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).