A few items, all involving Peter Scholze in one way or another:

- A seminar in Bonn on Scholze’s geometrization of real local Langlands is finishing up next week. This is working out details of ideas that Scholze presented at the IAS Emmy Noether lectures back in March. Until recently video of those lectures was all that was available (see here, here and here), but since April there’s also this overview of the Bonn Seminar, and now Scholze has made available a draft version of a paper on the subject.
- In three weeks there will be a conference in Bonn in honor of Faltings’ 70th birthday. Scholze’s planned talk is entitled “Are the real numbers perfectoid?”, with abstract

Rodriguez Camargo’s analytic de Rham stacks play a key role in the geometrization of “locally analytic” local Langlands both over the real and p-adic numbers. In both settings, one also uses a notion of perfectoid algebras, with the critical property being that “perfectoidization is adjoint to passing to analytic de Rham stacks”. This suggests a “global” definition of perfectoid rings. We will explain this definition, and present some partial results on the relation to the established p-adic notion. Two natural open questions are whether tilting works in this setting; and what perfectoid algebras over the real numbers look like.

- On the abc conjecture front, Kirti Joshi has a new document explaining his view of The status of the Scholze-Stix Report and an analysis of the Mochizuki-Scholze-Stix Controversy. To some extent what’s at issue is what was discussed by Scholze and others on my blog back in April 2020 (see here). Joshi is trying to make an argument that there is a way around the problem being discussed there, but I don’t think he has so far managed to convince others of his argument (Mochizuki refuses to even discuss with him). He ends with the following:

Meanwhile, Scholze and I are having a respectful and professional conversation (on going) as I work to clarify his questions; while I continue to wait for Mochizuki’s response to my emails.

He also clarifies that he has not yet finished a water-tight proof of abc along Mochizuki’s lines:

My position on whether or not Mochizuki has proved the abc-conjecture is still open (as my preprint [Joshi, 2024a] still remains under consideration). In other words, I’m currently neutral on the matter of the abc-conjecture. However, I continue to work on [Joshi, 2024b,a] to tie up all the loose ends.

**Update**: In the comments someone points to this conference at MIT next week, which will start off with a talk by Faltings on *Mordell past and present*. That conference will be followed by this one the following week.

**Update:** Erica Klareich at Quanta has a very nice article about the recent proof of Geometric Langlands. About the implications of this work, there’s a nice quote from Peter Scholze:

“I’m definitely one of the people who are now trying to translate all this geometric Langlands stuff,” Scholze said. With the rising sea having spilled over into thousands of pages of text, that is no easy matter. “I’m currently a few papers behind,” Scholze said, “trying to read what they did in around 2010.”

He’s not the only one struggling to understand what was known before this proof, and daunted at the prospect of trying to read the 800 pages of five papers (see here) that make up the full proof.

On the Mordell conjecture, closer to home: https://mordell.org/

Can someone clarify, for a math literate audience, what Scholze’s real local Langlands is doing?

You missed the final sentence of Scholze’s abstract: ” Two natural open questions are whether tilting works in this setting; and what perfectoid algebras over the real numbers look like.”

Peter see this…and maybe you can write something…

https://mathstodon.xyz/@tao/112557248794707738

Boris,

I fear that topic is one that I know very little about, and certainly couldn’t improve on Tao’s explanation. He points to talks at the Sarnak conference, for which videos should now be available.

Anon,

I’ll try to say something, but hoping one or more of my readers is willing to do a better job…

Real local langlands classifies representations of real Lie groups. For the simplest example, think of SL(2,R) and its discrete and principal series representations. This is done in terms of “Langlands parameters”, for which Scholze provides a new geometric interpretation, in terms of geometric Langlands, on the twistor P1.

This is the same classification of representations as before, but a new geometric point of view. Most dramatically, this is done with new methods replacing the usual geometry and analysis, methods that have developed out of work on the p-adic case. This provides some very new and different ways of looking at the real case, and a unified perspective on what is happening at the p-adic and real places. A speculative possibility is that this will lead to new global results (i.e. for Q, not Q_p and R), with Scholze’s upcoming talk suggesting “a “global” definition of perfectoid rings.”

Indeed, one can see the videos at https://www.youtube.com/watch?v=dIe5hqTuB4k and https://www.youtube.com/watch?v=diASDVdMaN0

The first talk (by James Maynard) goes over the statement of the result, prior work, applications, and a brief summary of the proof. The second talk (by Larry Guth) explains the ideas behind the proof. I would encourage everyone who hears that description and thinks they will understand the first talk but not the second to try watching it – it’s a masterpiece of finding the key ideas of a complicated proof and explaining them in simple terms.

There’s something weird in the poster of “conference in Bonn in honor of Faltings’ 70th birthday”. I see Peter Sarnak name as one of the speakers, but he’s not listed as the speaker in the abstract of the conference

Anon,

A likely explanation is that he was planning to attend the conference, so on the poster, but at some point changed plans and won’t be there, so, not on the program.

The conference for Faltings’ 70th birthday has updated their streaming policy, see main page at https://www.mpim-bonn.mpg.de/faltings70

“Due to the great interest in this conference, we will stream most of the talks live via Zoom. The program indicates which talks will be streamed. The Zoom link will follow shortly before the conference. The talks will not be uploaded on the website later.”

Excellent!

By the way, the Mordell Conference https://mordell.org site has been updated with slides of the talks, including that of Faltings (1st talk), and a talk on possible formalization by Michael Stoll (last talk), both fascinating in different ways.

Also see the photo of the participants, Faltings is near the top right….

@Oisin McGuinness

actually, not posting the videos afterwards is less good than just live-streaming. Far fewer people will get the watch the talks as a result.

@David+Roberts

“Far fewer people will get the watch the talks as a result.”

Agreed, but when I asked the organizers initially, some weeks back, they were not going to have anything, so what they are doing is an improvement.

It is a big shame (similarly for the recent Mordell Conjecture Conference) that recording is not standard anymore for many of these conferences. Some talks are clearly of possible historic import. Some foundational support is needed to support more recording! By contrast the IAS does a great job.

Are you listening Simons Foundation???

I might have missed, but was there a post on the proof of Geometric Langlands? I just read the Quanta article.

Sandeep,

I haven’t written about the Gaitsgory/Raskin et al proof, haven’t followed the technicalities. Thanks for mentioning the Quanta article, I just read it and it’s quite good. I’ll add a link to that.