDITOR Still making invidious comparisons Response from Edward F. Aboufadel In view of the Society’s recent efforts to increase the partic- Prof. Dawson is correct to note that STEM professional ipation and recognition of women in our field, I was dis- organizations, including the Society, have embarked on heartened to read, in the opening sentence of the memorial intentional efforts “to increase the participation and recog- tribute to Jane Cronin Smiley Scanlon in the October No- nition of women.” For that reason, in writing the tribute to tices, the description of her as “one of the most prominent my thesis advisor, I was deliberate in emphasizing her gen- female mathematicians of the twentieth century,” followed der. Recent research [M] shows that one reason that young later on by the section heading “Celebrating an Outstand- women lose interest in STEM fields is due to an insufficient ing Woman Mathematician” and, in the final sentence of number of female role models. Prof. Scanlon can serve as a historic role model, having persisted in a time when the tribute, the phrase “a star in the constellation of female women in STEM faced “barriers to advanced training, bias mathematicians.” It is clear from her name that Professor against women in hiring and promotion practices, and a Scanlon was a woman, and the description in the tribute lack of recognition of women’s accomplishments” [Q]. Yes, of her accomplishments leaves no doubt about her stature Scanlon was one of the great mathematicians of the 20th within the field. The qualifying adjectives “woman” and century.
    [Show full text]
  • Enumerative Geometry and Geometric Representation Theory
    Proceedings of Symposia in Pure Mathematics Volume 97.1, 2018 http://dx.doi.org/10.1090/pspum/097.1/01681 Enumerative geometry and geometric representation theory Andrei Okounkov Abstract. This is an introduction to: (1) the enumerative geometry of ra- tional curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute. 1. Introduction 1.1. These notes are written to accompany my lectures given in Salt Lake City in 2015. With modern technology, one should be able to access the materials from those lectures from anywhere in the world, so a text transcribing them is not really needed. Instead, I will try to spend more time on points that perhaps require too much notation for a broadly aimed series of talks, but which should ease the transition to reading detailed lecture notes like [90]. The fields from the title are vast and they intersect in many different ways. Here we will talk about a particular meeting point, the progress at which in the time since Seattle I find exciting enough to report in Salt Lake City. The advances both in the subject matter itself, and in my personal understanding of it, owe a lot to M. Aganagic, R. Bezrukavnikov, P. Etingof, D. Maulik, N. Nekrasov, and others, as will be clear from the narrative. 1.2. The basic question in representation theory is to describe the homomor- phisms (1) some algebra A → matrices , and the geometric representation theory aims to describe the source, the target, the map itself, or all of the above, geometrically.
    [Show full text]
  • The Work of Andrei Okounkov
    The work of Andrei Okounkov Giovanni Felder Andrei Okounkov’s initial area of research is group representation theory, with par- ticular emphasis on combinatorial and asymptotic aspects. He used this subject as a starting point to obtain spectacular results in many different areas of mathematics and mathematical physics, from complex and real algebraic geometry to statistical mechanics, dynamical systems, probability theory and topological string theory. The research of Okounkov has its roots in very basic notions such as partitions, which form a recurrent theme in his work. A partition λ of a natural number n is a non-increasing sequence of integers λ1 ≥ λ2 ≥···≥0 adding up to n. Partitions are a basic combi- natorial notion at the heart of the representation theory. Okounkov started his career in this field in Moscow where he worked with G. Olshanski, through whom he came in contact with A. Vershik and his school in St. Petersburg, in particular S. Kerov. The research programme of these mathematicians, to which Okounkov made substantial contributions, has at its core the idea that partitions and other notions of representa- tion theory should be considered as random objects with respect to natural probability measures. This idea was further developed by Okounkov, who showed that, together with insights from geometry and ideas of high energy physics, it can be applied to the most diverse areas of mathematics. This is an account of some of the highlights mostly of his recent research. I am grateful to Enrico Arbarello for explanations and for providing me with very useful notes on Okounkov’s work in algebraic geometry and its context.
    [Show full text]
  • View This Volume's Front and Back Matter
    CONTEMPORARY MATHEMATICS 413 Representations of Algebraic Groups ~ Quantum Groups ~ and Lie Algebras AMS-IMS-SIAM Joint Summer Research Conference July 11-15, 2004 Snowbird Resort, Snowbird, Utah Georgia Benkart Jens C. Jantzen Zongzhu Lin Daniel K. Nakano Brian J. Parshall Editors http://dx.doi.org/10.1090/conm/413 Representations of Algebraic Groups, Quantum Groups, and Lie Algebras Conference Group Photo Snowbird Resort July 2004 CoNTEMPORARY MATHEMATICS 413 Representations of Algebraic Groups, Quantum Groups, and Lie Algebras AMS-IMS-SIAM Joint Summer Research Conference July ll-15, 2004 Snowbird Resort, Snowbird, Utah Georgia Benkart Jens C. Jantzen Zongzhu Lin Daniel K. Nakano Brian J. Parshall Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Carlos Berenstein Andreas Blass Abel Klein This volume contains the proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, held at the Snowbird Resort, Snowbird, UT, from July 11-15, 2004, with support from the National Science Foundation, grant DMS-9973450. 2000 Mathematics Subject Classification. Primary 05E10, 14L17, 16G20, 17Bxx, 20C08, 20Gxx. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Library of Congress Cataloging-in-Publication Data AMS-IMS-SIAM Joint Summer Research Conference, Representations of Algebraic Groups, Quan- tum Groups, and Lie Algebras (2004 : Snowbird, Utah) Representations of algebraic groups, quantum groups, and Lie algebras : AMS-IMS-SIAM Joint Summer Research Conference, Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, July 11-15, 2004, Snowbird, Utah/ Georgia M.
    [Show full text]