Various Langlands program related news, starting with the man himself:

- For the latest from Langlands about the geometric theory, best if you read both Russian and Turkish. In that case you can read this and this. For the rest of us, all we get are this commentary on the Russian and Turkish documents and these last or very well last thoughts on them.
- In a couple of weeks there will be a conference celebrating the work of Langlands, organized in conjunction with his Abel Prize. Perhaps there will be live stream here.
- I hear that at his lecture at the CMI at 20 conference Scholze made some new conjectures about possible ways of getting the Langlands correspondence in certain cases of the number field case. I haven’t however seen anywhere that one can read or hear more about these. It would be great if the Clay Mathematics Institute could make available videos of the talks at that conference.
- Scholze will be giving the Chow lectures in Leipzig next week. The program there includes some preparatory talks by others, including my ex-Columbia colleague Daniel Litt (now at the IAS). I see that Daniel has at least posted a problem set you can get started on.
- Also coming up next week is the Breakthrough Prize Symposium at Berkeley, where Vincent Lafforgue will talk about his (valued at $3 million) work on the Langlands program Monday morning (live stream here). On the physics side, in the evening a group of prize-winning theorists will talk about “Is Time Travel Possible”, live stream here.
- A central idea conjecturally relating the geometric version of local Langlands to the number field version is the Fargues-Fontaine curve, which Jacob Lurie has been giving a course about at
~~Harvard~~UCSD this fall.This fall in Bangalore there will be a meeting devoted to the Fargues-Fontaine curve, about which the organizers tell us: “This field will unravel in the coming years…”

- On the local geometric Langlands front, there’s something new from Dennis Gaitsgory. I’ve always been fascinated by the way BRST appears in this story.
- I’m told by experts that one of the best recent results in the Langlands program is this work, which doesn’t seem to have yet made it to the arXiv, but was explained in some detail in a blog post last year by Frank Calegari.

**Update:** Slides from the Chow Lectures are becoming available, see here. Remarkable in particular is Peter Scholze’s wonderful introductory lecture on Numbers and geometry, which includes something one sees all too rarely, a set of drawings showing the sort of pictures arithmetic geometers have in their minds for how to think about number theory geometrically.

**Update:** I just watched Vincent Lafforgue’s talk at the Breakthrough symposium. It included basically thanks to the CNRS for providing him a permanent position with freedom, a survey of Langlands, mainly talking about the topology of algebraic varieties, and comments on the ecological crisis. He says he’ll put up the slides on his website

http://vlafforg.perso.math.cnrs.fr/

He made one (to me) very striking claim, that the functoriality conjecture could be thought of as a quantization problem, how to pass from a classical system to a quantum system. Can an expert enlighten me on what exactly he was referring to here?

**Update**: Lafforgue’s slides are here. James Milne has provided a Google-translated version of the Langlands Russian article here, with the comment:

This may help readers gain some idea of what the manuscript is about until there is an official translation. Given that even native Russian speakers (not just google) have trouble understanding Langlands’s Russian, this would best be done by the author.

**Update**: Edward Frenkel gave a talk at the Langlands Abel Prize conference discussing the geometric theory and a bit about recent ideas of Langlands on this topic. He has written up some detailed notes on his take on this, available here.

**Update**: Videos of the CMI-20 talks are available, with Scholze’s here.