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.