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In August 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known from the beginning that it would take experts months to understand his work enough to be able to verify the proof. Are there any updates on the validity of this proof?

nfdc23
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Ovi
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    Here is the last thing I've seen: notes by bcnrd on the recent workshop at oxford on IUTT --- http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/ – Vidit Nanda Feb 25 '16 at 05:01
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    Note that there are a small group of people, no more than three or so, who say they understand the papers and think them correct. Their best efforts to explain the theory, which is what people really want to know about, are described in the blog post Vidit links to. Let's say that there is another workshop coming later this year where people are hopeful of more progress. – David Roberts Feb 25 '16 at 08:59
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    @DavidRoberts: is this upcoming workshop publicly announced yet, and if so, can you point us to the announcement? – Peter LeFanu Lumsdaine Feb 25 '16 at 09:04
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    https://www.maths.nottingham.ac.uk/personal/ibf/files/kyoto.iut.html – Sylvain JULIEN Feb 25 '16 at 09:24
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    I've received another conference announcement: "We are happy to announce the conference Kummer classes and Anabelian Geometry, which will take place at the University of Vermont on September 10-11, 2016.

    The conference will consist in approximately eight talks (a full day on Saturday and a half day on Sunday) introducing concepts involved in Mochizuki’s work on the ABC conjecture. For more details and to register, please visit our website http://www.uvm.edu/~tdupuy/anabelian.html."

     – Gerry Myerson Jul 21 '16 at 00:33
  • (continued from previous comment) Sincerely, Taylor Dupuy, University of Vermont; Carl Pomerance, Dartmouth; Christelle Vincent, University of Vermont; John Voight, Dartmouth. – Gerry Myerson Jul 21 '16 at 00:34
  • @GerryMyerson Thank you for the update. – Ovi Jul 21 '16 at 00:58
  • Related: https://mathoverflow.net/questions/280043/on-a-proof-of-abc-conjecture-after-mochizuki – Watson Sep 12 '17 at 19:49
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    Mainichi Shinbun reports that Mochizuki's proof has been accepted for a special issue of "Publications of RIMS" (PRIMS) by a group of independent referees who have taken 8 years to arrive at their verdict that it is correct. https://mainichi.jp/articles/20200403/k00/00m/040/093000c – Daniel Moskovich Apr 03 '20 at 05:45
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    Mochizuki's papers are now published: https://www.ems-ph.org/journals/show_issue.php?issn=0034-5318&vol=57&iss=1 – Sam Hopkins Mar 12 '21 at 03:48

7 Answers7

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September 2018: There has been a back-and-forth in 2018 between Shinichi Mochizuki and Yuichiro Hoshi (MoHo) in Kyoto, and Peter Scholze and Jakob Stix (ScSt) in Germany, with ScSt spending a week in Kyoto in March 2018 to confer with MoHo.

ScSt have released a report saying they believe there is a gap in the proof of Corollary 3.12 in IUTT-3, and Mochizuki has posted a reply saying that ScSt are missing some understanding of the background theory. It sounds like ScSt are still skeptical, and at minimum further clarification is needed about proving this corollary.

Suvrit
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none
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    The article by Erica Klarreich was really good! – Bjørn Kjos-Hanssen Sep 21 '18 at 00:26
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    The only criticism of M that I can understand that has real bearing on the ScSt report is that he claims (Comment (Lin) in http://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-08.pdf) they assume certain maps between 1-dimensional ordered vector spaces over R (the commutative hexagon at the end) are linear, when they are not. I feel it would be most useful if these maps could be transparently defined so we can see where the problem lies. – David Roberts Sep 21 '18 at 05:26
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    @DavidRoberts: See this paper for a latest update. –  Sep 21 '18 at 13:59
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    A Corollary with a 9-page proof? Is that a record? – bof Sep 21 '18 at 23:51
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    @user170039 like I said, "that I understand that has real bearing". There are lots of things in that 40-odd page report that are irrelevancies from a structural, isomorphism-invariant point of view. Demanding specific symbols for objects in order to make the proof work is not a mathematical concern, but a psychological one. – David Roberts Sep 22 '18 at 00:43
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    To a complete outsider, the thinly veiled insults Mochizuki addresses to Scholze and Stix in that response are surprising, to say the least. – PseudoNeo Sep 22 '18 at 15:17
  • These results render the 2017-12 Wordpress post The ABC conjecture has (still) not been proved by Galois representations a little out of date, but it is still interesting, particularly the replies of 2017-12-21 by PS and 2017-12-22 by BCnrd. – PJTraill Oct 10 '18 at 13:57
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    See also "Comments on Mochizuki’s 2018 Report" by David Roberts : https://thehighergeometer.files.wordpress.com/2018/09/mochizuki_final1.pdf – jjcale Oct 17 '18 at 22:32
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    There's a RIMS workshop coming up in May 2020 which is one of four workshops of a special RIMS year on "Expanding Horizons of Inter-universal Teichmüller Theory." Notably, Jakob Stix is an invited speaker. – Timothy Chow Nov 22 '19 at 18:29
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    @TimothyChow I note that Stix's name has since been removed from the list of invited speakers on that workshop page. It was still there on Dec 5 2019, but was removed some time between then and Feb 8 2020. – estan Apr 04 '20 at 21:06
  • @estan Also, the workshop date seems to have been moved to September. – Emil Jeřábek Apr 07 '20 at 11:45
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In January, Vesselin Dimitrov posted to the arXiv a preprint showing that Mochizuki's work, if correct, would be effective. While this doesn't validate Mochizuki's work it does do a few things:

