Transcript

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

What did you think of the symposium?

FMS Blogger Richard Cerezo (by Luke Chang)

The inaugural Fields Medal Symposium has taken a good amount of running around on my part and I did not get a chance to do what I love doing, writing about math! Well I’ve been able to get things organized and develop a few new ways of keeping track of lectures and other things, thanks to the Fields Institute’s new Live video system. I was able to record the majority of talks using screen capture software.

In the coming days and weeks I will be reviewing these videos and making detailed posts about each lecture, so you can definitely look forward to that. But for now, you can check the Facebook page for photos (you don’t need Facebook to do this),

Click here to view Fields Medal Symposium photos by Richard Cerezo

Some quick impressions before they escape me. The FMS is the highlight of the 20th year anniversary of the Fields Institute and it truly marks a step forward into the next level of Fields Medal recognition. I have been involved with the institute for a number of years now and I have not been as motivated to learn new things as I currently feel.

Interacting with experts in many different fields and working with various members of the Fields staff has brought a certain kind of optimism to my perspective of the mathematical community. In particular, speaking with Ngô Bảo Châu, Edward Frenkel, James Arthur, and Peter Woit has given me the inspiration to continue doing what I do. Discovering mathematics, regardless of its newness or difficulty, strikes one’s innermost sense of curiosity. The ease of mathematical expression that I’ve witnessed during the week has shown me that these structures and ways of thinking are very natural if looked at properly. I hope this will come across in the video interview that I conducted with Ngô Bảo Châu, which should be ready in a week or so. I also hope this will translate nicely into blog posts about the mathematical talks.

I should note that it has been extremely rewarding to have been part of the organizing team. Seeing a full room of excited participants and getting questions from online participants is what kept us pushing harder and harder to make this the best week ever. I’m happy to say that we’ve gone far beyond our expectations!

What did you think of the symposium? Leave some comments for us!

-RJC

[Note: There will be much longer posts on each of the speakers mentioned in the videos.]

Public Event Recording, October 15, 2012

http://audability.com/AudabilityAdmin/Clients/FieldsInstitute/101343_1015201270000PM/registrationform.aspx?Event_ID=1343

The Langlands Program: Number Theory, Geometry and the Fundamental Lemma
James Arthur, University of Toronto

The Fundamental Lemma
Ngô Bào Châu, University of Chicago

With remarks given by: Ingrid Daubechies, His Excellency Le Sy Vuong Ha, The Honourable Bob Rae, and The Honourable Glen Murra

Student Event Recording, October 16, 2012

http://audability.com/AudabilityAdmin/Clients/FieldsInstitute/101344_1016201270000PM/registrationform.aspx?Event_ID=1344

Surfing with Wavelets
Ingrid Daubechies, President of the International Mathematical Union

Symmetry in Mathematics and Physics
Edward Frenkel, University of California, Berkeley

Mathematics on Stage and Behind the Scenes — To Infinity and Beyond – Panel Discussion
Moderated by Zach Paikin, Munk School of Global Affairs, University of Toronto
Panelists: Ngô Bào Châu, Ingrid Daubechies, Edward Frenkel, James Stewart

So this week has been a little busy, to say the least. Blogging about every single lecture in real time was not possible, however I will be blogging each day following the conference to highlight the work of one of the medalists.

Some highlights from the week so far, the Public Event was a huge success with a turnout of nearly 400 people. We received a message that Ontario’s Premier resigned early in Monday evening, but nonetheless our political guests Bob Rae and Glen Murray showed great support for the initiatives of the Fields Institute. Profs. Ngo and Arthur made the ideas of the Langlands Programs quite presentable that even my brother, cousin, and mother enjoyed watching! The links will be soon posted here.

The Student Event sold out of tickets and we had about 100 online viewers. We hope that you enjoyed the event and we’re very happy for our stars. A special thank you goes to Zach Paikin who kept the panel discussion very lively.Video links for this will also be posted here within the next day or so.

There is still a bit of running around to do today, but I should be able to begin blogging about the lectures starting on Thursday.

Why did Langlands call them Endoscopic Groups?

I don’t have a good answer to this question, but I will do a bit of digging. This is one question I will ask Prof. Shelstad tomorrow.

Since the average length of papers relating to the Langlands program is over 50 pages, asking is the most efficient way to this kind of knowledge. However, these papers seem to be mostly self-contained.

This paper has a very good general overview of the definitions of endoscopy and lead all the way up to a concise statement of the Fundamental Lemma.

But to see beyond the basics and to look at things like Ngo’s 2009 proof, we’d probably need months of work and multiple graduates courses to even begin reading, not to mention fluency in mathematical French.

Those of you coming to the symposium are in for a nice treat. We are going to be surveying a lot of material in such a short period of time. The line-up has been put together very nicely and I expect to at least get a sense of the questions that are being asked.

