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. 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 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: , 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 , 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 , 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 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 in nite symmetric group. not exist in 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 there representation theory inspired has been a strong tradition by the in nite-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), . ’ 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 of ce 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 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 in nite 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 in nite 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). ingredients, so this was the total collapse, and traveling book with me for the subway So I had some familiarity with start. From the beginning, it to the West was basically the ride, also in NYC. the subject, but of course no was clear that this project was only source of income for real technical knowledge. going to work; sometimes most Russian scientists. In my Collaboration with Rahul And then, after moving collaborations just have such case, my wife Inna started Pandharipande on Quantum to California, Rahul wrote a a lucky beginning. (Later, for a business and provided for Cohomology paper on various implications example, when we proved our family, while I had plenty Nakajima: Now I want to ask of the then conjectural Toda those Toda equations, or of time to think about math you about collaborations with equations of Eguchi, Hori, in some of the GW=DT

26 Kavli IPMU News No. 40 December 2017 (Gromov-Witten/Donaldson- Thomas correspondence) papers, we really had to try many different things before we found the right mix of ideas.) Now, I don’t think anybody reads that rst paper of ours, even though Rahul’s part contains an excellent introduction to virtual classes and virtual localization, among other things. But this is okay, the subject has reached much higher heights since then.

Collaboration with Nakajima: Next, collaboration curves. It seems that you had continuum of literature on Anyway, once the right with Nikita (Nikita A. a good understanding from any given popular topic. Very neurons red, it was obvious Nekrasov). Nikita understand the beginning. fortunately, there are people that the SW limit is just about mathematics very well, but Okounkov: I think like Nikita, in whose head it is the law of large numbers for he is still a physicist. I do mathematics and theoretical all ordered and all the voids partitions and the SW curve is not know many examples physics grow very differently. are lled in. I never had any just the associated limit shape of mathematicians, other In math, we spend a lot problems understanding him. (which is a Vershik-Kerov- than you, who have joint of time rethinking our Sometime in the spring style math that I knew really papers with physicists. foundations and count of 2002, I was in Paris and well), and it became a purely Many mathematicians are it as progress when a Nikita gave me a draft of his mathematical problem, as a interested in physics these phenomenon is presented in “Seiberg-Witten Prepotential physicist would say, to work days (and it is the reason why its most general form, with all from Counting” out the details. we have IPMU), but it is not essential details highlighted that contained the Nekrasov You see, a power series is that easy to overcome the and accidental features partition function Z, as we just an integral over natural communication barrier yet. In removed. As a result, our know it now. Among other numbers N, or over the reals particular, I do not have many subject is solid, not just in the things, Z is a sum over R with respect to a measure examples of collaboration sense of rigor, but also in the partitions, and I admit I like μ supported on the natural between physicists and sense of our knowledge lling partitions and nd their numbers. Sometimes, the mathematicians. How do you a certain volume, with a well- elementary geometry very asymptotics of the series can communicate with Nikita? de ned boundary, beyond comforting. Nikita knew that. be computed by the Laplace More concretely how did you which lies the unknown. In To get to the SW prepotential, method, i.e. by looking at the get your result on instanton physics, it seems to me, it one has to take a certain limit largest terms, which means counting? is very important to be the in Z, and at rst I was sure the law of large numbers for I have a personal interest, as rst to add some new key that it had to be something suitably rescaled μ. This works I had a different proof of the bit to the tip of the current very subtle since, after all, for much more general spaces, same result with Yoshioka. In research focus, a bit like we are talking some of the e.g., for the space of Lipschitz my case, we did not originally a DLA (Diffusion-limited deepest structures known functions on R instead of intend to give a proof. We had aggregation) growth, or a to man at that time. So I R with the diagrams of Interview an unsuccessful joint project discussion in a social network, didn’t give it much thought partitions instead of N. on on blowup may my physics colleagues at rst until Nikita came to (Following Vershik and Kerov, before, and we wanted to forgive the comparison. As visit Princeton the following Russians draw partitions with correct it by using instanton we know, this grows faster, winter. It is remarkable how 45° axes, which saves space in counting. Quite unexpectedly but also results in very fractal much in science depends the paper by the factor of √2 we were able to nd a proof. dendriform structures, in on social aspects, because and also makes the boundary Nevertheless, it took us some which, for a mathematician, humans are social animals of a diagram a function with time to understand the it is very hard to trace the and our brains are wired Lipschitz constant 1.) The meaning of Seiberg-Witten boundaries or to navigate the for interaction with others. largest term, that is, the limit

