Robert Langlands will be speaking at Yale in a couple weeks at a day-long Mostowfest of lectures in honor of Dan Mostow. His title is “The search for a mathematically satisfying geometric theory of automorphic forms” and he has already posted some notes for the lecture. A much longer set of reflections on the same topic was finished late last year and published in a volume in memory of Jonathan Rogawski. It’s available at the IAS Langlands site as A prologue to functoriality and reciprocity: Part 1. There’s no part 2 yet, but an earlier version of the full document is here, based on some lectures by Langlands in 2011 at the Institute, one of which is available on video here.
In all of these, Langlands is struggling with various ideas about “geometric Langlands”, meaning analogs of the Langlands program in the case of Riemann surfaces instead of number fields or function fields (functions on a curve over a finite field). One approach to this question, starting with Beilinson and Drinfeld around 25 years ago, has been extremely active and I’ve often written about this here. For the latest from this point of view, you can consult Dennis Gaitsgory’s web-site here. Langlands doesn’t find this often very abstract point of view to his taste, so has been trying various more concrete things. In particular, he’s quite interested in the connection to quantum field theory. I don’t think he’s actually found a satisfying line of attack on this problem, but it’s fascinating to see what he’s thinking about. There are all sorts of very deep questions in play here about the relationship of quantum field theory, representation theory, number theory and algebraic geometry. Langlands himself describes what he has as just “still provisional reflections on the geometric theory”, and says about his upcoming lecture:
The best I can offer in the way of a geometric theory with which I would be pleased is a sketch of the principal difficulties to be overcome. There are many. The importance for me is the very strong analytic flavour of the theory I hope to construct or see constructed.