There’s an interesting development in the math-physics overlap, with a significant number of physicists getting interested in the theory of automorphic forms, often motivated by the problem of computing string scattering amplitudes. This has led to a group of them writing up a very long and quite good expository treatment of Eisenstein series and automorphic representations, which recently appeared on the arXiv. The emphasis is not on the physics applications (which an introduction explains come about when one is dealing with systems with discrete symmetries like the modular group or higher dimensional generalizations), but on the calculational details of the mathematics. There are quite a few expositions of this material in the mathematics literature but many (mathematicians included), may find the detailed treatment here very helpful.
Another aspect of this area is some overlap with the interesting of mathematicians studying Eisenstein series in the context of Kac-Moody groups. There’s a conference this week bringing together mathematicians and physicists around this topic.
Turning to recent developments in the Langlands correspondence itself, which relates automorphic forms to Galois representations, when I discussed David Nadler’s talk at the Breakthrough Prize symposium (the video is available here), I forgot to mention one thing he talked about that was new to me, the Fargues-Fontaine curve. Nadler explained that Fargues has recently conjectured that the local Langlands correspondence can be understood in terms of ideas from the geometric Langlands correspondence, using the Fargues-Fontaine curve. For more about this from Fargues himself, see materials at his website, which include lecture notes, links to videos of talks at the IAS and MSRI, and this recent survey article. Also informative is some explanation from David Ben-Zvi at MathOverflow.
In April there will be a workshop in Oberwolfach on geometric Langlands that will include this topic, for details of the planned discussions, see here.
Fargues was here today at Columbia, and gave a talk on “p-adic twistors”. Nothing much about Langlands, this was about the question of what the analog is for the Fargues-Fontaine curve when you take the real numbers as your field (the use of “twistors” is that of Simpson’s, see here, not the common use in physics, which is quite different).
I won’t display my extremely limited understanding of this subject by trying to provide my own explanations here. A big problem is that this is mainly about the p-adic Langlands correspondence, something I’ve never been able to understand much about. After making a renewed attempt the last few days, I at least started to get some idea of what are the biggest problematic holes in my knowledge of the math background. Interestingly, it seems many if not most of them have Tate’s name attached (Hodge-Tate, Lubin-Tate, etc, etc…). One pleasant discovery I made is that there are now some excellent expository pieces on this material available, often courtesy of some talented graduate students. One wonderful source I ran into is Alex Youcis’s blog Hard Arithmetic, which has given me some hope that with his help I might soon make a little progress on learning more about this kind of mathematics. I don’t know what’s in the water at Berkeley, but something there keeps producing high-quality blogging by mathematics students, another example is here.