27 shape, is determined by a Miwa-Faddeev-Reshetikhin-... ask you some questions. As spaces (called quiver varieties certain variational problem, style math really easy to someone who has studied after that) were related to which in this instance is very absorb, because, rst, it is, Nakajima quiver varieties for representation theory, and elegantly solved by the SW fundamentally, representation many years, I am naturally study went on smoothly curve. This is all really basic theory with a mix of statistical very curious about their origin afterwards. probability, except for the mechanics, two subjects to and early history. How did it Okounkov: You worked in part in which one can actually which I can relate really well. happen? both differential and algebraic solve the variational problem But maybe more importantly, Nakajima: I was impressed geometry; which one did explicitly using algebraic it both answers some by Mukai’s work on moduli you enjoy more? How do geometry. With Rick Kenyon, fundamental enumerative spaces of holomorphic vector these two different kinds of we later developed some geometry questions and bundles on K3 surfaces in geometry compare for you? general theory about this. also is illuminated in a new 1988, and started to study Nakajima: I studied One of the simplest among and, I think, simplifying similar problems for ALE differential geometry, these limit shapes, the limit light by these geometric spaces, which are noncompact especially nonlinear PDE shape for a uniformly random considerations. version of K3 surfaces. Then I on manifolds, when I was 3-dimensional partition, I've spent a very long collaborated with Kronheimer a student. Since Kobayashi- was at some serendipitous time doing and analyzing to give the ADHM description Hitchin correspondence moment recognized as various enumerative (or quiver description) of was one of the hot topics identical to the Hori-Vafa computations, which is really holomorphic vector bundles, in the area at that time, I mirror of C3. That was the a very hard subject. It may or instantons on ALE spaces. had seminars with algebraic start of whole GW=DT story... be a fashionable topic to This happened in summer of geometers. I was also discuss, but to actually work 1989 at Berkeley. Kronheimer interested in Kaehler-Einstein Relation between Quiver out a solution to a modern changed his interests to metrics on Fano manifolds. Varieties and Quantum enumerative question is a applications of the gauge Since these problems were Integrable Systems different matter. There may be theory to topology, but I related to geometric invariant Nakajima: You explained a trivial case you can do from continued to study these theory, I gradually learned the relation between quiver the de nition, a couple more particular moduli spaces. In it. On the other hand, the varieties and quantum with some cleverness and 1990 at ICM (International minimal model program was integrable systems in your tricks, perhaps a bunch more Congress of Mathematicians) the central topic for most lectures for students. They with the help of a computer, Kyoto, I heard Lusztig’s plenary Japanese algebraic geometers. were very interesting, and maybe a lucky guess, and talk and knew that quivers It looked dif cult to me, I was impressed that you then at some point some appeared in his works. I and I classi ed myself as a understood the works of miracle needs to happen. So started to study his works, but differential geometer at that Jimbo-Miwa and others any time there is a framework it was hard as they were very time. very well. How did you learn to explain a certain totality of far away from my background After I had analyzed quiver quantum integrable systems? answers, your brain is already at that time. Meanwhile, I varieties for several years, I Okounkov: I am very glad prepared to esh it out with found that Slodowy slices needed algebraic geometry you liked it, even though concrete data and features. I appear as moduli spaces, and more and more. For example, I am surely still very much certainly felt that immediately knew that Hotta-Springer I wrote differential geometric behind people like Jimbo and after Nikita and Samson (and also Hotta-Shimomura) aspects in the quiver variety Miwa. I nd our brains really (Samson L. Shatashvili) had computed their Betti numbers paper written in 1994, but repel some parts of math their vision for how curve in the context of Springer not in the paper in 1998. while very easily absorbing counts in Nakajima varieties representations. In 1991, I This shows a shift in my others. This depends maybe and related geometries should found that their computation interest. Finally, I found on some inborn qualities, be tied with the quantum in top degrees gave weight smooth quiver varieties were but also very much on integrable systems. Things multiplicities of irreducible best understood as moduli what you already know and that I sort of understood from representations in type A, spaces of sheaves and Hilbert understand. I always tell my one side suddenly shone in a and it gave a link to Lusztig’s schemes of points, rather than students to go with what new light from the other̶a works. This observation moduli of holomorphic vector comes naturally... Maybe this great feeling. occurred when I read a paper bundles and instantons. This is what Kirillov meant by his by Kashiwara-Nakashima, and was the time when I stopped saying that“ math can be Andrei Asks Hiraku I still remember that I was my interest in differential only learned adiabatically.” Questions very excited at that time. Thus geometry. Anyway, I nd the Jimbo- Okounkov: Now, let me I understood how moduli Nevertheless, I feel that