  1. It shows that people are understanding more of the proof.

  2. It gives another avenue through which to check whether Mochizuki's work is invalid.

  3. It makes Mochizuki's work that much more important.

Pace Nielsen
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    Dimitrov's paper treats Mochizuki's IUT ideas and results as a black box, replacing the appeal to a proof in one of Mochizuki's much earlier pre-IUT papers (reference [8] in Dimitrov's paper), so unfortunately it doesn't involve #1 or #2 (in terms of the core material which has not been disseminating; the material in [8] hasn't been related to the difficulties that have arisen). But it very much contributes in the direction of #3, which is of course a very good thing! – nfdc23 Feb 25 '16 at 15:47
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    @nfdc23 I think you misunderstood my comment. Regarding #2, since (at least in principle) Mochizuki's work is now effective, it may be possible to find counter-examples to some of his claims. Of course, one of the criticisms I've seen of the work is the lack of motivating examples, so this might just be a theoretical rather than practical consideration. – Pace Nielsen Feb 25 '16 at 20:04
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    Thanks for clarifying the intent of #2. My understanding from discussing this stuff with Dimitrov is that making explicit the "effective" constants he gets is a daunting task, and that most likely such explicit constants will not be practical (i.e., not suitable for testing against examples). – nfdc23 Feb 26 '16 at 05:39
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    That has been my experience when making things effective as well. Of course, if Mochizuki's work does check out, I can imagine lots of people will be very interested in accomplishing that "daunting task"! – Pace Nielsen Feb 26 '16 at 17:18
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    Does an effective abc conjecture give an effective Mordell conjecture? –  Dec 18 '17 at 05:08
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    @TKe : The short answer is yes. See https://mathoverflow.net/a/58890/3106 especially Silverman's comments. – Timothy Chow Nov 23 '18 at 00:45
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Today (3 April 2020) his papers have been accepted for publication on RIMS journal.

