On Mochizuki’s “Report on discussions…”

(Edit 22nd October: I have updated my notes below).

In March 2018 Peter Scholze and Jacob Stix travelled to Japan to visit Shinichi Mochizuki to discuss with him his claimed proof of the abc conjecture. In documents released in September 2018, Scholze–Stix claimed the key Lemma~3.12 of Mochizuki’s third Inter-Universal Teichmüller Theory (IUTT) paper reduced to a trivial inequality under certain harmless simplifications, invalidating the claimed proof. Scholze apparently had concerns about the proof of Lemma 3.12 for some time; it has been reported that a number of other arithmetic geometers independently arrived at the same conclusion. Mochizuki agreed with the conclusion that under the given simplifications the result became trivial, but not that the simplifications were harmless. However, Scholze and Stix were not convinced by the arguments as to why their simplifications drastically altered the theory, and we stand at an impasse.

The documents released by both sides include two versions of a report by Scholze–Stix, titled Why abc is still a conjecture, each with an accompanying reply by Mochizuki, as well as a 41-page article, Report on discussions, held during the period March 15 — 20, 2018, concerning Inter-Universal Teichmüller Theory (IUTCH). This latter document is written in a style consistent with Mochizuki’s IUTT papers, and his other documents concerning IUTT. As such, it can be difficult (at least for me) to extract concrete and precisely-defined mathematical results that aren’t mere analogies or metaphors. Rather than analogies, one should strive to express the necessary ideas or objections in as precise terms as possible, and I argue that one should use category theory to clean up the parts of the arguments that are not actual number theory or arithmetic geometry.

I made some more detailed notes about this hereNEW !! (2018-10-22)

Edit (4th October): Ivan Fesenko has released a strongly pro-IUTT document (which you can find linked to from Peter Woit’s recent post on Scholze-Stix’s report) that claims

This oversimplification strikes as incorrect even people far from number theory, e.g. math physicists and categorists.

where “this oversimplification” refers to the paper of Scholze and Stix. I don’t know another category theorist who has made such comments, and I certainly don’t say Scholze and Stix are incorrect. It is just unclear how much effect their simplifications to Mochizuki’s work has had.