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.
- 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.
