Andrei Yuryevich Okounkov works on representation theory and its applications to a wide range of mathematics, including algebraic geometry, mathematical physics, probability theory and special functions. He is currently a professor at Columbia University and an academic advisor of the Institute of Mathematics, Academia Sinica. He won the EMS Prize in 2004, and received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry" in 2006.

YPL: First of all, thank you very much, Andrei, for agreeing to be interviewed by Math Media. As I said Math Media is the only popular math magazine in Taiwan that I know of, and also the one I read as a high school student. I don't know if they still have it, but they used to have some math puzzles you could solve.

HHT: I actually published an article in MathMedia when I was a high school student.

ICF: There was something like that in Russia, right?

AO: Yes, there were lots of resources for high-school students, for instance the famous Kvant. For myself, I liked the “Kvant library” the most. These books were in this Russian style: very short and straight to the point. Somehow, the paper was precious when I was growing up; you would think Russia has hardly any forests. For research papers, this was really bad. Many classic papers from those times are only a couple of pages long and read like very puzzling telegrams. But the parsimony of very compact books is wonderful. These Kvant books are gems.

ICF: There was also another one called Matematicheskaya Skolastika (Математическая Сколастика) or something like that?

AO: I don't know. I don't think any of those translated into English.

ICF: No, no, no, they were in Russian, so I would have to ask somebody to translate the problems for me.

AO: There were about 100 books published in that Kvant series. I don't know why nobody thinks about translating them. I mean, there were plans like that in connection with the St Petersburg ICM, but…

AK: So, how did you get interested as a child? Was it partially through these kinds of magazines?

AO: Actually, I wasn't so focused on mathematics as a child, and I had a different career before math.

HHT: Oh, please tell us. I knew you were in the military but…

AO: I was a student in the economics department and, after two years of it, I had to go serve in the military. Somehow, after the military service, most people find that they want to switch from mathematics to economics, and in my year I was the only person who wanted to switch in the other direction. The university administration was very supportive of that. To be objective, there was a lot of mathematics already at the economics department. Russia has a very developed school of economic thought and there were lots of very interesting people working in the department of Economics. But in terms of scientific freedom, at that time, it was easier to be in math economics, work with mathematical models… There was this joke: the central question of history is ‘when?’, the central question of geography is ‘where?’, and the central question of political economy is ‘in which volume and on what page?’ [Laughter]

AO: The economics department had three sections: political economy, planning, and mathematical economics. My future wife and I were both in the latter.

AK: And then what made you switch and actually go to the department of mathematics?

AO: I guess I felt I liked pure mathematics better.

ICF: There wasn't any particular figure, any particular person that you interacted with before who sort of convinced you?

AO: The way it is organized in Russia, a department consists of different chairs. A chair is not a person, it is a group of several, in fact, many people. And so, in our mathematical economics chair, many professors had a very strong connection to the mathematics department. When I switched to mathematics, I chose representation theory as my field of specialization and A.A. Kirillov became my advisor. While this was a free choice made on the basis of my own interests and my own taste in mathematics, it was a very easy choice to make given that Kirillov’s wife Louisa worked in the chair of mathematical economics, as did Kirillov’s former student I.A. Kostrikin, with whom I took many many classes and other activities. I.A. Kostrikin, by the way, is the son of A.I. Kostrikin, of the Burnside problem fame. We also had A.V. Kochergin and V.F. Pakhomov with background in dynamical systems and supersymmetry, respectively, and they both very much expanded my mathematical and other horizons.

YPL: Just to follow up on what Adeel asked, so how did you decide to work in this area, and with Kirillov?

AO: When I came to the mathematics department, it was 1989. By that time, the golden days of Russian mathematics were over, and everybody had more or less left. But there were some excellent seminars that kept going: Arnold’s seminar, Sinai’s seminar, Vinberg’s seminar, and also Kirillov’s seminar. Vinberg was there all the time, the others would come for some part of the year. There was also the Gelfand seminar without Gelfand — A.N. Rudakov was doing it.

ICF: They were doing only derived categories at the time? [Laughter]

AO: I never participated in the original Gelfand seminar, but my impression is that there were two important components to it. On the one hand, there was an infinite curiosity about all kinds of math, on the other hand, there was an anthropologically interesting way of satisfying it. Rudakov was one of the sweetest people ever, so his approach to curiosity was probably opposite to Gelfand’s original approach. He would very politely insist on finding out the truth, as opposed to getting the truth out of the speaker the hard way. And so, I enjoyed all these seminars but Kirillov’s seemed like the most fun. Kirillov is really kind of a fun person.

