There’s various news to report on the geometric Langlands front, spanning number theory to quantum field theory:
Minhyong Kim has been running an Online Mini-Conference on the Geometric Langlands Correspondence for the past month, and Dennis Gaitsgory has been doing something similar since last spring at his Geometric Langlands Office Hours.
Very recently Edward Frenkel has given talks in both places (see talks here, here and here, slides here and here). He’s been talking about joint work with Etingof and Kazhdan on a function-theoretic (as opposed to sheaf-theoretic) version of geometric Langlands. They have a paper out here, are working on two more.
This work to some extent has its origins in attempts by Langlands to come up with his own version of such a function-theoretic approach. Frenkel was asked to discuss this topic by the organizers of the Abel Conference in honor of Langlands. I wrote about what happened here. Frenkel came to the conclusion that what Langlands was suggesting could not work (Langlands vehemently disagreed…), but this led him to the current research he is pursuing with Etingof and Kazhdan. For a written version of Frenkel’s talk explaining all this, see here.
On the quantum field theory front, Witten and Gaiotto have been working on relating older ideas of Gukov-Witten about using branes as a general method of quantization, applying this to geometric Langlands, in the new context that Frenkel’s talks discuss. Witten talked about this last week in the Kim seminar (video here, slides here). Gaiotto last week also spoke about this at a Kansas State seminar, video here, slides here.
The original 2008 Gukov-Witten paper on branes and quantization is here, Gukov’s 2010 Takagi lectures on this are written up here. The problem of how to quantize a general symplectic manifold is a fascinating one, and at the time I was very interested to see this proposal. It does however invoke a very sophisticated set of ideas about quantum field theories in order to deal with what one would think are much simpler examples of the quantization problem. Perhaps this program would come into its own in this new case, where the quantization problem involves similarly sophisticated mathematical constructions.
From another side of the geometric Langlands world, Peter Scholze is continuing his lectures on his ongoing work with Laurent Fargues that reformulates the local Langlands correspondence in terms of geometric Langlands on the Fargues-Fontaine curve. There are associated discussion sections, with a web-page here.
Announcement: I’d been reading about how the hot new idea for authors on the internet is Substack, where all sorts of interesting material can now be found. After thinking about this “back to the email newsletter” model for a minute, I realized that I should try and see if I could get email subscriptions to this blog working. There’s now a place over on the right where you can ask for an email subscription. No experience with this yet, so I can’t guarantee either that it works or that problems won’t turn up that will cause me to have to turn that feature off.
Update: For another talk by Witten about this from today (Feb. 11) see here.