Month: September 2018

On Mochizuki’s “Report on discussions…”

~ David Roberts

(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 here. NEW !! (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.

More ICM2018 videos online

~ David Roberts ~ 2 Comments

The Laudatio videos for the Fields and Chern medals, the Nevanlinna, Gauss and Leelavati prizes, the Emmy Noether and Abel lectures as well as the medal and prize winners’ lectures are now on YouTube! Handy links:

One thing I learned from the Laudatio for Peter Scholze is that Serre’s Deligne’s weight monodromy conjecture is, approximately, the p-adic analogue of the Riemann hypothesis part of the Weil conjectures (which Serre Deligne had proved for function fields over finite fields). I somehow missed this whenever I’d read statements that Scholze had proved (cases of) this conjecture. You can read Saito’s technical article on Scholze’s work on this here.

Update 20 September: And all the plenary talks are up! See this playlist.

A case of induction

~ David Roberts ~ 1 Comment

I have finally proved the induction step for the construction of a Fréchet manifold that is a limit of a large diagram of such manifolds, by a very carefully chosen iterated sequence of pullbacks of submersions. The base case requires one to construct by hand, in the notation of the picture below, the manifolds (I) for |I|=1,2,3.

Screen Shot 2018-09-10 at 5.12.03 pm

Here I is a finite set, with elements i,j,\ldots, indexing the sets in a closed cover \{V_i\}_{i\in I} of a compact manifold M, and the subsets J=I\setminus i etc index ‘partial subcovers’. That is, the closed sets corresponding to the elements of that subset, which do not themselves form a cover. Let V_J := \coprod_{j\in J} V_j. The manifolds (J) are a priori the diffeological spaces of functors \check{C}(V_J) \to X, where X is a fixed finite-dimensional Lie groupoid, and here \check{C}(V_J) is the diffeological Cech groupoid of V_J \to M. The aim here is to show that (I) = \mathrm{Hom}(\check{C}(V_I),X) is in fact a Fréchet manifold, by induction on the size of I. This result is Proposition 5 of my MATRIX Annals note with Raymond Vozzo, The smooth Hom-stack of an orbifold (publisher link, arXiv), with the proof there only stating

This diffeological space is what we show is a Fréchet manifold, by carefully writing the limit as an iterated pullback of diagrams involving maps that are guaranteed to be submersions by Proposition 3 and Theorem 4, and using the fact that X is appropriately coskeletal

“Carefully writing the limit” indeed! Past me was highly constrained by page limits…