The last couple months I’ve heard reports from several people claiming that arithmetic geometers Peter Scholze and Jakob Stix had identified a serious problem with Mochizuki’s claimed proof of the abc conjecture. These reports indicated that Scholze and Stix had traveled to Kyoto to discuss this with Mochizuki, and that they were writing a manuscript, to appear sometime this summer. It seemed best then to not publicize this here, better to give Mochizuki, Scholze and Stix the time to sort out the mathematics and wait for them to have something to say publicly.
Today though I saw that Ivan Fesenko has put out a document entitled Remarks on Aspects of Modern Pioneering Mathematical Research. It refers in footnote 18 to:
two recent texts by Sh. Mochizuki, ‘Report on discussions, held during the period March 15–20, 2018, concerning inter-universal Teichmüller theory (IUTCH)’and ‘Comments on the manuscript by Scholze–Stix concerning inter-universal Teichmüller theory (IUTCH)’, July 2018
I haven’t seen these two texts, or the Scholze-Stix manuscript. What I have heard about them is that Scholze-Stix identify what they see as a specific, serious flaw in the proof, and that Mochizuki denies that this is a problem or that his manuscript needs to be revised. Presumably, after the two sides try and sort this out amongst themselves, at some point we’ll see something publicly available describing the details of their disagreement.
Fesenko’s document has a lot of unpleasant things to say about those who have written anything at all skeptical concerning Mochizuki’s claimed proof, mostly without naming names. He refers to journalists and “US bloggers” as having produced “ignorant absurd articles and posts”, presumably has someone other than me in mind since the information posted here about this I believe has been quite accurate and of reasonably high quality. The one negative reference to identified mathematicians is in the text with footnote 18 pointing to Scholze and Stix, which says:
Several researchers, who could have become potential learners of IUT and then progressed to become experts, declined invitations to participate in the IUT workshops. Some, affected by negative emotions, broke professional rules of conduct and made public their ignorant and sometimes intolerant opinions. Tellingly, the only questions produced were shallow and misplaced and they were communicated only after several years of requests to do so.
Peter Scholze is by far the most talented arithmetic geometer of his generation, a sure thing to receive a Fields Medal at the ICM in a couple weeks. That his questions about Mochizuki’s proof were “shallow” seems highly unlikely, to me at least.
Much of Fesenko’s article concerns the question of whether contemporary mathematical research is too narrow and unambitious, devoted to minor improvements and producing lots of publications. This is a serious issue, one though where other fields than arithmetic geometry (e.g. fundamental physics) are in a much worse state. Fesenko tries to make the difficulties mathematicians have had with Mochizuki’s claims about his IUT research an exemplar of this problem, but this seems to me misguided. There are quite good reasons for why experts have been skeptical about IUT and the supposed abc proof, reasons which will be conclusively vindicated if Scholze and Stix turn out to be right. Ironically, an excellent example of the kind of fundamental breakthrough that Fesenko is asking for is Scholze’s own ground-breaking work over the past few years.