AK: So, was it more the personality than perhaps the math?

AO: Field-wise also. When I was in the economics department, I always liked matrices. [Laughter] In fact I tried to tell this story to G. Strang when I eventually met him. In Strang’s linear algebra, which is a book I enjoyed very much, there’s a line which I remember as saying something like, “the commutator of matrices plays an important role in quantum mechanics”. This is where he talks about matrix multiplication being noncommutative, so this is certainly a relevant remark. I was very curious about that line but the book didn’t say anything further. Eventually, of course, I found Kostrikin & Manin’s “Linear Algebra and Geometry,” which is just really a great book, also about linear algebra so to speak [laughter]. This is the same A.I. Kostrikin I mentioned earlier. Is this book translated into Chinese?

HHT: I'm not aware of that. I didn't know they had written books like this.

AO: But that linear algebra book discusses for instance the Euler characteristic of coherent sheaves, if I remember correctly. [laughter]

YPL: In Taiwan, the mathematics textbooks on college-level or above are all in English.

AO: I see. Which linear algebra book do you use?

HHT: The course I took, I think the book was called “Linear Algebra” by Larry Smith.

AO: For calculus, Zorich’s book is amazing. I don't know if you guys know Zorich, but in mainland China everybody knows him. Sadly, he just passed away. At some later point, I TA’d for him. He was such a nice person, such a fantastic person. Anton Zorich, his son, as you may know, is a mathematician. His daughter is married to Khesin. Somehow, I know many people in this family a little bit, super nice people. So that’s just an amazing book, Zorich, there’s two volumes. For linear algebra I forgot which textbook we were supposed to use. In economics, linear algebra is the main thing, but Strang's book, which I found by chance in the library, was the first linear algebra book I enjoyed.

ICF: By the way, it's not surprising that you know that Kostrikin and Manin discuss Euler characteristic of coherent sheaves, I mean Beilinson's paper was about problems in linear algebra. [Laughter]

AO: They could have done derived categories, exactly. [Laughter]

AK: So, you mentioned this Gelfand seminar. I was reading this note that Beilinson has on arXiv, reminiscing about this seminar and talking about the influence that it had on generations of Russian mathematicians. So I was curious, because it seems you came maybe slightly after the original Gelfand seminar, but do you feel that you were also influenced by the legacy of Gelfand and Manin?

AO: I was definitely very much influenced by V.I. Arnold, Y.G. Sinai, A.A. Kirillov, A.N. Rudakov, E.B.Vinberg, and, of course, G.I. Olshanski. When Kirillov wasn't in town, Olshanski would run his seminar, and then also during my studies I interacted much more with Olshanski. And later, another seminar which had a very important influence on me was the Dobrushin seminar. But that was happening outside of the university. Yuri Ivanovich [Manin] wasn't so frequently in person in Moscow when I was studying. But one semester Sasha [Beilinson] and Boris Feigin were giving a seminar on conformal field theory and that was a lot of fun.

AO: In general, people weren't around so much. When I finished my undergraduate degree, the situation was so dire that there was no point in enrolling in graduate school. The normal career path before the collapse was to enroll in a full-time graduate program, supported by a very modest fellowship. By 1993, however, there were big problems with payment of that fellowship and nobody was around to actually supervise students. And so Olshanski suggested I consider some kind of graduate school by correspondence and go work in Dobrushin’s institute instead, as a research assistant. In principle, I already had something that could have been considered a thesis. As we already discussed, journals and books in Russia had a strong preference for brevity and it was very difficult to publish anything which was longer than 2 pages. Since I just happened to have solved a certain problem, I had 10 pages in “Functional Analysis and its Applications”, which was considered to be almost like a whole volume of Annals of Mathematics. [laughter]

HHT: So, I'm actually somewhat curious. That was the time when Russia was already sort of in contact with the Western countries. Did people submit their papers out to Western journals? Because then, I don't think there's a page limitation.

AO: At that time, email would be something quite exotic. You had to go somewhere to send email.

HHT: Right. You had to postmark your submission to send it.

