I have updated my notes on Mochizuki’s recent Report on discussions…, new copy available at my previous blog post. I’m more than happy to discuss these in the comments (and would welcome some feedback, especially from people who are more expert in arithmetic geometry).


Burritos for the hungry mathematician

This is just to provide a link to this literary, culinary and mathematic-y masterpiece

Ed Morehouse, Burritos for the Hungry Mathematician, 2015 (pdf)

Abstract: The advent of fast-casual Mexican-style dining establishments, such as Chipotle and Qdoba, has greatly improved the productivity of research mathematicians and theoretical computer scientists in recent years. Still, many experience confusion upon encountering burritos for the first time.
Numerous burrito tutorials (of varying quality) are to be found on the Internet. Some describe a burrito as the image of a crêpe under the action of the new-world functor. But such characterizations merely serve to reindex the confusion contravariantly. Others insist that the only way to really understand burritos is to eat many different kinds of burrito, until the common underlying concept becomes apparent.
It has been recently remarked by Yorgey [9] that a burrito can be regarded as an instance of a universally-understood concept, namely, that of monad. It is this characterization that we intend to explicate here. To wit, a burrito is just a strong monad in the symmetric monoidal category of food, what’s the problem?

Possibly useful documents on IUTT

Just a short post so that I can find these again. Chung Pang Mok has some notes on the first two IUTT papers that distill Mochizuki’s wordiness down to what seems like the essential minimum. They are pdf scans of hand-written notes (of high quality):

Together they are 86 pages, but this is shorter than the original, at least, with no ‘motivating’ paragraphs.

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.