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.

Peter,

For the email notifications, will the notification say who is doing the post, or will it just say “new post in topic “X”?

Jim Akerlund

Jim Akerlund,

When I post a new blog entry, you’d get an email from “Not Even Wrong”, with a subject line like

[New post] Geometric Langlands News

and then the full text of the blog entry as an email. So you can read the blog entry in your email, and get it write when it’s posted.

It’s actually very similar to the Substack model, where subscribers get content in their email exactly this way. Only difference is that subscriptions are free!

This is being done by a WordPress feature called JetPack, and everything is being managed at wordpress.com. My only concern about this is that it’s something I don’t have full control over, since it is managed by wordpress.com. On the other hand, it does seem convenient for people, and I don’t want to manage subscriptions + sending out bulk emails myself.

As reassurance, I’ve had people subscribing by email via JetPack to my blog for several years. There is one issue that’s come up a few times, and that is that if someone replies to the email, that reply is automatically submitted as a comment on the blog. I had a few family members reply thinking it was just going to me, only for it to end up in moderation as a comment instead. But as long as your readers don’t confuse the blog emails with your personal email you shouldn’t have any problems.

Sad news from MIT: Isadore Singer died yesterday.

Further recent progress in geometric Langlands theory is the work of Arinkin, Gaitsgory, Khazdan, Raskin, Rozenblyum, and Varshavsky (https://arxiv.org/abs/2010.01906), which like the work of Etingof, Frenkel, and Khazdan, is devoted to finding concrete, function-theoretic consequences of geometric Langlands, but in a different direction: they give a new statement of the geometric Langlands correspondence which formally implies, and doesn’t just suggest, the (everywhere unramified) Langlands correspondence for function fields over finite fields.

Thanks Will!

Looking forward to the day when it will again be possible for you to help explain this sort of thing to me over lunch…

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