AO: Regular mail from Russia was barely working. If you wanted to communicate with somebody, then, since there were many people traveling abroad, you would give your letters or papers to somebody and that person would mail them from abroad.

YPL: That’s like ’93?

AO: Yeah, around ’93. When I came to Moscow State, the economics department had a mainframe machine with 40 megabytes of total disk memory. Everybody would be allocated some number of kilobytes for their personal storage. A bit later, we got GDR-made machines called Robotron, each had, I believe, 48k of RAM. When the operating system would load, that would leave maybe around 12k for RAM for applications to run. When I came to the US and someone found out that I could program in assembly, they were really impressed.

HHT: You could talk to the machine.

AK: It’s funny that when I was growing up, we had these TI graphing calculators. And if you wanted to write simple programs there, you could use some kind of BASIC language. But if you wanted to make more advanced things like games and so on, you would actually need to use assembly. So I actually learned a bit of assembly just so I could make some games on this graphing calculator. It’s funny that after so many years, it sort of cycled back to assembly.

AK: Anyway, even though you were perhaps trained as a representation theorist, in those years, you ended up working in much broader areas. And my impression is that's sort of characteristic of many of the great Russian mathematicians including yourself.

AO: Perhaps I can quote Kirillov. Kirillov was a bit like Confucius — I mean Confucius was even more concise, but Kirillov would often speak almost like in Analects. And he would say, for instance, that representation theory, like any other important area of mathematics, includes all of mathematics. [laughter]

AO: Another quote I remember is: it’s much easier to generalize an example than to specialize a theory. [laughter]

AK: Yeah, I agree with this.

AO: Also, when somebody would come up to him and start talking about some math, he would often say: “Please start your story from the end, what’s your main formula?”

HHT: My advisor’s typical question for me back in the day was, what's your theorem? Is that the kind of tradition going on in Russian schools?

AO: But Kirillov was even more concrete. For him, a theorem was not as interesting as a formula. What's your main formula, and then we can start talking about this. [laughter]

AK: Do you think this is some kind of style of Russian mathematics?

AO: This was certainly the style, yes. Human thinking and preferences are very much influenced by our formative years, which is why we use the word “formative”. I grew up in such an atmosphere, and I like it. For instance, in Kirillov’s seminar, Gelfand's seminar, or pretty much any seminar, it was normal for people to interrupt the speaker—not in an impolite way, but just as part of the discussion. Maybe that’s where I picked up this bad habit of trying to answer questions that other members of the audience ask the speaker. In Russia it was normal that, when a question is asked, then it wouldn’t necessarily be the speaker who answers it.

HHT: Just about finding out the answer, not necessarily whatever.

AO: Yes, and also finding out the answer in the sense that nobody was interested in the most abstract formulation. Everybody wanted to understand something concrete, and then you can generalize it later if you want. It’s kind of remarkable how concrete all math was back then. Well, perhaps with the exception of Sasha Beilinson. He’s a fantastic mathematician, but it takes a while to project what he says onto the plane in which everyone else lies…

HHT: How does the degree program work in Moscow? So there was the Ph.D., and then…

AO: In this PhD by correspondence program, I had my thesis from the start, but it took about two years just to assemble the Ph.D. committee. This is because everybody was away. The Ph.D. committee in Russia is a rather big committee with maybe twenty members, and at least some percentage of them has to be in attendance.

ICF: Did you have in mind to stay in Russia initially?

AO: Hard to say…

ICF: There are people like A.I. Bondal who had a hard time, but eventually…

AO: Bondal is older than I. He was already established by the time…

ICF: I know, but I’m talking about that sort of frame of mind: I would rather endure the hardships here for the scientific milieu that you were describing…

AO: But that old world was just all falling apart. Kirillov wasn't around for my defense, for instance. He was between Russia and Philadelphia for a while but then moved to Philadelphia. And similarly for Sinai… Sinai is now very old and his health has been a serious concern, but until recently he would come every summer. Still, it's only in summer.

ICF: So, it was basically disintegrating too fast.

AO: Maybe there were a few centers. One was the Dobrushin institute, another was the Sharafevich group. But at the time I felt so far away from algebraic geometry — somehow, it is really sad, I never even saw Shafarevich. If I was doing something else, perhaps it would make some sense for me to stay, but among the representation theorists, more or less all of them were gone, with the exception of Olshanski. And of course, none of these jobs paid a salary.

