Yesterday I was at Peter Scholze’s Seminaire Bourbaki talk, his write-up is available here. It was advertised as an exposition of the recent proof of the geometric Langlands conjecture, but Pierre Colmez accurately describes it as something much more: the Langlands program 2.0.
In his talk, Scholze described the situation of the geometric Langlands program as for many years getting further and further away from the original arithmetic Langlands program. Recently there has been a dramatic change, as things have turned around and geometric Langlands is having an impact on arithmetic Langlands. He gives a new version of the basic conjecture of the arithmetic Langlands program (Conjecture 1.5), in a form similar to that of geometric Langlands. He describes this conjecture though as merely a “template for a conjecture”, since some of the terms remain undefined.
I learned from his references about something perhaps more accessible, David Ben-Zvi’s talk What is the Geometric Langlands Correspondence About? at the AMS meeting this past January. The blurb for the talk is:
Number theorists found the tusks. Physicists found the tail. Now geometric Langlands tells us it’s an elephant.
Update: In a comment David Ben-Zvi explains that there’s a revised version of his talk write-up available at this website (soon to be on the arxiv), and a video of the talk should soon appear on the AMS Joint Mathematics Meetings youtube playlist.
Thank you Peter for posting this – I wouldn’t have seen the PS paper for a while otherwise. I always find your math links useful!
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Just glanced at the paper. Footnote 9 was amusing. (ymmv, but having an analogy sharp enough to cut you – tell you that it isn’t helpful to solve the problem – works for me.)
I look forward to reading how (if?) his gestalten formalism reformulates technicalities about the spectral actions.
Fascinating. Are you going to attend Serre’s 100th birthday conference. It would be cool way to spend some time in Paris.
Recently Sam Raskin has given a 6-part series of lectures at IHES on the geometric Langlands program, which also might be of interest to anyone interested in this: https://www.youtube.com/watch?v=n20XG8S97Yc
Peter – thanks for the link! I just put my Geometric Langlands survey article on my website, should post to the arXiv soon (it’s been significantly revised since the version that can be found on the AMS website). It is indeed meant for a much broader audience than Scholze’s (incredible) Bourbaki talk. The talk should be even more accessible – I’m waiting for the video to appear on the AMS Joint Mathematics Meetings youtube playlist, hopefully in the next couple of weeks.
About Langlands 2.0: Scholze provides a really new perspective on some of the key structures in the Langlands program in his article – I’m particularly excited about the formulation using the Gestalt of Langlands parameters, which gets around what was an extremely uncomfortable issue up til now (the need for automorphic sheaves with nilpotent singular support).
However I think the statement of Conjecture 1.5 is not meant to be radically new — it is an “obvious” reformulation for number fields of one of the main topics of his article, the very powerful formulation of the (arithmetic) Langlands conjectures over function fields due to Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum and Varshavsky (much of which has now been proven). What IS radical is that Conj 1.5 is getting closer by the day to being made precise, using a great accumulation of Scholze and collaborators’ work over the past several years (in particular Gestalts).
Do we know where the video of Scholze’s talk might be posted?
Aaron Cofer,
Presumably it will be posted at some point here
https://www.carmin.tv/en/collections/seminaire-bourbaki