At the Centre of the Mathematical Universe

AV: Thank you.

AV: I was born in Delhi. My parents are Tamilians. We moved to Australia when I was a little under three, so I grew up in Australia.

AV: To be honest, I don’t really know the answer. I probably should, but I don’t. I never discussed with my parents why exactly they moved. Of course, in many ways, life would have been easier in Australia than in India. But I don’t know specifically what prompted the move.

AV: That’s right, my mother is a computer scientist. She completed her PhD after we moved to Australia, after which she became an academic.

AV: I believe my first grade teacher, Joan Hollingsworth, had already talked to my parents, saying—I don’t exactly recall what was said—that I was good at math. I don’t know about the Australian system in general but the school that I went to was very flexible. It allowed me to take classes that were usually taken in upper years. So they were very nice to me. I stayed with my own age group for many years but the school allowed me to take advanced mathematics and physics courses.

AV: I can’t remember not being interested in math and mathematically intensive subjects. So I can’t think of a more convincing answer.

AV: I don’t think my mother or father thought that way. I don’t think it occurred to them. I don’t know if they had thought that I was particularly talented. But I know that my first grade teacher went and talked to my parents and told them that it would be good to expose me to something a little more advanced in school.

AV: That is true.

AV: There were various enrichment programs and problem-solving competitions that were around. I don’t think it was as developed there as it seems to be, for example, in the US where it seems to be a very structured system. I don’t remember how I ended up at one of these contests. But then, there were some programs, I think at the universities, for students who were interested in math that led to this involvement in the Olympiads. The truth is that I don’t remember exactly how I found those.

AV: Well, it seemed like a natural path at the time. I think that was the avenue that was available for students who were interested in that type of thing. It could be that the high school teachers at my school pointed me towards these competitions.

AV: My father is a mechanical engineer, so my parents had many books from their college years. I learned a lot of things from their college math textbooks. In a direct sense, I think I was probably influenced more by conversations with my father about mathematics and physics than with my mother.

For example, looking at an aeroplane, he would tell me things about the engine, things like that. He was very fond of gas turbines. So I think that had an influence on me, and led me to develop an interest in physics. Also, I remember, at some point when I was learning Euclidean geometry and I had some trouble, my father helped me a lot with it.

AV: Definitely, I had good teachers. My mathematics teacher—his name is Neville Punch. I think he was the head mathematics teacher. And there were several others as well but maybe he’s the one I remember most. I had a very nice school. They were very supportive and I think I enjoyed being taught by the teachers there.

AV: I don’t remember when I actually had classes with him. But I interacted with him. You know, he probably had to go out of his way to make arrangements for me to take classes that were ahead of my grade and so on. So he certainly helped.

AV: I don’t think there was anyone else at my school taking upper courses at that time. Or, at least, not that I remember.

AV: I can’t quite talk about the situation outside of math. Australia has produced a lot of good mathematicians, and I have wondered why that is. I am sure there are many factors, but it seems to me that it’s a less aggressively competitive place than the US. If I look at the undergraduates at many of the top US schools, I think they are discouraged to see all these brilliant people around them. I don’t think that results in the best outcome in the long term.

I could easily have imagined being in that atmosphere and not finding it a net positive. In Australia, there aren’t very many students who are interested in pure mathematics. So you don’t have a sense of competition, and you have time to develop at your own pace.

Some people develop at different rates to other people. You don’t necessarily want the ones who develop fastest at the start to have an advantage. Maybe I was such a person, but in the grand scheme of things, this is not the ideal scenario.

AV: Actually, I enrolled initially in a course that was science and engineering. I was planning to study math, physics and electrical engineering. After a little while, I decided I didn’t like electrical engineering. I did study physics but I did an honours degree in math. I don’t think there was much of a reason for that. Maybe it could have gone another way. When I went to university I seem to remember that I was most interested in learning physics, but I’m not sure.

AV: No, no. I think I could have easily gone into physics or maybe even other mathematical fields and I would have enjoyed doing them. At some point, I had made a decision. I don’t really remember why, but in the first couple of years, I took both math and physics classes.