AO: V.A. Ginzburg offered me a postdoc in Chicago and I was very happy to go. It turned out to be a great decision because I met Rahul Pandharipande, I met Sasha Eskin, I met all these people. In my final year, even Beilinson and Vladimir Drinfeld came. I didn't really interact with them so much in Chicago, because by the time they started their famous seminar, it was already time for me to go.

AO: It’s funny, Rahul was my next door neighbor, but while in Chicago we only interacted superficially. I enjoyed attending Bill Fulton’s algebraic geometry seminar. Bill’s an amazing person, absolutely amazing. And at the time he was very interested in enumerative things. Rahul was a very active participant in the seminar and that’s how I became aware of the subject. The actual point of contact was through Spencer Bloch. Spencer's curiosity was ignited by certain zeta values and related numbers appearing in representation theory, and we wrote a paper about this circle of questions (which eventually turned out to have solved one of the problems that Eskin was trying to solve). One day, Carel Faber came to visit and was speaking about Hodge integrals in Fulton’s seminar. He and Rahul have just proven some formulas for Hodge integrals in terms of Bernoulli numbers. Se, he writes these Bernoulli number formulas on the board, and Spencer gets up and says: those are exactly the numbers that Andrei and I are getting in our computation. Eventually, these Bernoulli numbers were explained by Rahul and I as the constant terms of the completed cycles, but this was just a little bit after we both left Chicago.

HHT: That’s the starting point of the Gromov-Witten/Hurwitz correspondence?

AO: Not really the starting point. It was just one of the many pieces of the puzzle. And it wasn’t clear at the moment that the puzzle would really assemble into a coherent picture. I mean, Bernoulli numbers appear everywhere.

ICF: Todd class. There must be a Todd class. [laughter]

AO: Exactly, that’s a classic quote from Victor Kac. He observed Bernoulli polynomials in certain vertex algebra computations, and makes a remark that it must be the Todd class. [Laughter]

AO: Anyway, the first real point of contact with Rahul was that he published this paper about the implications of the at-the-time conjectural Toda equation for the Hurwitz numbers. And I looked at it and thought: ok, this is something I should know how to do. [laughter] So this is how everything started. The completed cycles story became clear later, when we realized — the three of us, Rahul, Eskin, and I — that we were working on the same subject. Our original way of thinking about them was much more combinatorial. In the end, we published only the most effective geometric and algebraic constructions, and their applications. There is, in fact, only one paper with all three of us as authors, the one about the pillowcase orbifold. It was fun to see people rediscovering our original combinatorial constructions later.

AO: Our older daughter was born in Moscow. In fact, I was an undergrad when she was born. But our younger daughter was born in Chicago. It was sort of a dramatic moment because there was a huge snowstorm. Our car wouldn’t start, we had to somehow get to the hospital, so in the end we decided to walk. It was very difficult and slow, with piles of snow in our way and Inna already very much in labor. Anna was born, maybe, 45 minutes after we entered the hospital. And just before all of these events, I received an email from Eskin saying that he asked everybody in the department whether they can do some kind of summation, some kind of Hurwitz problem, and nobody could. That’s how that started… When we got home from the hospital I wrote back explaining that our paper with Spencer Bloch solves this problem.

HHT: I always knew that you started a family early, but I didn't know it was that early.

AO: In Russia, that was normal.

YPL: Is that still the case currently?

AO: One has to look at the numbers, which I don’t have.

HHT: I think for the younger generation it would be helpful if you share your experience of how to manage work and life, and family life especially. I had the fortune or misfortune depending on how I look at it, to raise children myself—you know, diaper duties and stuff. I already had a stable job at the time, but I couldn't imagine myself doing that when I was an undergrad. Even for graduate students that’s kind of impossible. Of course Y.P. did that.

ICF: You need to be lucky to have a wife that is really supportive.

AO: Well, in our family, I was considered the stay-home dad. [laughter] Of course, we got a lot of help from the grandparents. When our older daughter was born in ’92, it was the time of a complete economic collapse. It's very normal for Russian grandparents to participate very actively in helping to bring up grandchildren, that must be the same in Taiwan.

HHT: When you moved to Chicago and had your second daughter, then you two were by yourselves?

