I’ve seen reports today (see here and here) that indicate that Mochizui’s IUT papers, which are supposed to contain a proof of the abc conjecture, have been accepted by the journal Publications of the RIMS. Some of the sources for this are in Japanese (e.g. this and this) and Google Translate has its limitations, so perhaps Japanese speaking readers can let us know if this is a misunderstanding.
If this is true, I think we’ll be seeing something historically unparalleled in mathematics: a claim by a well-respected journal that they have vetted the proof of an extremely well-known conjecture, while most experts in the field who have looked into this have been unable to understand the proof. For background on this story, see my last long blog posting about this (and an earlier one here).
What follows is my very much non-expert understanding of what the current situation of this proof is. It seems likely that there will soon be more stories in the press, and I hope we’ll be hearing from those who best understand the mathematics.
The papers at issue are Inter-universal Teichmuller Theory I, II, III, IV, available in preprint form since September 2012 (I blogged about them first here). Evidently they were submitted to the journal around that time, and it has taken over 5 years to referee them. During this 5 year period Mochizuki has logged the changes he has made to the papers here. Mochizuki has written survey articles here and here, and Go Yamashita has written up his own version of the proof, a 400 page document that is available here.
My understanding is that the crucial result needed for abc is the inequality in Corollary 3.12 of IUT III, which is a corollary of Theorem 3.11, the statement of which covers five and a half pages. The proof of Theorem 3.11 essentially just says “The various assertions of Theorem 3.11 follow immediately from the definitions and the references quoted in the statements of these assertions”. In Yamashita’s version, this is Theorem 13.12, listed as the “main theorem” of IUT. There its statement takes 6 pages and the proof, in toto, is “Theorem follows from the definitions.” Anyone trying to understand Mochizuki’s proof thus needs to make their way through either 350 pages of Yamashita’s version, or IUT I, IUT II and the first 125 pages of IUT III (a total of nearly 500 pages). In addition, Yamashita explains that the IUT papers are mostly “trivial”, what they do is interpret and combine results from two preparatory papers (this one from 2008, and this one from 2015, last of a three part series.):
in summary, it seems to the author that, if one ignores the delicate considerations that occur in the course of interpreting and combining the main results of the preparatory papers, together with the ideas and insights that underlie the theory of these preparatory papers, then, in some sense, the only nontrivial mathematical ingredient in inter-universal Teichmueller theory is the classical result [pGC], which was already known in the last century!
Looking at these documents, the daunting task facing experts trying to understand and check this proof is quite clear. I don’t know of any other sources where details are written down (there are two survey articles in Japanese by Yuichiro Hoshi available here).
As far as I know, the current situation of understanding of the proof has not changed significantly since last year, with this seminar in Nottingham the only event bringing people together for talks on the subject. A small number of those close to Mochizuki claim to understand the proof, but they have had little success in explaining their understanding to others. The usual mechanisms by which understanding of new ideas in mathematics gets transmitted to others seem to have failed completely in this case.
The news that the papers have gone through a confidential refereeing process I think does nothing at all to change this situation (and the fact that it is being published in a journal whose editor-in-chief is Mochizuki himself doesn’t help). Until there are either mathematicians who both understand the proof and are able to explain it to others, or a more accessible written version of the proof, I don’t think this proof will be accepted by the larger math community. Those designing rules for the Millennium prizes (abc could easily have been chosen as on the prize list) faced this question of what it takes to be sure a proof is correct. You can read their rules here. A journal publication just starts the process. The next step is a waiting period, such that the proof must “have general acceptance in the mathematics community two years after” publication. Only then does a prize committee take up the question. Unfortunately I think we’re still a long ways from meeting the “general acceptance” criterion in this case.
One problem with following this story for most of us is the extent to which relevant information is sometimes only available in Japanese. For instance, it appears that Mochizuki has been maintaining a diary/blog in Japanese, available here. Perhaps those who read the language can help inform the rest of us about this Japanese-only material. As usual, comments from those well-informed about the topic are welcome, comments from those who want to discuss/argue about issues they’re not well-informed about are discouraged.
Update: Frank Calegari has a long blog post about this here, which I think reflects accurately the point of view of most experts (some of whom chime in at his comment section).
Update: It occurred to me that I hadn’t linked here to the best source for anyone trying to appreciate why experts are having trouble understanding this material, Brian Conrad’s 2015 report on the Oxford IUT workshop.
Update: Curiouser and curiouser. Davide Castelvecchi of Nature writes here in a comment:
Got an email from the journal PRIMS : “The papers of Prof. Motizuki on inter-universal Teichmuller theory have not yet been accepted in a journal, and so we are sorry but RIMS have no comment on it.”
Update: Peter Scholze has posted a comment on Frank Calegari’s blog, agreeing that the Mochizuki papers do not yet provide a proof of abc. In addition, he identifies a particular point in the proof of Conjecture 3.12 of IUT III where he is “entirely unable to follow the logic”, despite having asked other experts about it. Others have told him either that they don’t understand this either, or if they do claim to understand it, have been unable to explain it/unwilling to acknowledge that more explanation is necessary. Interestingly, he notes that he has no problem with the many proofs listed as “follows trivially from the definitions” since the needed arguments are trivial. It is in the proof of Corollary 3.12, which is non-trivial and supposedly given in detail, that he identifies a potential problem.
Update: Ivan Fesenko has posted on Facebook an email to Peter Scholze complaining about his criticism of the Mochizuki proof. I suppose this makes clear why the refereeing process for dealing with evaluating a paper and its arguments is usually a confidential one.