AV: I’m not sure I was convinced back then. And I’m not even sure if I’m convinced of that now [laughs]. I kept doing math and I was quite good at it and it seemed to work out well. Especially at that time in university, I never felt that I was really destined to be a mathematician. But I was enjoying it and it was going well, so I kept doing it.

AV: Oh yes. I was interested in other things. I had a normal set of non-academic interests. But work-wise, I never seriously considered options very different to math. But I certainly didn’t do math all the time.

AV: Oh yeah. Not in any serious way but I’ve played many times in our backyard. When I was in my honours year in math, I used to play. We had a little room. We used to play cricket in this room and then very hastily hide all the equipment when the professor would come in. I used to play squash too.

AV: I never had any trouble. I had a great experience. Everyone treated me very well. From the little I’ve spoken to other people with similar experiences, I’d say that’s actually a fairly normal outcome. People are actually very willing to accommodate differences, including of age. So I never felt really out of place.

We have, of course, age-segregated classes, but if you look at children of different ages, they are pretty willing to play with each other. So I think people are perfectly okay with age differences. It is just that with structured teaching… it doesn’t happen so often here.

AV: In retrospect, compared to the information available now, I knew very little. I applied to a few more places.

In Australia, the previous generation of academics went to England to do their PhD. So, for this reason I didn’t have a lot of advice about choices in the US. I didn’t know that many places in the US. Nowadays you can find anything you want on the internet. But I knew one mathematician in the US, namely Charles Ryavec at University of California, Santa Barbara. He’d always said, and gave me the impression, that Princeton was the centre of the mathematical universe. So, that’s how I picked it.

When I was admitted, I got an email from Charlie Fefferman.¹ And his email said something like, “I was in a very similar position to you at a similar age, and I came to Princeton and I’ve never regretted it.” I remember this left a certain impact on me.

AV: Yes. But I didn’t know who these people were. I wanted to study number theory. There was no one doing number theory where I was an undergraduate in Australia. And, of course, I knew who Andrew Wiles was but, you know, I don’t think I knew enough math to be aware of many of the other famous mathematicians there.

AV: I don’t know where. I think everyone connected to math knew about that. Yeah, it was quite a big news at the time.

AV: Again, I’m not sure. I think number theory is easy to appreciate when you’re an undergraduate. I knew some of the theorems of complex multiplication that I was impressed by. I found the fact that e^\pi \times \sqrt{163} is almost an integer was tremendously interesting. I understood that it was meant to be related to the class group, and I found that tremendously interesting . When I was coming to graduate school, I don’t think I had any understanding of the proofs, but I understood some of the statements. They seemed amazing.

AV: Yeah. I was trying to teach myself number theory. I hadn’t had classes in number theory—not beyond an elementary level. I remember a book of David Cox titled Primes of the Form x^2+ny^2, which I tried to read. It was too difficult for me, but I developed some appreciation for it, and thought that it was an exciting subject.

AV: I think I talked to both Andrew Wiles and Peter Sarnak. Perhaps they both suggested things for me to think about. But I was just not able to get started on what Wiles suggested, whereas what Peter suggested—I somehow got my foot in the door and I ended up working with him.

AV: Sarnak is a very positive person. You would go in to see him and, of course, you’d be depressed because you hadn’t been able to do anything. But you’d come out of his office feeling cheerful. You’d come out feeling you’re working on the right thing. I think that’s actually very valuable—that he was able to kind of make you feel positive. He was able to motivate you. That was really valuable for me. For a graduate student it can be very difficult. It’s very easy to get discouraged if you have no real understanding that you’re going to be stuck for a very long time.

I could very easily have imagined myself getting discouraged and dropping out of graduate school. That could have easily happened to me under different circumstances.

Graduate school is a different environment and then you start to wonder—“Am I really good enough to do research-level math?” It’s hard, in fact, to tell. You see all these people who are doing great things around you and then you think, well, maybe you’re not cut out for this. So I think it’s important [long pause] to have someone who believes in you.

AV: Oh, I think I certainly considered not continuing. Because Sarnak was such a supportive adviser, I was going to finish my PhD, but I was seriously considering going and doing something else at the end of it.

It’s a different experience when you try to do mathematics professionally. Generally, I suppose that when you are working on a problem, more often than not you find yourself stuck. These are perhaps the moments when you wonder whether to go on or not.