AO: So, of course, the limitation to that is that grandparents sometimes have their own lives and so forth. But at the time the economic situation was just so bad that both grandmothers just quit their jobs because there was no income from those jobs anyway, and they became full-time grandmothers. And as visas would allow, they were also extremely happy to come visit us in Chicago. And also we were coming from a background where you’re expected to do a lot of things around the house. For instance, in my thesis defense I came with my left arm in a bandage. Nobody asked any questions. The reason was that we didn't have a washing machine and we didn't have disposable diapers, so we had to do it the old-fashioned way. Part of the procedure was that you have a big vat on the stove and the diapers boil there. On the day of my defense, while emptying this vat, I scalded my arm and so I was all a big bandage. And so when we came to America there were disposable diapers… [All laughing]

ICF: I know very well what you’re talking about.

AO: Yeah, exactly. It's like what could be easier than that? Or, I would typically do other things, for example: our family somehow finds a really great virtue in everything being very clean. And so, after diapers have boiled and so forth, I would still iron them.

AO: So, I finish this ironing and I get up maybe 5 in the morning, and I get a big backpack and I go around Moscow finding something to eat when the food shops open. In Chicago, we come and there is a supermarket and there is food in it. That’s another task that became suddenly just so easy. An idiot could do it – you just go there.

HHT: I think it's very encouraging to know that you can still do good math while raising children.

AO: I think it's true.

HHT: Yeah, honestly, I worried about it being true for a while, but it turned out okay.

AO: There are many good things about having kids when you're still young and the grandparents are also. My parents were in their early 50s when our older daughter was born, and so they were just full of energy.

AO: Now, I sit all day, stare at a poor piece of paper and I think to myself, how did I manage to accomplish so much when I was young? Something good to be said about youth.

YPL: One question suggested to us was whether there are particular books that inspired you. Mainly math books, but books in any area are all good.

AO: I really liked reading books and I was very fortunate that many excellent books were available and cheap. I already mentioned Zorich’s calculus book and my favorite linear algebra books earlier. For algebra in general, not necessarily linear, it is hard to imagine anyone can surpass Shafarevich’s “Basic concepts of algebra”. It appeared in this green VINITI encyclopedia of mathematics as the first volume of their algebra sequence. In English translations these are yellow Springer books and they are insanely expensive (the Russian originals are now free on mathnet.ru). There are many great works in the same series by other authors. Obviously, I read Kirilov’s and Vinberg’s books in this series particularly closely. I think they are the best expositions of the Lie theory and representation theory. It is interesting how some book series sometimes share certain great features across multiple authors and topics. I already mentioned the Kvant library. The VINITI encyclopedia is another great example. Outside of Russia, I really like Iwanami Series, translated from Japanese, in particular Kato’s et al Number Theory books. One trait that unites all these series is how succinct they are. The authors obviously thought a great deal not just about the subject, but also about how to present its very essence in the most direct way. Outside of any series, anything by Arnold is a really fun reading. Not sure how relevant this is now, but I very much enjoyed reading many classic computer science books, about efficient algorithms, data structures etc. Maybe this is how I developed a strong liking for combinatorics. This was taken to another level by reading Stanley’s “Enumerative combinatorics”. I could easily go on, and on, and on. Coming back to textbooks, I believe Zorich’s book is extremely popular on the Mainland.

YPL: The calculus book commonly used in Taiwan is by Courant and John.

HHT: I used that too.

AO: German math books tradition is different, but it's also very interesting. I really enjoyed many classical German books, such as Hurwitz and Courant’s function theory. I loved them, and many of them were translated. Wait, here you guys read it all in English or?

YPL: Yes, the books were all in English. We instructed in Chinese, but the special mathematical terms were spoken in English.

AO: I see.

HHT: Yeah. I think like "scheme”, we don't have a Chinese name for scheme.

ICF: Yeah. I noticed while walking downstairs that people are lecturing in Chinese, while writing on the board in English. It seems very odd to me how one can do that.

ICF: Like, "Let F inside E be a field extension”, that's what I saw on the board and I heard the guy saying it in Chinese as he was writing.

AK: A random question. You were mentioning that sometimes you're staring at the paper all day and feeling frustrated. In general, what's your attitude when you get stuck mathematically? Is it frustrating, or do you regard this as normal? Different people seem to have very different attitudes.