It might help to refer to look things up on Wikipedia before coming to these lectures such as Endoscopic Group, Representation, Automorphic Representation, Unitary Representation, Reductive Group, Fibration/Fibre Bundle, Hitchin Fibration

I’m really looking forward to hearing how the speakers describe these objects tomorrow, especially Professor Shelstad who cleaned up many of Langlands’ conjectures. In particular, the Langlands-Shelstad Fundamental Lemma, which Ngo solved.

I will be taking notes and making posts on things that I feel are worthy to share, but this may mean a lot of work. I also need to get LaTeX on this blog!

-RJC

This October, the inaugural Fields Medal Symposium will mark the repatriation of the Fields Medal back to its Canadian origins. The Fields Medal was established by Canadian mathematician John Charles Fields in the early 1900′s. It is the international mathematical community’s highest award for distinction in mathematical research, specifically for young mathematicians.

At the 2010 International Congress for Mathematicians in India, Professor Bierstone negotiated the Fields Institute as the permanent home for the symposium. This means that every year, the Institute will host a one-week long symposium in honour of one of the most recent recipients of the Fields Medal. The Fields Medal has not received the same level of recognition as the Nobel Prizes. However, the medal has had a very rich history within the mathematical and scientific communities. We are hoping that the inaugural symposium can help the global community rediscover the medal’s Canadian roots.

Tomorrow there will be a talk given by Craig Fraser about some of these interesting facts,

http://www.fields.utoronto.ca/programs/scientific/12-13/historyofmath/

 History of Mathematics Lecture

Mathematics on the Eve of the Great War:
Background to the Career of John Charles Fields
Craig Fraser, University of Toronto

October 11, 2012 at 3:30 p.m.

It will also be live streamed here,

http://www.fields.utoronto.ca/live

Why is the Fundamental Lemma, ‘fundamental’?

A large class of mathematical results rely on Chau’s award winning proof of the ‘fundamental lemma’. The proof ensures that a set of essential conditions, forming the basis of other mathematicians’ work, are true. More simply, without the proof of the ‘fundamental lemma’ many parts of mathematics would remain undecidable. That is, their truth, or their falsehood, could not be determined.

A non-mathematical example for undecidability is going skydiving without checking your parachute. Imagine how important it would be to know that you have a working parachute before jumping out of a plane. The parachute in this analogy is the ‘proof of the fundamental lemma’. Many mathematicians have taken the risk and have jumped without checking their parachutes. Luckily, Chau’s work has eased everyone’s worries. Also, this mental image should be taken with a grain of salt; no known mathematicians have been that reckless, in real life.

Photo cred: Owen King (www.defenceimages.mod.uk)

Chau’s result is revolutionary since it strikes at the foundations of mathematics. It provides confidence that many mathematicians are working in the right direction and ensures that their mathematical machinery is working properly. The methods of his proof were mainly algebro-geometric, but the consequences have a significant impact on all mathematical fields, from analytic number theory to mathematical physics. One reason is that the methods are transferable and can be used to produce other results. It may even provide the proper technique for many questions that have not even been asked yet. Another reason is that it bridges the gap between fields and essentially serves as a translator back and forth between the languages of number theory and harmonic analysis. [See Edward Frenkel's Interview]

Another post will deal with the fact that results similar to Chau’s have been posed many times in the past and I’ll explore questions have been asked systematically in the Langlands Program. As mentioned in a previous post, a half-dozen Fields Medalists have been involved with these kinds of questions and many more have reaped the benefits of these breakthroughs.

The researchers invited to attend the inaugural Fields Medal Symposium have made significant contributions to Ngo Bao Chau’s areas of research. In particular, Chau’s research spans the mathematical areas of algebraic geometry, group theory, and automorphic representations.

Another very good treatment of Chau’s result is given in the ICM2010 Archives

A technical summary, The laudations by James Arthur

A non-technical summary, The work profile by Julie Rehmeyer

Professor Edward Frenkel (http://math.berkeley.edu/~frenkel/) is one of the main scientific organizers for the inaugural Fields Medal Symposium. The main focus of his research is symmetry in mathematics and quantum physics. Many of the questions that he is currently tackling have their origins in the Langlands Program. He is Professor of Mathematics at the University of California, Berkeley.

At this year’s Fields Medal Symposium, Frenkel will be giving a general survey lecture on the Langlands Program, and he will also sit on a panel to talk about the world of mathematical research. Prior to the panel he will be giving another short talk to provide context for the discussion. Frenkel is a strong advocate for improving the public image and understanding of mathematics.

Frenkel has agreed to contribute regularly to our blog. His first post deals with some general questions from Richard Cerezo about the Langlands Program.