You are stuck in what you’re working on, but also you ask yourself—“I have learned a lot of math now. Could I really be producing math at this level?” And the answer to this, at that time, it seemed to me, was not at all. And then I was wondering that, maybe, I shouldn’t be doing this.

AV: Yeah, you definitely need that. Perhaps also some belief that you would like to do this. You have to be interested in what you’re doing.

AV: Firstly, I don’t see myself as anything but a number theorist. I was always motivated by problems in number theory. I have no idea if the things I’ve done are interesting from a dynamics point of view; it’s interesting from a number theory point of view. That’s my motivation.

There is some path between the various topics I have worked on—each one borders on the next and so on. So it’s not like I jumped; more like I drifted. But number theory borders on a lot of different things, and it seems like I drifted along these boundaries a lot. I don’t think that I, in any way, set out to do things that involved those ideas.

For me, at least, it is important to keep moving. If you stay working on the same thing for a long time, you can become technically very strong but I felt it important to keep moving. I’ve enjoyed that. I’ve been able to learn new things. In some ways I think that if you become too technically adept it interferes a little with being original as well.

AV: It’s a nice thing in number theory that sometimes you have proofs that combine ideas of different natures. It’s interesting that you often find that a simple idea of analysis would go a long way in algebraic contexts or vice versa. It’s often a very simple idea, not something very sophisticated. There are really different sources of intuition and techniques in number theory. Sometimes they complement each other very well.

AV: It should be. Unfortunately, the way things work in practice is that people tend to specialize, and you’re forced to specialize because of the structure of the system. You have to do a PhD. You have to get a job. It’s hard to do those things without specialization.

AV: You’re right. It’s nice, it’s enjoyable and interesting to use these interfaces with other fields. But the motivation for me was really number theory.

AV: Yeah, it’s an interesting example. Sometimes the problems which are solved using methods from other fields then achieve a greater prominence, as a result, than they had before.

AV: There’s this field called the geometry of numbers. One way you can describe it is that it was realized, by Hermann Minkowski and others, that many properties of integers can be deduced from the geometric properties of grids. I kind of view this dynamics on the space of lattices as a modern update of the geometry of numbers, where the idea is you can flow from one grid to another. I think the geometry of numbers is something that is sometimes seen as it was in the past, but I think it is kind of also reincarnated in various ways, and one of them is this dynamics on the space of lattices. So I view it as a successor to the geometry of numbers.

AV: I don’t know enough to give you an answer. There are several different interfaces: there is the school of Hillel Furstenberg at the Hebrew University of Jerusalem, and then there’s the school of [the late] Aleksandr Khinchin at Moscow State University. I think they are different sources, points of contact. I don’t know if I can summarize them.

AV: Linnik’s work is something that was very influential for me. I think it’s very remarkable and it was not well appreciated, perhaps because it is not so easy to understand. But in modern terms, Linnik was making an entropy argument—that was before the concept of the entropy of a measure-preserving transformation had been introduced. He’s asking how many essentially different trajectories does this system have, and he’s asking that at a very fine scale. Then he uses that information to bootstrap to a larger scale.

Philippe Michel and I were studying Linnik’s arguments together with Manfred Einsiedler² and Elon Lindenstrauss.³ Together with Anatole Katok,⁴ they had made very effective use of entropy arguments in studying the dynamics of homogeneous spaces. So we were a little bit stunned to find that Linnik had used entropy in exactly the same context from the 1940s, although the specifics are very different, and the ideas appear in a disguised way. Linnik’s work is fantastic. I found it very impressive.

AV: My collaborator Philippe Michel,⁵ spoke, but we wrote up the talk jointly. Yeah, there we tried to trace some of the ideas of Linnik to the modern day.

