I just got back from a few days in Toronto, where I attended the Fields Medal Symposium on Fundamentals of the Langlands Program. This is the first of a planned yearly series to be held a the Fields Institute, with the idea that each Symposium will focus on an area of mathematics crucial to the work of one of the recent Fields medalists. In this one, Ngô and his work proving the Fundamental Lemma in Langlands theory was the center of attention.

The talks were recorded, and I believe that video of them will soon appear. An effort was made to get speakers to give talks aimed at non-specialists, and the results were quite good. Among the talks I attended, I can highly recommend those of Sophie Morel, Edward Frenkel, Nigel Hitchin and Edward Witten, which covered some of the huge diversity of fundamental mathematics that goes into this subject. Unfortunately I only got to Toronto midday Tuesday, so missed all the Monday and Tuesday morning talks. I heard that the Tuesday morning talks of Richard Taylor and Michael Harris gave excellent introductory surveys on the number-theory Langlands program. Monday was devoted to more specialized talks on endoscopy and the fundamental lemma.

Ngô’s talk was about some new ideas on how to go “Beyond Endoscopy”, to extend previously successful uses of the trace formula to prove Langlands functoriality to a wider range of examples than those covered by the fundamental lemma. Another example of this sort of ongoing work mentioned by a couple speakers was work by Ali Altug, who has just finished up as a student at Princeton and started teaching here at Columbia this fall. Witten’s talk surveyed the relationship between QFT and geometric Langlands, motivating clearly why the N=4, d=4 SYM theory appears. For more details about much of the more advanced material covered in his talk, see the write-up here of his lecture at Atiyah’s 80th birthday conference.

Monday evening there was a big evening program for the public (which I watched some of from New York via web-cast), and Tuesday evening there was a special program for high school and college students, with Ingrid Daubechies and Frenkel giving talks, as well as a panel discussion with them, Ngô and James Stewart. A lot of students attended, and many stayed on for almost an hour to talk with the speakers. Ngô has a popular book out in Vietnam, which evidently has been a huge success. Frenkel has a book entitled *Love and Math* coming out next year, a chapter of it is available here.

Panelist Jim Stewart has them both beaten as a successful author. His excellent Calculus textbook may be the most widely-used one in the US, and evidently the financial rewards have been significant. He was one of the financial supporters of the symposium, and Wednesday night had many participants out to his amazing home in Rosedale for a banquet. It’s a spectacular, award-winning piece of architecture he calls “Integral House”, and its five stories and 18,000 square feet of space are perched over a ravine not far from downtown Toronto. Evidently it cost him about $24 million, as well as about ten years of his life in design and execution. For more about Stewart and Integral House, see here, here, and here.

Richard Cerezo was taking lots of pictures and has been posting on the Symposium blog here. A short video of me, Frenkel and Hitchin discussing the Symposium topic may appear there at some point.

**Update**: The conversation with Frenkel and Hitchin is now available here.

Peter,

Sounds like fun, not that it’s even vaguely related to what I do. On another note, Stewart’s calculus excellent, really? I mean it’s not too bad for a modern calculus book, but it’s way too long, with all sorts of useless stuff in it, and it doesn’t emphasize the important stuff nearly enough. I guess the “proofs,” such as they are, are better than many other current books, since from what I remember they aren’t actually incorrect, just often sloppy and incomplete. It’s fine to have an incomplete proof, but if you do that you should say “this is meant to give you an idea of how the actual proof works.” I guess that most of this isn’t Stewart’s fault, especially at this point, but still. You want a truly great calculus book, use Spivak.

Yes! Spivak’s calculus book is probably one of the greatest math textbooks ever written!

If one wants a proof-based calculus book, yes, Stewart’s not the way to go. But, I’d really rather not host a discussion here of Calculus books, unless it’s somehow closely related to the Langlands Program, which is the topic of the posting….

Peter,

OK, Langlands program. A quite well known mathematician once told me that he hated getting reference letters from Langlands, since all they ever talked about was how the candidate had advanced the Langlands program

Hi Peter,

Thanks for the post. I’m so glad that you enjoyed the symposium and that we were able to have some conversations. I’m glad to see that your blog pops up among the first few when I look for coverage of this FMS! There were quite a few reporters there on Monday evening and hopefully I can find something in English as most of them were Vietnamese news reporters.

Unfortunately I missed Sarnak’s talk and I heard that it was also very nice.

As an aside, Spivak’s Calculus is really quite phenomenal for a proof course, but as far as I know he didn’t have that very impressive wall of his own books!

Richard is referring to this

http://thestar.smgmedia.topscms.com/images/c6/0d/c868f86e476a931df0f06e7c012c.jpeg

Guests at Stewart’s got to see his huge bookcase of Calculus and other course books, which this is a part of, the part containing his own books..

Peter,

I have an undergrad math background, but I know almost nothing about the Langlands program. Not to sound sarcastic, but what’s so special about this topic anyway? I took a look at its Wiki page, but I couldn’t quite understand it. And what applications (if any) does it have to physics?

Paul,

The public program talks tried to give some motivation for the Langlands program, I think they’re online. Richard Taylor’s talk I think was also pretty general, should be online at some point.

To oversimplify dramatically, the Langlands program grew out of number theory, where it is a basic insight into the structure of numbers (relating the Galois group to Lie groups). The proof of Fermat’s last theorem is based on it. It also has some geometric analogs, known as Geometric Langlands, and this geometry has turned out to be related to certain very special quantum field theories. Witten’s talk covered that, but it’s a long and rather complicated story.

Thanks for posting the link! What did you think of the video? I have picked up a very nice book dedicated to Hitchin called ‘The Many Facets of Geometry: A tribute to Nigel Hitchin’. It has a very nice article written by Witten on the Geometric Langlands! I will include my comments on it when I write about Hitchin’s talk.

Hi Richard,

I thought the video of us came out very well, thanks for arranging it and producing the great video.

The Witten article you mention is a nice review article, written earlier on and probably worth reading before the one that I mentioned, which corresponds more to his talk at the conference.

Looking forward to videos of the rest of the talks, I hope we get to see those!

Thanks alot Peter for the link to Edward Frenkel’s story on jews’ treatment in 1980s USSR. It is fantastic -the recounting. I always wondered how Frenkel could develop the way he has, he is quite a unique mathematician.

Anyone have a link to the talks?

The talks are available here

http://www.fields.utoronto.ca/video-archive/event/108

Thanks Peter! I know what I’m doing for the next 10 hours . . .