AO: So, there are two answers to that. First, very fortunately, in my line of work, if you get stuck, you can try to compute examples. In fact, you can be very systematic and write code to really explore the examples, and any time I had a coding project I would always learn something from the coding itself and, obviously, from the data obtained. I would certainly encourage people to spend more time doing examples, concrete computations. To quote Kirillov again, he said that only the new-fashioned mathematicians start their day by trying to prove a theorem. The classical mathematicians, they started their day by doing a computation.

HHT: I think it's really true. I mean for myself, the older I get, the more basic examples I'm interested in, concrete examples. Like I was going to ask, I know you are big on swimming.

AO: No, it's something I started just relatively recently just because…

HHT: When you visited us (at Ohio State), you were so impressed by our swimming pool.

AO: Yeah, but…

HHT: But that was a while ago.

AO: Yeah. But that's…

HHT: Anyway, I was just curious if swimming helps you get over something.

AO: I'm still not good at swimming. Maybe I’ll try to guess the direction of your next question and say this. Computing an example is one thing and just letting it sit in your head for a while is another important thing. I mean, just our heads work this way. An idea has to rotate in many different ways before it actually falls into the keyhole. And what was really best for me over the years in this respect was going to shoot hoops by myself. That was really good. That kind of activity, I think, is really conducive to thinking. Bike rides are okay, but with bike rides, you have to somehow pay more attention to your surroundings, especially if you’re on the road with cars or pedestrians. And with swimming, unfortunately, I don't know if I'm ever going to get to the point where I can swim and think about something else.

HHT: I mean, the reason I asked this is because I remember reading some…

AO: You guys have amazing…

HHT: Basketball court.

AO: Yeah. And the number and the quality of them is just absolutely amazing. My dad used to be a professional athlete. He was primarily a track and field athlete, but he also played basketball. In particular, he played for his Republic, Kazakhstan. And so, I absolutely had to send him a picture of you guys’ basketball facilities.

HHT: Okay. I'll take one for you.

AO: Yeah. It's pretty amazing.

HHT: Yeah, I remember reading some interviews of Vladimir Arnold. And I think he said something like, if I got stuck on something, I’ll just put on my ski and go out cross-country or whatever, and then after a while, problem solved or something. This is sort of like some kind of different or maybe physical or not otherwise activity could help you think. So, that's why I asked whether you had such activities.

AO: Generally speaking, any kind of activity that brings oxygen to your brain is good.

HHT: Right.

AO: It is true that there's a lot of subconsciousness in research in mathematics or in human life in general. Subconscious plays a role, so you have to let your subconscious and consciousness interact for a while, and sports or outdoor activities can be very good for that. The person's level of physical fitness varies. Somebody wants to go for a walk. Somebody wants to go paragliding. Arnold was pretty fit. Actually, that was rather typical. There are a lot of Russian mathematicians who were extremely good at challenging outdoor things like alpinism. They would go to really challenging places. And Kirillov, he was really good at diving, volleyball, and many other things.

AO: He didn’t take his seminar diving, but we would often go play volleyball. And Sinai’s seminar would play soccer. I’m not sure what Arnold’s seminar did. Kolmogorov used to take his students skiing. Maybe that's how, I don’t know, Arnold got into skiing. But skiing is good. I mean, of course, as long as it's cross-country.

AK: So are you saying that, by playing basketball and by computing examples, you've managed to avoid ever getting stuck? Or do you still get stuck sometimes?

AO: I'm still stuck on so many problems. It's okay.

YPL: So, do you find yourself juggling between multiple projects and you get stuck in project one…

HHT: Oh yeah, that's like the work attitude, right?

AO: Maybe when I was young, that was possible. But right now, I just like…

ICF: Do you actually have problems that obsess you now? That you really want to…like you can’t sort of take your mind off them?

AO: Sure. But somehow, it feels like if I put a problem aside now, then I'll more or less forget everything I knew about that problem. That's annoying. I need to organize my work better, write down more things before I forget them.

AK: Yeah, that's one reason I actually like writing papers. I tend to forget everything after 1 or 2 years, so I have to look back and read my own papers to see how some things were supposed to work. Otherwise, I have to recreate everything from scratch.

AO: You know, math becomes always more complicated somehow. It never gets easier, right? And so, you have some kind of sense of…you sort of…It’s like you're in some kind of complicated building and then there's some sort of dead-end here and then some dead-end there, somehow, but maybe this way there’s maybe some kind of hope.

ICF: Yeah, the most annoying thing is when you make the same mistake twice, like 4 years apart.