28 Kavli IPMU News No. 40 December 2017 my differential geometric of ICM 90), but his talks were outside Kyoto.) And we do them many years... background is useful when very different from anyone not have classes teaching Okay, I really enjoyed I read physics papers. I like else’s. He usually started integrable systems. A similar talking with you. Thank you. joint works with Yoshioka on with easy examples, and thing happened on algebraic Okounkov: Thank you. instanton counting on blow- gave some computation, but topology, where it was up. He is an actual algebraic suddenly said something very popular in Kyoto at some geometer and very strong mysterious but interesting period, but only a very few in moduli theory. Thus I towards the end of lectures. people remain now. On concentrated on looking for They were very hard to follow the other hand, algebraic relevant physics literature, and as I was not used to hear un- geometry, number theory, found the paper on the RG organized talks like his. Also probability, and many other equation. it was impossible for me to areas are taught in regular Okounkov: You are a understand from where he classes. Their research frequent visitor to Moscow got his ideas. His thinking groups keep the same size, now, but what things looked very mysterious. or even grow. There was no surprised you the most I had an idea for a symplectic geometry when I at rst? Do you see many long time that all Russian was a student. But we have similarities and differences mathematicians gave talks a strong group in Tokyo and between mathematics in like him, and Russian students Kyoto. Moscow and in Japan? were accustomed to learn As far as I understand, these Nakajima: When I was a things from such talks. I met changes and preservations student, we did not have other Russian mathematicians, did not happen by plan. The many chances to hear talks and gradually understood that number of faculty member by foreign mathematicians. Feigin is unique even among is xed (in fact, decreasing Since Japanese professors Russians, and most people recently), and we need to hire could cover limited areas in are not so different from good researchers in newly mathematics, we learned us. Therefore, when I rst born elds, like symplectic many things from written visited Moscow in 2013, I was geometry, as we cannot texts and papers. We were not surprised at all. My rst keep up the level of research encouraged to read many encounter with Feigin was a otherwise. Hence some papers in detail. There were much bigger surprise. elds shrink in turn. Another also many expository talks Okounkov: Okay, this is factor is the availability of where new preprints sent my last question. In Japan, textbooks. There are many from abroad (by ordinary mail) many things are very carefully good Japanese textbooks were introduced. The situation preserved, while many other from the undergraduate level might be different in other things are very dynamic. What to advanced ones in algebra. elds, where more Japanese is your sense of the balance We have a few good books mathematicians were of tradition and innovation on integrable systems, but working, like number theory between the generations in certainly not enough. It is and algebraic geometry. But Japanese mathematics? dif cult for students to learn I got my basic knowledge Nakajima: Last year Takeuchi integrable systems. through papers, rather than published a text book on Since I have successfully direct communications from D-modules (in Japanese), and changed my elds in my professors when I was young. he wrote that he regrets that career, I enjoy discussion When I met many people who D-modules theory did not with people with a different Interview learn many things from talks become popular in Japan background. So I like the in US and other places, I was despite the fact that it was dynamic changes of my surprised. born in this country. Also surroundings. On the other Feigin visited Kyoto every as you observed, quantum hand, I understand that I summer starting around integrable systems are not should write textbooks for 1990. Opportunities to hear popular in Kyoto any more future generations, but it is talks by foreign people had since Miwa retired. (It is not easy for various reasons. already drastically increased partly because researchers I promised to write three at that time (partly because in integrable systems spread books, but I cannot nish

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