A few items, all involving Peter Scholze in one way or another:<\/p>\n
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.<\/p><\/blockquote>\n<\/li>\n
- On the abc conjecture<\/a> 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<\/a>. To some extent what’s at issue is what was discussed by Scholze and others on my blog back in April 2020 (see here<\/a>). 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:
\nMeanwhile, 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\u2019s response to my emails.<\/p><\/blockquote>\n
He also clarifies that he has not yet finished a water-tight proof of abc along Mochizuki’s lines:<\/p>\n
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\u2019m 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. <\/p><\/blockquote>\n<\/li>\n<\/ul>\n
Update<\/strong>: In the comments someone points to this conference at MIT next week<\/a>, which will start off with a talk by Faltings on Mordell past and present<\/em>. That conference will be followed by this one<\/a> the following week.<\/p>\n
Update:<\/strong> Erica Klareich at Quanta has a very nice article about the recent proof of Geometric Langlands<\/a>. About the implications of this work, there’s a nice quote from Peter Scholze:<\/p>\n
\u201cI\u2019m definitely one of the people who are now trying to translate all this geometric Langlands stuff,\u201d Scholze said. With the rising sea having spilled over into thousands of pages of text, that is no easy matter. \u201cI\u2019m currently a few papers behind,\u201d Scholze said, \u201ctrying to read what they did in around 2010.\u201d<\/p><\/blockquote>\n
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<\/a>) that make up the full proof.<\/p>\n","protected":false},"excerpt":{"rendered":"
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 … Continue reading