Would you characterize the work being done in the Langlands Program as the rigorous development of mathematical language that relates number theory and mathematical analysis?

Yes. And there is also much more to it.

Could you please elaborate?

In creating the Langlands Program, Robert Langlands was motivated, in the late 1960s, primarily by hard questions in number theory.

What kinds of questions?

Let’s say you have an equation such as y^2 = x^3 + 5x +3 and you look for solutions x, y that are not real or complex numbers, but elements of the finite field Z/pZ, where p is a prime number. In other words, you look for x and y from the set of integers {0,1,2,…,p-1} such that when you substitute them into the equation, the left hand side is equal to the right hand side up to a multiple of p — we then say that the equations is satisfied “modulo p”. For instance, let p=5, and take x=1, y=2. Then the left hand side is equal to 4, and the right hand side is equal to 9. The difference is 5. So x=1,y=2 is a solution “modulo 5″.

There are plenty of equations of this kind in number theory. And we want to know, for example, how many solutions a given equation has modulo p, for all possible primes p. This turns out to be a very hard question.

Langlands’ deep and surprising insight was that these mysterious numbers of solutions could be read off objects of an entirely different world, called “harmonic analysis”.

What is harmonic analysis?

It is the study of functions of a particular kind. For instance, we all know trigonometric functions (of a variable x), sin(x) and cos(x). Let us consider also sin(nx) and cos(nx) for all integers n. The idea, going back to Fourier in the 19th century, is that almost all functions that are periodic can be written as superpositions of these very basic functions. This is a remarkable statement! Imagine that we have a signal that is represented by a function. Writing it as the sum of trigonometric functions represents the decomposition of the signal into “elementary harmonics”. This is what harmonic analysis is about: funding some elementary harmonics, like sin(nx) and cos(nx), but in a much more general situation, and finding ways to decompose general functions in terms of these harmonics.

This is a beautiful theory, and note that from the outset it seems to be very far away from number theory.

Here comes a surprise: Langlands conjectured that these two worlds: number theory and harmonic analysis, are inextricably linked. More precisely, he conjectured that the questions in number theory like finding numbers of solutions of equations modulo primes can be solved by harmonic analysis. For example, there is a harmonic function that “knows” about the numbers of
solutions of the above equation modulo all primes (or maybe all but finitely many primes, but that’s a technical point).

This is absolutely mind-boggling, kind of black magic! And that’s why people got so excited about the Langlands Program: first of all, because it gives us a way to solve what look like intractable problems. And second, because it points to some deep and fundamental connections between different areas of mathematics. So we want to know what is really going on here, why it is like this? And we still don’t fully understand it.

So these are the origins of the Langlands Program. But then what happened is that people realized that the same mysterious patterns may be observed in other areas of mathematics, such as geometry, and even in quantum physics. I have previously written, partly tongue-in-cheek, that the Langlands Program is a Grand Unified Theory of Mathematics. What I meant by this is that the Langlands Program points to some universal phenomena and connections between these phenomena across different fields of mathematics. And I believe that it holds the keys to understanding what mathematics is really about.

Can you talk further about the idea that the Langlands Program might serve as mathematics’ Grand Unified Theory?

The Langlands Program is a vast subject. Hence there is a large community of people. But, as I said, the ideas of the Langlands Program have penetrated into many areas of mathematics. So you have people working in number theory, or in harmonic analysis, or in geometry, or in mathematical physics, working with very different objects, but observing similar phenomena. What’s most interesting to me is to see how the same patterns play out in all of those different domains and understanding how these different areas are connected.

It’s like you have different languages and you have sentences from those different languages which you know mean the same thing. You put them next to each other and little by little you start developing a dictionary that allows you to translate between different areas of mathematics.

In other words, I view the Langlands Program not as an “area” of mathematics, but as a “meta-area”, in the sense that it is something that is observed throughout mathematics.

Is there a way to look at the Geometric Langlands Program (GLP) informally or intuitively? If so, and if possible, could you please give us your everyday version of the GLP?

I have tried to give above a “pedestrian” version of the Langlands Program, the way it was originally formulated.

When people say “geometric Langlands Program”, they mean a set of similar ideas playing out in geometry. In geometry, instead of the algebraic equations modulo primes that I talked about earlier, we have objects that at first glance look very different: namely, the so-called Riemann surfaces. The simplest one is the sphere, then we have the surface of the donut (with one “hole”, so to speak), then there is the surface of a Danish pastry (with two “holes”), and so on. Why these geometric objects are analogous to the equations modulo primes requires a separate explanation, which I won’t give you. Let’s just say that this is something that mathematicians have known and exploited for a long time and let’s trust that there is this analogy.

So then the natural question is: what should be the objects on the other side, what are the analogues of harmonic functions? This is by no means obvious, and this was understood much later, in the 1980s, in the works of several great mathematicians: Deligne, Drinfeld, Laumon, Beilinson, and others.

