Not Even Wrong

The site at UBC collecting the work of Robert Langlands is now no longer being maintained. There’s a new site now at the IAS. It includes some interesting recent short articles of various kinds that I hadn’t seen before, including a short autobiographical memoir, an expository piece written for Pour La Science, and another piece which includes extensive speculative remarks about his current thinking on the topic of the “Langlands Program”.

The expository piece includes remarks about the remarkable centrality of representation theory both in number theory and quantum theory:

La leçon que nous voulons tirer de ce dicton, “il se trouve derrière tout nombre quantique une representation d’un groupe”, c’est que tomber en mathématiques ou en physique sur les représentations d’un groupe, c’est souvent tomber sur une veine d’or à laquelle il faut tenir corps et âme.

(”The lesson we would like to draw from this motto [due to Weyl] ‘behind every quantum number is a group representation’, is that when one comes upon group representations in mathematics or physics, one has often come upon a vein of gold, which one must pursue body and soul.”)

On the geometric Langlands front, earlier this month the Clay Mathematics Institute organized a series of talks at RIMS in Kyoto by Bezrukavnikov, Gaitsgory and Nakajima about various aspects of the subject. Unfortunately notes from the lectures don’t seem to be available anywhere that I have looked.

Last week the KITP in Santa Barbara hosted a mini-conference on Dualities in Physics and Mathematics, with some of the talks devoted to topics relating geometric Langlands and quantum field theory.

At the KITP in Santa Barbara there’s a wonderful program on Gauge Theory and Langlands Duality starting up this week, with some of the talks beginning to become available. The main topic will be the relations between S-duality in quantum field theory and geometric Langlands duality that Witten and collaborators have been working on the past few years.

I started trying to watch the talks, but the fact that the video quality is such that one can almost but not quite tell what is being written on the blackboard makes this a bit of a trial. I’m hoping that David Ben-Zvi or someone else will make available notes, which would help a lot. I did very much like Edward Frenkel’s description of the Langlands story as a “Grand Unified Theory of Mathematics”, and was interested to hear that he still feels that there are two different stories about the relation to QFT here, whose relationship is not at all understood (S-duality in 4d QFT is one, 2d CFT and vertex algebras is the other). It seems that A.J. Tolland is there, maybe he or someone else will do some blogging. As I get time to take in the lectures, I hope to write some more about them here.

Update: Notes for the talks are now also being posted, making following them on-line much more feasible. The quality of the talks is excellent, with Ed Frenkel so far giving a beautiful introduction to the roots of the Langlands program in number theory, David Ben-Zvi explaining the structures in topological quantum field theory that mathematicians are trying to exploit, David Morrison and Paul Aspinwall explaining mirror symmetry, D-branes, and the relation to N=(2,2) superconformal field theory, with examples, and Anton Kapustin starting on the 4d N=4 TQFT used to turn S-duality into a mirror symmetry.