You look at a three-dimensional sphere, and you look at all the points with integer coordinates. As the sphere gets bigger and bigger, there are more of these points, most of the time. You can ask if they spread themselves more or less uniformly on the sphere. If you experiment a little bit, the answer is clearly yes. But proving that is quite subtle and difficult. It can be approached from different methods. It’s related on the one hand to the theory of L-functions. On the other hand, Linnik observed that you could approach it by dynamics, namely, you can go from one point to another. His idea was to follow trajectories of this process, and he essentially tries to measure how many trajectories there are. That’s his proto version of the idea of entropy. He shows that effectively there are very many—as many different trajectories as you could reasonably expect. He deduces that from what he calls his basic lemma. This basic lemma comes from the work of Gauss. Gauss was actually the first person to study this problem and he understood that, to study integer vectors on the sphere, you also have to study pairs of integer vectors at the same time. I’m not sure of the best way to explain this last point. Probably that will become too technical.

Anyway, Linnik showed that there is a very large number of essentially different trajectories, and from that he was able to go backward to show that the initial points themselves are uniformly distributed. If they weren’t somehow, there wouldn’t be enough allowed trajectories. So that was Linnik’s entropy argument.

Therefore there are two very different seeming approaches to this question, via L-functions or via dynamics. And I kind of got a lot of mileage trying to go back and forth and understand whether these approaches are really different, or if they are related. I think that led to a somewhat broader connection between dynamics and L-functions.

AV: My guess is not at all. I think this goes back to the correspondence principle, something that Niels Bohr formulated—that in the limit [of large quantum numbers] quantum physics should approach classical physics. That’s a sort of mathematically precise statement in many settings. The work of Lindenstrauss and Soundararajan is about one specific setting where you can extract mathematically precise statements from the correspondence principle. These are mathematically interesting statements, but they are extremely specific from the physics point of view. Of course, you should ask a physicist this question but my guess is that the systems we study are so rigid and specific that they are not of any real world significance that I know of. So my guess is, not at all.

AV: This was a problem that had a lot of prominence coming from the work of William Duke, John Friedlander and Henryk Iwaniec. It’s hard to say in a totally non-technical way. L-functions are objects like the Riemann zeta function. The subconvexity problem is concerned with giving upper bounds for them. I think that is from, some points of view, a technical problem. The cases that were proved in the eighties and nineties then turned out to be connected to a surprising number of other things. I think before that it was considered a kind of technical problem. But then some cases were proven and it was found that they actually controlled the behaviour of other questions. Then people got much more interested in it.

It remains to be seen whether there are more such applications or not. I don’t think at this point it’s completely clear. It would be unfortunate if we have exhausted the most interesting applications of it. So I’m saying it’s an interesting mathematical problem where collection of cases prove to have surprising connections. The other thing I would say about it is that, thus far, studying it has led to the development of a lot of useful and valuable techniques. So both of those things suggest it’s an interesting problem to study.

AV: I would imagine that I’m enormously influenced by them, mostly in ways that I don’t specifically attribute to them, but their work is integrated into the whole structure of math. The most influential ideas are ones that, in a way, are not attributed to the people who came up with them. They are assimilated. It seems like they were always there. The works of Selberg or Weyl have this nature.

AV: I think I have many. I really admire the work of a lot of both past and current mathematicians.

AV: No, I don’t know if there is. I do have a great admiration for the Russian school of math though, like the school of Israel Gelfand. That seems like something I wish I could have attended and seen. I have a great admiration for what he created and what came out of it. It seemed like he had a very broad vision of what math is and should be. I don’t want to single out Gelfand but the school around Gelfand seems like one of the wonders of the mathematical world.

AV: I’ve started to work with my graduate students a lot more. I used to suggest problems; now I work with them. I found that that seems to work better both for me and for them. I’m not very convinced that I know how to be a good graduate adviser. It’s difficult. I can’t think of anything that I’m adding and giving to my students. Some of them are good and they do fine, but I don’t think I did anything for them. I don’t think I’ve really figured out how to be a good adviser.

AV: I’m interested in music. And I enjoy it a lot. I don’t play any instrument, but I sing songs in a safe isolation.

AV: Whatever my daughters sing, I sing with them. I like many kinds [of music], and I certainly very much like Indian classical music. Also, Western classical and pop music too.

AV: Yeah, both Hindustani and Carnatic. My father used to listen to Carnatic music, and later I started to listen to Hindustani and found that in many ways, there are many things I like about it. For example, the singers often sing much more slowly, and they are willing to sing slowly for a long time, which I appreciate. In Carnatic music, I enjoy M.D. Ramanathan. I don’t know Hindustani so well. I like rock music too. My wife has very broad musical tastes. I get exposed to a lot of things, so I’m not picky.

