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.
Do we know what exactly Gaitsgory et al. have achieved in the direction of geometric langlands ?
On an other topic : I hope we will soon get a “latests from ABC” post, I think i’m not only speaking for myself when I’m saying that its crazy that we have no information whatsoever on how the reading of the proof is going !
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The latest from Gaitsgory et al. on geometric Langlands that I know about is this
which outlines a proof of the “categorical” version of geometric Langlands, for the case of GL(2). There is a school on the topic planned for Jerusalem in March
This has become a large subject though, and there may very well be more interesting things going on using these ideas than that focused on the conjectured equivalence of categories.
Unfortunately, from the QFT point of view, I don’t think there’s much activity.
On Mochizuki, I think the situation remains that most experts have given up as hopeless the task of making their way through the details of his papers. He has a relatively new “panoramic overview of Inter-universal Teichmuller Theory”
but I think this doesn’t provide the kind of detailed outline of a proof that experts are looking for.
One interesting question I don’t know the answer to concerns the status of refereeing of Mochizuki’s papers. I’ve heard rumors some of them have been submitted for publication, haven’t heard what happened. It does appear that he has been going through them with the help of Go Yamashita and others, checking things and making periodic revisions, see
It’s undoubtedly a problem to find referees who feel competent and willing to go through the papers and understand exactly what is going on well enough to judge whether the details were correct.