https://www.nature.com/articles/d41586-020-00998-2

manifold
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    A new version of IUT 2 and IUT 3 was posted to Mochizuki's website on March 22, and IUT 1 on March 10th. Since according to the article, the paper was accepted on February 5th, I would guess these are the final accepted versions. – Will Sawin Apr 03 '20 at 21:26
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    What a disgrace! – Lucia Apr 03 '20 at 22:52
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    @Lucia: According to the link to Nature: 'mathematicians often publish papers in journals where they are editors. As long as the authors recuse themselves from the peer-review process ... and Mochizuki had recused himself from the review process, and had not attended any of the editorial board meetings about the paper. The journal has previously published papers from other members of the journals’ editorial board ... – Mozibur Ullah Apr 04 '20 at 01:25
  • ... [moreover] In the world of mathematics, a journal’s seal of approval is often not the end of the peer-review process. An important result only truly becomes an accepted theorem after the community has reached a consensus that it is correct, and this can go on for years after a paper’s official publication.' It looks like this will be case for this 'proof'. – Mozibur Ullah Apr 04 '20 at 01:26
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    @Lucia: I'm not claiming that I can understand Mochizukis proof, nor am I an expert in number theory but I do understand English and what the article in Nature is saying is plain enough. – Mozibur Ullah Apr 04 '20 at 01:32
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    @MoziburUllah: Journals don't publish questionable papers and hope that the community sorts itself out. At least no decent journal would willingly choose to do that. Something seriously wrong has happened here, and I can't imagine any editorial board being happy with this. This is not to say that journals won't make mistakes -- that'll of course happen -- just that no journal would/should walk into a situation like this. – Lucia Apr 04 '20 at 01:33
  • @Lucia: I don't think the paper is questionable in any easy or simple way. That the paper is being published after eight years shows just how complex and difficult certain parts of modern mathematics has become. Moreover, the editors and referees are in a very difficult position themselves when the mathematical community as a whole can't come to a consensus. – Mozibur Ullah Apr 04 '20 at 01:46
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    @MoziburUllah: You don't really know what you're talking about here. No one in the number theory community believes this result -- apart from acolytes of Mochizuki in Nottingham and Japan. And I don't think this sorry state of affairs has been seen in any of the other breakthroughs in mathematics that have happened over the last 20 years -- many of them quite complicated. – Lucia Apr 04 '20 at 02:00
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    @Lucia: Have you heard about String Theory? How many thousands of man-hours and hundreds of doctorates have been devoted to that theory without there being a shred of real evidence of its truth, physically speaking? – Mozibur Ullah Apr 04 '20 at 03:15
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    @MoziburUllah: No one wants to be like string theory! And math doesn't have to go down that path. Anyway, I'm done with responding. – Lucia Apr 04 '20 at 03:25
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    @Lucia: I happen to think that string theory is a reasonable idea - I'm just not interested in the hype; the point I'm making is that at the frontiers of science its difficult to make judgement calls. – Mozibur Ullah Apr 04 '20 at 03:31
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    @WillSawin do the new versions of the papers at least attempt to fix the problems called out in the Scholze/Stix report? MoziburUllah, the string theory comparison is bogus because physics papers are allowed some mathematical unsoundness but math papers are not. A mathematically unsound physics paper can still be interesting if it somehow describes nature in a new way, like QFT did. But math like this isn't about describing nature, so if it's unsound it's just not interesting. – none Apr 04 '20 at 03:39
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    @none I'm not an expert and I only looked very quickly, but my impression was the new information was less than contained in Mochizuki's response to Scholze/Stix. As you (hopefully) already know, there was nothing that can be described as an attempt to fix the problems in that response, but rather a a series of arguements-by-analogy that the problem does not exist. – Will Sawin Apr 04 '20 at 12:04
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    @none: not at all an expert either, but back in late october (2019) the very long remark 3.9.5 (over 17 pages) was added to IUTpartIII and refered to at several places. It seems to be meant to adress details of the proof of corollary 3.12. Could this be what convinced the referee? – Thomas Sauvaget Apr 06 '20 at 10:55
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    On Woit's blog, there is a very interesting comment by Peter Scholze that he has made in the light of the current press coverage. – Timothy Chow Apr 06 '20 at 14:40
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    The above link doesn't work properly now that the comments have spilled over to a second page. Use this link instead to go directly to Scholze's initial comment. – Timothy Chow Apr 10 '20 at 14:17
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    The situation is pretty clear. Within Japan the abc conjecture is now a Theorem. Outside of Japan it is still a conjecture. So if you want to publish counterexamples this is only possible in non-Japanese Journals. Corollaries on the other hand can only appear in Japanese Journals. I am glad this entire matter is finally settled. – santker heboln Apr 15 '20 at 22:41
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    Maybe this preprint of Taylor Dupuy and Anton Hilado is interesting : "THE STATEMENT OF MOCHIZUKI’S COROLLARY 3.12, INITIAL THETA DATA, AND THE FIRST TWO INDETERMINACIES", https://arxiv.org/pdf/2004.13228.pdf . – jjcale Apr 29 '20 at 18:52
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I think that not much has changed since 2012, in terms of general consensus within the mathematical community.

There's some very interesting opinions and notes on the topic (see for example the one by Brian Conrad mentioned in the comments above, or this one by Ivan Fesenko), but not a lot of people seem to have a strong opinion yet as to whether IUT implies Szpiro's conjecture or not.

On the other hand, Mochizuki has two reports on the progress of the verification process, which have a lot of information that you might find helpful.

Myshkin
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What's interesting with the Scholze-Stix rebuttal is that (staring from mathematically a long way away) there is a reasonable proof strategy which would fit the Scholze-Stix rebuttal and Mochizuki rejoinder well. The obvious objection to it being right is: well, Scholze-Stix would have seen it, and even if somehow not Mochizuki would have explained it, right? But maybe it is worth posting here, in order that someone explains why it is not what is going on and not correct. So here goes...

Very caricatured, the proof of Mochizuki's Corollary 3.12 is supposed to give two different (complicated) transforms from a set $S$ to a set $T$, along with inequalities regarding an associated parameter $f(t)$, and what comes out for a given $s\in S$ is the inequality $c(x)f(t)\ge d(x)f(t')$. Here $x$ is the arithmetic information which Mochizuki wants to get some control of, and $c$ and $d$ are (`simple') functions which depend on the transforms chosen but not on the $s\in S$.

The obvious way to get something useful out of this is to ask that $t=t'$; this is insisting that the Scholze-Stix diagram is commutative. Then you can cancel the $f(t)$ factor and get an inequality involving $x$. This looks like it's what Mochizuki wants to do (he says the images are the same). One way to get $t=t'$ is to choose a couple of spaces equal (this choice fixes the transforms).