AO: Somehow, to recreate that sense of how to record that sense of understanding, it's not clear to me because it's at some point in time.

HHT: Yeah, personally, I try to write perhaps too much detail in papers so that it'll help me remember stuff afterwards, but that probably isn't helpful in terms of making the paper more readable.

ICF: You should keep notes of things that you've calculated and whatever that you don't actually put in papers. Then if somebody asks you, you know where to find the notes.

AO: Yeah, yeah. But it's a little hard to… It’s super annoying that a lot of these notes are tied to specific platforms. I mean, if I write a TeX note, that's somehow okay. I can maybe just find it with the internal Mac search. Anything else, it’s just so super annoying that it is locked into specific platforms. For example, a lot of stuff I've written for myself in this Windows Journal, which is some software that somebody just decided that they will abandon. Because I didn’t convert it to PDF at the time, now I don't know how to open it.

ICF: Bumsig (Kim) used to write these notes and then scan and email them to me. But now, I don't have that resource anymore, unfortunately… Yeah, he used to be very good at, kind of, recording whatever he was thinking, our discussions…, even though somehow those notes would never make it into the finished product, but, somehow, it was easy to go back and find.

AK: So I think that different people find writing to involve varying levels of pain. Like for some people, the process of writing their results as an actual paper is just something they really don't enjoy, whereas for others it can actually be enjoyable. For me, it's a mix of both things, but I do enjoy the aspect of forcing myself to think through things. How is writing papers for you?

AO: So, as usual, we spend most of the time doing things we don't know. There are some parts of math with which I may feel a certain familiarity, a certain proficiency. They may form the core part of the argument, the part that I find easy to write. But then there’s are some auxiliary statement which come from some other field of math, and to get those right may be is quite painful. That’s something you haven’t thought through and then you try to either emulate somebody's argument, or just use somebody's result and try to make sure that you're really doing this correctly. That can be quite frustrating. Unfortunately, very often, most of the work goes into that, not in the core thing. Do you have that sort of experience?

ICF: Somewhat, but I also feel that I'd rather not write something if I don't feel that I don't understand the guts of it somehow.

AO: Yeah, but sometimes you need something. And I could see this situation in other people. We wrote one paper with Rahul that had random trees in it. Perhaps I thought that the probability theory involved was nearly obvious, but I could clearly see that Rahul would prefer to have a very detailed argument spelled out in the paper.

AO: The same way, more often, I'm kind of receiving it. Like there's somebody's result which I want to somehow use correctly or…

ICF: I actually learned a lot about quasimaps by reading your notes for students. Because you actually do examples there, honest computations that I never did on my own. So, I… [A.O. laughing] No, I'm actually serious about this. And sometimes like, you know, I had a student trying to do something and I said, don't read my paper, go read Andrei's notes because that's where things are done sort of the right way.

AO: Oh, I see, thank you ! I am really honored and happy to hear this.

HHT: Speaking of students, you know, just sort of curious, what is your attitude towards, like do you have…some people got so popular, they have like criteria for selecting graduate students, to be their advisors and stuff. What would be your attitude?

AO: No, I’ve never been there, I've never achieved that status. But, it's…

ICF: You actually had more students at Columbia than at Princeton, right?

AO: Well, of course. Because at Princeton, the competition is just so stiff. I think I had about two and a half students throughout my whole time at Princeton. … I mean, if I were a student at Princeton, I would certainly go work with Rahul rather than myself.

HHT: People that were interested in representation theory, they would not…

AO: This depends on what people mean by “Representation theory”. In Kirillov’s seminar, representation theory meant that you are answering questions, often rather concrete, about actual representations. Olshansky was particularly interested in asymptotic questions, when the group gets larger and larger, for instance, one may study the unitary group U(n) as n grows to infinity. In the U.S. this didn’t really exist at the time, and now is probably seen as a part of probability theory, akin to random matrices. Now there are active groups at MIT around Borodin, at Columbia around Corwin, also at other places, but at the time, at Princeton, students weren’t really interested in this.

ICF: Langlands…

AO: Yes and, in particular, they have to do with unitary representation, spectral decompositions, and so forth, for both real and p-adic groups. And if you want to do that, Peter Sarnak or somebody like Peter is a much more natural advisor. I don’t mean to complain, I was very happy with my colleagues and students at Princeton, but I was deservingly in a deep shadow of advisers like Peter Sarnak, Andrew Wiles, Janos Kollar, Rahul Pandharipande, et cetera.