Roughly speaking, the objects on the other side may be expressed in terms of what we call “D-modules”. D-modules are, roughly speaking, mathematical objects representing systems of partial differential equations. So now the surprising link discovered originally by Langlands between number theory and harmonic analysis becomes a link between some objects related to Riemann surfaces and D-modules. This link is as fascinating as the original Langlands conjectures.

At other conferences similar to the Fields Medal Symposium, was work on the proof of the Fundamental Lemma the centre of attention? Did you or any of your collaborators take a shot at proving that particular result?

Let me say a few words about the Fundamental Lemma. In pursuing his Program, Langlands came up with certain mathematical formulas that should hold for his Program to work. He called the statement that these formulas are true “the Fundamental Lemma”. Why did he call it a lemma, and not a theorem? I guess he thought that this was a rather technical statement that one could prove in a fairly straightforward way. Well, unfortunately, that was not the case. This “lemma” resisted attempts by many mathematicians to prove, until finally Ngo Bao Chau came up with a brilliant proof. His proof uses entirely new geometric ideas (some of which were introduced earlier by Goresky, Kottwitz, and MacPherson, as well as Laumon — partially, in collaboration with Ngo himself).

Of course, everybody knew that the Fundamental Lemma was one of the central statements of the Langlands Program. People worked on it and there were meetings and conferences at which the subject was discussed. But there is something that you have to understand. The Fundamental Lemma is a statement in the “original” Langlands Program; more precisely, it is a statement in harmonic analysis. So people working in the geometric Langlands Program, like myself, did not pay much attention to it. But a remarkable aspect of Ngo’s proof is that it is geometric — and moreover, it uses many of the objects that are the staples, so to speak, of the geometric Langlands Program. So what Ngo’s work has done (in addition to proving a very important and long-standing result) is that it brought together people from different fields.

The program of our Fields Symposium is a testimony to this: we will have lectures by the top experts across different disciplines: number theory, harmonic analysis, geometry, and physics. This is what Ngo’s work has done. And this is very important in my view.

There seem to be a number of very distinguished people working inside the Langlands Program and it has a good amount of well-deserved attention. Could you please give us your reasons for continuing to work in this Program?

The more we know, the more we understand how little we know. As I said, the beauty of the Langlands Program is that it points to mysterious connections between different fields of mathematics. And the biggest question, in my mind, is why these connections exist, what is the mechanism behind them. We still don’t know, but we are working on it. For example, in my recent joint work with Langlands and Ngo, we suggested a path to proving the so-called Arthur–Selberg trace formula using methods similar to those used by Ngo, as well as ideas from the geometric Langlands Program. Ngo’s work has matured us. We now understand better how different pieces of the puzzle fit together. But we need new, fresh ideas. I hope that young people who come to our Symposium, who follow it online, will get interested in this subject and will lead the next revolution in the Langlands Program.

The obverse of the Fields Medal (Wikipedia)

51 Fields Medals have been awarded, 43 Fields Medalists are still alive, 1 Fields Medal was declined in 2006, and 1 silver plate was awarded in 1998 in place of a Fields Medal.

Since 1936, medals have been given at every International Congress for Mathematicians, also known as ICM. Before the first medals were awarded, there were 9 ICMs held previously, with the first in 1897. John Charles Fields, the medal’s namesake, died before the first medal was awarded. Since the 1924 ICM in Toronto and until his death Fields worked towards the establishment of an ‘international medal for research in mathematics’. He was opposed to having the award named after any particular donor, including himself. Currently the award has a financial value of $15,000.

The reverse of the Fields Medal (Wikipedia)

6 Fields Medalists have received the medal for work related to the Langlands Program. This includes mathematicians Atle Selberg (1950), Alexander Grothendieck (1966), Edward Witten (1990), Vladimir Drinfeld (1990), Laurent Lafforgue (2002), and Ngo Bao Chau (2010). There are a few mathematicians that worked indirectly in the Langlands Program including Pierre Deligne (1978) and Sir Michael Atiyah (1966).

The international community of mathematicians, which is much more difficult to quantify, has been deeply interested in the work of these mathematicians. There are likely hundreds of professional mathematicians working on directions of research related to the Langlands Program.

Part II of this post will go in depth with the historical directions of the Langlands Program. Also, rigorous estimates of mathematicians working in the Langlands Program will be made.

In the video Professor James Arthur explains the historical figures on the poster of the Fields Medal Symposium. He talks about a graphical representation of the Fundamental Lemma. He also describes the work of each of mathematician on the poster. (from left to right) Hermann Weyl, Erich Hecke, Emil Artin, Atle Selberg, Harish-Chandra, Alexander Grothendieck, and Robert Langlands.