AV: Yeah, it did. I think when I went to graduate school I really liked music. At that time I tried to learn the violin. I took lessons for four years and I would practice. After graduate school, I just didn’t have time to keep that up. But it would have been nice to have done that. I think I didn’t realize I liked music until I was about 16.

AV: Yeah. I don’t remember why. I took what I could find. I think there was a place where I could take violin lessons at the university. The other thing is that I did learn violin for a short time when I was young—when I was in second or third grade—and at that time I didn’t really practice. I didn’t really think I particularly enjoyed it then.

My children really enjoy music—both of them. My elder daughter, who is eight, is, in particular, very willing to listen to music. The younger daughter, who is only four listens to children’s songs but the older daughter—she’ll listen to Bob Dylan and is quite open minded. In fact, I think a few of my collaborations are based on a common love of Bob Dylan.

AV: Math collaborations. I have some math collaborators and we listen to Bob Dylan together.

AV: That is true in math. I don’t know the history of faculty in the other subjects.

AV: Oh, Harish-Chandra’s work is quite foundational. In fact, one of the first things I learned was representation theory. My adviser told me, “you have to learn the representation theory of semi-simple Lie groups.” And so I did, you know, in the way that Harish-Chandra developed it. It’s a marvelous theory and it’s part of the foundation of the theory of automorphic forms. So it kind of plays a role in my everyday life.

AV: Oh, I like running. I try to read fiction too.

AV: Pretty much what I pick up at the library. Sometimes I pick it up and don’t get very far. I can tell you that I haven’t finished the last round of reading I tried to do. The most recent book I read was a book of science fiction short stories by Ted Chiang. There’s a science fiction movie called Arrival. It’s based on a short story, Story of Your Life, that he wrote. That’s the most recent thing I read.

AV: There’s some part which is aesthetically very satisfying but there’s always a lot of work involved too. Like, you have to check a lot of details and so on. But somehow the core of it is very satisfying aesthetically, although I think that people, often students, don’t really understand how much unpleasant work is involved in between there and completing something.

AV: I’m trying hard to find that. I spend a lot of time figuring out what to think about. I try to choose carefully what to work on because you have limited time and you have to make a choice about what you think will prove to be interesting, without knowing in advance what’s there. So I have some ideas which are kind of continuations of what I’ve been working on in the past. I’m also trying to learn some ideas from physics, from gauge theory. I kind of hope they’ll be useful to me.

AV: I don’t spend so much time trying to solve specific problems now, which I used to do earlier in my career. It’s more like I feel I look for interesting phenomena, and then there are problems that come up consequently. That’s shifted over time.

AV: Oh sure. Fortunately in number theory you can do a lot of experiments. There’s a postdoc here, Preston Wake. He and I have been doing experiments about modular forms, specifically, we’ve been trying to generalize certain results from modular forms to Hilbert modular forms. What we found first, by doing computer experiments, is that they generalize in a funny way—sometimes the analogous results are true, and in other cases they fail completely. So the next challenge was to figure out what controlled this. And we were able to guess the answer—there is a single readily computable number that controls whether it works or not. Again, there was a lot of experiment involved in finding this. The next challenge is now to understand, to prove, that things work the way we have guessed. So I really enjoy the computational aspect and fishing around, so to speak.

It wasn’t something we expected. We were just looking around, then we saw this. I like that kind of thing. Now there’s a question: How interesting is this? Maybe there’s a simple explanation and maybe there’s not. It seems at least somewhat subtle to me and so we keep pushing and see how much of this is behind it.

AV: I haven’t gone as much since my children were born. I used to go more often. Both my grandmothers live in India and I always enjoyed visiting with them a lot—one lives in Bangalore and the other lives in Coimbatore. I used to go almost every year for quite a few years after I came to the United States. I enjoyed going but after getting married, having children, I haven’t gone as much. It’s harder.

AV: I visited for the ICM but I was there for only four days. My daughter was born just three weeks before, so I visited only briefly. I hope to visit the TIFR in Mumbai. I’ve been there once and hope to come for a longer visit.