HHT: I'm curious, having the Fields Medal afterwards, does that change things in terms of attracting students?

AO: I'm not sure it helps. I think, if anything, that maybe it could scare away people.

YPL: So how did the Medal change your math or your life in any way?

AO: Well, it depends.

YPL: For example, you must be asked to sit on many more committees.

AO: People have different attitudes in life towards anything. For myself, I try to somehow balance things. I did accept a lot of administrative responsibility. Maybe most importantly, the St Petersburg ICM was an enormous amount of work. It was 6 years of more or less 24 hours a day work. And it amounted to very little, perhaps less than zero. Okay, maybe forget it.

AO: Something I tell my students is that, while math is an abstract subject, it exists inside the society. Therefore, math depends on somebody within the math community to communicate with the society, to build institutes, find funding, make sure the math departments are open, and make sure the students are taken care of. It is fine for one individual person to just want to focus on math and not to worry about people, because people are maybe not as beautiful as math… But somebody has to do it. I felt I had my share of this responsibility and I was happy to do that work. Some people may take this attitude to the next level, and some may ignore it completely. Some parts of the ICM work I genuinely enjoyed, like the design of the website, souvenirs, et cetera.

AK: Thinking about it that way helped.

AO: But it's not just thinking. I think I always try to make this point. It's just that I'm very fortunate to somehow have met all these people who certainly enriched me and enriched my understanding of math.

HHT: Among the results that you have obtained, do you have a favorite one at this point.

AO: No, I don’t know.

HHT: That's fine, yeah.

AO: It’s not clear. I know which one is the most cited one.

HHT: I don't know which one.

AO: It's about the "body”.

HHT: The Okounkov body?

AO: Yes, that's the one.

AK: So, as we were saying, math is so large and there's so many fields. How have you navigated which directions to go in or what problems to work on?

AO: I don’t know. I think math is а discovery, and you just follow clues. What really helps is that you can interact with other people and learn from them. You can have this “aha” moment and you are suddenly seeing the solution on the horizon. This is why people need math institutes and other structures that bring people together. Earlier, I already told the story of how Rahul and I were led to a few discoveries through a fortuitous Bernoulli numbers coincidence. Another example is the work that I have been doing with Kazhdan recently. I was lecturing, over Zoom, in Kazhdan’s seminar about elliptic envelopes and how they help you obtain integral solutions to the Knizhnik-Zamolodchikov (KZ) equations. KZ equations, by now, is a classical area of mathematical physics, of interest to many people.

AO: People first found them in conformal field theory, and then in integrable deformation of conformal field theory. Of course, the theory itself is completely nonlinear, but the quantities of interest turn out to solve remarkable linear differential or difference equations in many variables. There is a somewhat unexpected geometric way to write down integral solutions to these questions and I was talking about it in Kazhdan’s seminar. And, somehow, Kazhdan’s intuition leads him to ask the following question. If we have a geometric way to deal with these integrals, perhaps we can deal with somewhat similar-looking integrals that appear in the spectral decomposition of Eisenstein series. And, of course, he was right. I never thought I was ever going to do any kind of automorphic forms until I got into this project.

HHT: So, you would encourage, for example, young people to not be too shy and talk to people and go out of their comfort zone, just attend seminars that may not be in your subject necessarily?

AO: Absolutely, and, in fact, this has to work both ways. I think it is great to be curious about all things mathematical, and conversely, if you are giving a public talk, you should try to really explain the material. In my seminar, I always encourage the speaker to focus on one key idea, and to remove as much packaging from this idea as possible. First explain it in the simplest nontrivial example, then maybe in a just slightly more advanced example. Don’t start with the most general statement possible. Simple examples are what usually clicks with people, and help connect seemingly different areas of math. For instance, in our work with Kazhdan, the basic connection lies in the fact that there is a geometric alternative to computing integrals by residues, and this can be explained in very simple examples, without scaring people away by the actual integrals one is interested in. Voila, Bernoulli numbers…

ICF: It's a Todd class…

AO: Todd class, yeah, exactly.

• Yuan-Pin Lee, Ionut Ciocan-Fontanine, and Adeel Khan are faculty members at the Institute of Mathematics, Academia Sinica.
• Hsian-Hua Tseng is a faculty member at the Ohio State University.