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.

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Lurie’s course is actually happening at UCSD, even though the website is Harvard.

Strangely, the live stream for “Is Time Travel Possible” indicates a starting time at 26 October. The website affirms that it’s Monday 5 November.

Itai, this not strange at all if the answer to the title question is affirmative.

Possibly naive question. With all these “great advances” in the Langlands Program, how far along are we towards the final goal (at least as outlined by Langlands, fields that make progress tend to open new questions I guess). 90%? 50%? 10%?

Rookie,

I’d be interested to hear a better answer from someone more expert, but my understanding is that that many of the original questions about the number field case raised by Langlands still remain open, with new insights needed to make progress on them. On the other hand, many of the Langlands program conjectures about number fields have been proved, with a high point the Taylor-Wiles proof of modularity, and recent progress that of the last item. I don’t think though there’s any sensible metric on these questions which would allow one to assign percentage completion numbers.

Since the original work of Langlands there have been huge extensions of his original ideas, providing a much larger vision unifying different areas of mathematics, often with proofs, not just conjectures. In particular the geometric theory as far as I know was not originally in the picture. For the current take from Langlands on that, all you need is to be able to read Russian and Turkish…

I was visiting the Middle East Technical University in Ankara, early September and was surprised to find out that Langlands knew Turkish (he had spent some time there early on in his career) and was scheduled to give the talk in Turkish. Too bad I had to be in London by the date of the talk.

Rookie, Peter. Just to remark that the Modularity Conjecture (proved by Taylor-Wiles and others), even if it can be read as part of the Langlands program, it was made before the Langlands conjectures, in fact part of the program may be considered a generalization of Modularity. So in a way, none of the principal conjectures that Langlands made have been proved in the number field case. I think the main result in that respect is still the work of Ngo, who proved the Fundamental Lemma.

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?

The slides for Vincent’s talk in Berkeley are up. Direct link:

http://vlafforg.perso.math.cnrs.fr/files/beamer-ted.pdf

Btw, he gave a pretty good talk at ICM2018. Even I managed to understand a few words here and there (mostly “is” and “the”) 🙂

Video here:

https://www.youtube.com/watch?v=lD8jE3NK8fw

Slide set:

http://vlafforg.perso.math.cnrs.fr/files/beamer-chtoucas-ICM-adelique.pdf

Hi Peter, off topic but I think you would be interested in Chris Fuchs’ review of Adam Becker’s book here: https://arxiv.org/pdf/1809.05147.pdf

The live-stream for today’s Abel conference is online at the indicated page, with the first talk scheduled to start at 9:30.