Peter Woit: Well there is something that I was thinking about while watching these talks. I felt it was amazing hearing a couple of talks per day. One thing that was amazing was a kind of sweep of these things, there’s so much mathematics, physics, so much going on in your [Ed Frenkel] talk, and Ed Witten’s, and your [Nigel Hitchin] talk. It’s kind of the opposite of what you normally think of the history of what’s happening to these fields, about becoming more and more specialized and people becoming more and more isolated from each other. These are subjects where, you know, all of these different things are coming together and I was wondering what you thought about that and there was a question Richard was asking about, what kind of advice is there for students or people that want to learn about this? How do you absorb this fantastic amount of material?
Nigel Hitchin: Yeah well, it’s difficult for students, I mean, they want a topic to work on so they have to focus but they need a certain amount of background too. But, there’s a degree of specialization that has to take place for students to write their theses but it has to be certain in this point of their background. How you actually integrate all the other influences into, say, a student’s work is very difficult. One student will come on day one and say “What do I have to know to be able to do this?” It’s difficult to actually explain that. But these types of meetings is going somewhat towards that.
Peter Woit: Also, I had a question about what you see, having lived through these last few decades of this program, what do you see happening? Are we going to be seeing this further unification? You like to talk about grand unification.
Edward Frenkel: Right, I wrote at some point that I considered the Langlands Program as a kind of grand unified theory of mathematics, partly tounge in cheek but it’s true that it connects so many different areas, but to me this is the lure of the students because I can tell about my story, how did I come into the Langlands Program. I was studying representations of infinite dimensional Lie algebras and I didn’t think of, but I heard about Langlands Program, that was back in the early 90s and I was fortunate that at exactly that time Drinfeld was thinking about his new approach to Langlands Program which used my work with Feigin on representations of Kac-Moody algebras. And, when he told me about this, that made me curious, and that was my entry point. And that’s the remarkable thing about Langlands Program that everyone can find, a lot of people can find their own entry point and once I was inside I was just so fascinated with the fact that there are so many connections and then I was curious by it and I studied them. And so that provided additional motivation for me. And that’s what I think, that for students you don’t need to know everything all at once, you need to find your own entry point. And that’s a beautiful subject which has so many possibilities and portals into the subject.
Peter Woit: Interesting, how do you find the experience of coming at it from one point of view or area and then all of a sudden there’s this completely different kind of mathematics which is relevant like Number Theory? How do you encounter Number Theory?
Edward Frenkel: So I just started to learn. At that time I was at Harvard, my first year at Harvard. Drinfeld came and was there for a semester so I actually just had this great resource that I could just knock on the door of his office and ask him some questions. And then I was just reading and attending conferences like this and talking to people. Just gradually I got more and more interested. So today I talked about these three different columns, three different threads of the Langlands Program: Number Theory, Curves over Finite Fields, and Complex Curves. And so, I started from the rightmost column so to speak. But then I moved gradually to the middle and the left one. And so, one doesn’t have to do that, I was just very curious I guess and in general I’m very interested in when different subjects are connected. But, I think one can find plenty of stuff even within one particular part of the Langlands Programs. But oftentimes, it’s really advantageous to try to take an objects from one of them and use it in another one. And a case-in-point which we discussed today is Nigel, giving a beautiful lecture on how he invented his moduli space, and he was motivated by questions in differential geometry and physics, right. Hyperkahler manifolds.
Nigel Hitchin: Yeah, that’s right. I mean, we all have different attitudes when we do research. I tend to be a problem solver but underneath it all, it’s the same thing, trying to understand why something happens and how it happens. So, it was focusing on particular equations that got me into this, but it turned out those equations, I felt that there was something important, I felt that there was something there. Something in 4 and 3 dimensions, there is something in 2 dimensions which has to be even more important because it’s 2-2 rather than 3-1. And so, gradually, it emerged. I didn’t know what I was getting into, I just had a feeling that something was there.
Edward Frenkel: But what I’m trying to say here also is, that your work that was squarely in the realm of complex differential geometry. What an amazing energy came out from taking your moduli spaces to the realm of curves over finite fields and using them so effectively for solving the decades long conjecture, the Fundamental Lemma. And that sort of re-advanced us in the sense that we now see that this object which looked to be so far away from each other are in fact not that far away.
Peter Woit: So, how do you feel about the use [of the Hitchin Fibration] in the Fundamental Lemma by Ngo, and how much are you able to take it and appreciate it or understand it?
Nigel Hitchin: Well, I’m totally amazed by it. But I mean, it’s far away, it’s use is far away from it’s origins of it from my point of view. …But, the idea of systematic compactification of abelian varieties which came into this. I guess it was something which once I’d seen the integrable system I think something really good was going on there. But I had no idea it could be transported to an area as far away as…
Edward Frenkel: And I think it indicates that there’s a lot more to come. So I think on this note, we should probably go back and hear the last talk of today’s lectures.
[all] Thank you
Transcribed by Richard Cerezo