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).

New Scientist has a story here. There’s still a lack of clarity about the status of the paper, whether it is “accepted” or “expected to be accepted”, see the exchange here.

**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 : βThe 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.β

**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.

BCnrd,

fair enough :-). Thanks for all your input both here and there.

David Roberts makes a good point in one way or another: it would have reflected much more positively on the mathematical community at large considering the media circus all about this work, if we had something to hold up publicly and say “yes well these manuscripts are all good and well, but part XYZ doesn’t make sense and must be addressed by the author”. Articles like “mathematicians tormented by proof” may have been prevented etc. (although I suppose one shouldn’t underestimate mass-media’s ability to misunderstand/misrepresent research-level mathematics)

The answer to that point is Remark 3.12.2 (ii) of newest version of IUT-III

As I understand it, Corollary 3.12 in IUT 3 (“main inequality”) is central to the whole proof of the abc conjecture. Previous sections of IUT (totalling hundreds of pages) just set up all the definitions and trivial proofs needed to prove Corollary 3.12, and when the corollary is proven abc conjecture is pretty trivial to prove based on that corollary. So what is needed to Mochizui to explain more in detail how one arrives in to the “main inequality”.

Is this correct understanding?

mJ:

Thank you for your comment. It came to my attention this morning from someone else via email that Remark 3.12.2 has been very much expanded since earlier versions (I don’t know when that change occurred), and that this Remark (not just its part (ii)) should address some aspects of how 3.12 follows from 3.11. The wider awareness about this due to the discussion on Frank Calegari’s blog and this one has helped in this direction. I immediately brought this to the attention of several other people who have looked a lot into the IUT papers; hopefully it will clarify things. But the coming days are a time of travel and vacation for many people, so don’t hold your breath. π

All,

This blog entry isn’t the place for trying to resolve the mathematical questions raised about the proof. I’m completely incompetent to moderate such a discussion and the set of people interested in participating in this on a blog may have zero intersection with the small set of people competent to engage productively in it. Mathoverflow is a site more set up for this sort of thing, it’s an interesting question whether it would work over there.

For settling mathematical issues like this one, there really are very few people in the world with the necessary expertise and talent, capable of telling what claims are unproblematic, identifying where possible problems lie. What is supposed to happen is discussion between the author and this small group of experts, until they understand the proof and the problems they find are resolved. This has failed to happen in this case, for a complicated set of reasons, and anyone who feels like it can try and blame participants X, Y, and Z for not doing more. I think the point Brian Conrad is trying to make is that the math community puts the main responsibility in a case like this on the refereeing process: it is the job of editors to pick referees up to the task (an extremely difficult and important one in this case), and it is the job of those referees to do everything they can to ensure proofs are solid and written in an understandable way. For good reasons this process is conventionally done confidentially. Unfortunately, in this case there is evidence of a failed refereeing process (claims the papers have been refereed, while an expert is pointing to problems that have not been addressed, and many experts have trouble with readability of the papers), and confidentiality makes it impossible for other than a small number of people to know what went wrong.

Addendum: I just saw the latest comment indicating that maybe the comment from “mJ” is highly relevant, and possibly it is blog discussions that will end up having a role in moving this process forward. If so, that’s a very unusual way (in comparison to private discussions between experts, the author, and referees) for mathematics to get done.

Ivan Fesenko has replied to Peter Scholze. It’s, uh, interesting.

https://www.facebook.com/ivan.fesenko.37/posts/1128469910617882

Recently mentioned by @math_jin on twitter is a series of talks at the end of this month at the Southern University of Science and Technology in Shenzhen http://math.sustc.edu.cn/event/10808.html to be given by Fucheng Tan (who apparently works at RIMS http://www.kurims.kyoto-u.ac.jp/en/list/tan.html )

I am no mathematician but active scientific PI and I follow this case since Castelvecchis first Nature article in utter fascination. Please forgive me to express my uneducated 5c.

I feel the situation is slowly approaching a state were expert statements get more and more explicit. In both directions and notably with some quite renowned critics. Lets suppose IUTT I-IV are complete and correct and prove abc. Then already now there is the question: Will Mochizuki ever/in our life times get full credits for his discovery? I fear due to the growing critics(!) it will not be possible anymore. I think the case just transgressed a point of no return for the maths community.

Merle Aucoin,

I think your perception of “growing criticism” misses what is going on here. What experts are now saying publicly is not different than what they have been saying privately for a long time now. What changed things was the news (the accuracy of which is still unclear) that the journal of Mochizuki’s institution, of which he is editor in chief, was about to publish the IUT proof, claiming it as properly refereed and a correct proof. This news made experts who felt that the proof had not been understood and checked to their satisfaction feel that they needed to say something in public, even though this is, for good reason, not the kind of discussion usually held publicly.

Mochizuki is a well-respected mathematician, and skepticism about this proof is not something personal. It’s just a fact that the usual way in which understanding of a proof transmits from an author to the rest of the math community has not happened here, and experts are just making clear that fact. There is a lot of attention now being paid to what has gone wrong, including more focus on a specific part of the proof that experts haven’t been able to follow. If this gets clarified and experts start to understand better the proof and be able to check that it works, Mochizuki will certainly get credit for proving abc.

Dear Peter, I see. Thank you for clarifying my misperceptions. I like also to draw your attention to the Wikipedia article on Shinichi Mochizuki, it seems there have been substantial changes and additions since the end of last year. In my eyes most likely from a very close proximity of SM if not from himself. Especially the section on IUTT seems notable, since it contains an attempt to explain the gist of it to even laymen:

https://en.wikipedia.org/wiki/Shinichi_Mochizuki .

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