The Canadian publication The Walrus today has a wonderful article about Robert Langlands, focusing on his attitude towards the geometric Langlands program and its talented proponent Edward Frenkel. I watched Frenkel’s talk at the ongoing Minnesota conference via streaming video (hopefully the video will be posted soon), and it was an amazing performance on multiple levels. A large part of it was a beautiful explanation of the history and basic conception of what has come to be known as geometric Langlands. He then went on to explain carefully some of the ideas in the recent Russian paper by Langlands, basically saying that they worked in the Abelian case, but could not work in the non-Abelian case. He ended by describing some alternate ideas that he is working on with David Kazhdan. Langlands was in the audience and at the end of the talk rose to comment extensively, but I couldn’t hear his side of this since he had no microphone (that Frenkel was sticking to his guns though was clear).
Besides giving the talk, Frenkel has made available a manuscript which gives a much more detailed version of the talk. See section 3.5 for an explanation of what he sees as the fundamental problem with what Langlands is trying to do: even in the simpler case of G/B over the complex field, you can’t successfully define a Hecke algebra in the way that Langlands wants.
The conference is finishing up right now, with final remarks by Langlands coming up later this afternoon.
A few more items, mostly involving my Columbia math department colleagues:
- If you connect quickly to the streaming video from Minnesota, you may be able to catch Michael Harris’s talk on local Langlands.
- Quanta magazine has an article about a recent proof of an old conjecture by Dorian Goldfeld about ranks of elliptic curves. This is due to Alexander Smith, now a
thirdfourth year graduate student at Harvard (he started working on this while an undergrad at Princeton, with Shouwu Zhang). His twin brother Geoffrey is also a math grad student at Harvard.
- Andrei Okounkov has been giving some talks recently at various places about developments in geometric representation theory with some connection to physics, under the title New worlds for Lie Theory. The slides from the ICM version of the talk are here.
- For those more interested in physics than mathematics the new issue of Inference has some articles you might enjoy. In particular, Sheldon Glashow is no fan (neither is Chris Fuchs) of the book I reviewed here
Update: Michael Harris is appearing via Skype from his home near here, since transportation out of NYC yesterday was mostly shut down (very early season unprecedented snowstorm, during rush hour…).
Update: I’m listening to the closing talk by Langlands. He is explaining his version of geometric Langlands, responds to criticism from Frenkel with “As far as I know there are no errors in the paper, no matter what you may see elsewhere”. He ends his talk with something like “At the last page I threw down my pen… It works and it works by a miracle. Don’t doubt it, it does work!”
Update: Another livestream, starting in moments: Alice and Bob Meet the Wall of Fire, a panel discussion with Quanta writers at the Simons Foundation.
Update: For another expository piece about the Langlands program, one that I somehow missed when it came out recently, see Sol Friedberg’s What is the Langlands Program? in the AMS Notices.
Update: An updated version of Frenkel’s notes is now available at the arXiv. Highly recommended for its lucid explanation of the form the geometric Langlands program has taken.