Scholze and Stix find that in this case you get a trivial inequality, and claim that anything else which gets $t=t'$ is likely to give the same result. Mochizuki agrees, and says that the reason is that in this case his transforms don't do anything interesting (he also says the Scholze-Stix choice is essentially the only way to get $t=t'$). This is consistent with Scholze-Stix saying that Mochizuki's use of anabelian geometry doesn't seem to be doing anything.

The other two things Scholze and Stix simplify are `polymorphism' to morphism, which in this caricature means they consider one $s\in S$ as above, where Mochizuki wants to consider all $s\in S$ (polymorphism). And averaging over the result, which is meaningless if you have only one morphism.

But one can also work as follows. Consider all $s\in S$, and you get a collection of inequalities $c(x)f(t)\ge d(x)f(t')$, where $t$ and $t'$ are images of $s$ under Mochizuki's two transforms. If as $s$ ranges over $S$, you get the same collection of elements appearing as $t$ and as $t'$, just permuted, then this is exactly what Mochizuki means by saying the polymorphism images are the same (as sets, even though the individual morphism images aren't the same). In this case, when you average the collection of inequalities, as Mochizuki wants to do, you get an inequality which is useful: the average of the $f(t)$ equals the average of the $f(t')$, because they're the same sum permuted, so you can cancel it and get $c(x)\ge d(x)$, this time (Mochizuki claims) with different $c$ and $d$ and hence meaningful content.

This is entirely consistent with Scholze-Stix saying that polymorphisms and averages don't appear to play a role - in this caricature, they would be playing no role in 400+ pages, except exactly at this point.

user36212
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    Has someone considered sharing this answer with Scholze/Stix? – Sam Hopkins Nov 14 '18 at 20:11
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    I was hoping someone expert would point out quickly why it is wrong..! – user36212 Nov 15 '18 at 22:50
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    Having asked an expert, it seems that at best Mochizuki's proof isn't clear enough to decide whether the above is part of the strategy. More likely, the above is simply nonsense (or, a coincidental resemblance to a proof strategy that's not what's intended). – user36212 Nov 19 '18 at 21:03
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I just read on Google+ that the paper will be published in 2018 in a Japanese journal whose editor-in-chief is Mochizuki himself. See https://plus.google.com/+johncbaez999/posts/DWtbKSG9BWD

Sylvain JULIEN
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New versions of Mochizuki's 4 IUTT papers (dated 2017-12-14) are available at Papers of Shinichi Mochizuki. Part III includes an expanded Remark 3.12.2 (starting at manuscript page 143) regarding Corollary 3.12 (Log-volume Estimates for $\Theta$-Pilot Objects) (starting at page 133), which has been troubling some experts.

See Peter Woit's blog post Latest on abc and comments thereto.

Progress?

Update

For the record, from the revision history maintained by Mochizuki here, major revisions to Remark 3.12.2 over the last year occurred on:

2017-06-30: "Rewrote 3.12.2 (ii)"

2017-08-18: "Remark 3.12.2, (ii), slightly modified the items "(e^{itw})" and ("e^{toy})" and added items "(f^{itw})" and "(f^{toy})"

Update (2018-09-21)

Back in March 2018, Peter Scholze and Jakob Stix met with Mochizuki and his associate Hoshi for a week of abc discussions. Scholze and Stix concluded that it remains a conjecture and identified specific issues, but Mochizuki sees no problems with the proof. Mochizuki has posted Scholze and Stix' report and his responses on his website ("discussions conducted at RIMS in March 2018 concerning IUTeich"). See also coverage by Peter Woit (where I learned of this development).

Art Brown
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    I believe the updated versions might be related to the fact that those papers are to be published (see Sylvain's answer above), given that the dates almost coincide. – Wojowu Dec 24 '17 at 19:09
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    @Wojowu apparently not... – David Roberts Dec 24 '17 at 22:58
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    No, not progress. As your update notes, those changes happened months ago. The concerns remain. – nfdc23 Jan 08 '18 at 04:38
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    https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ – Deane Yang Jan 08 '18 at 04:44
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    @nfdc23 I was under the impression, given the comment sections of the blogs, that those june/august changes had not been noticed by at least some of the experts, and do provide some insight (I do not have background in this area, but at least the toy model is pretty clearly explaining how one is meant to get the main inequality at all). Was your "concerns remain" made to mean that you, or someone you know, had (or now have) parsed those changes and still find gaps in the proof? – Archie Jan 08 '18 at 06:36
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    @user2802238 Let's replace your phrase "still find gaps in the proof" with the less provocative "still have the same concerns as before". Then the answer to your question is "yes" (entailing input from multiple people). – nfdc23 Jan 08 '18 at 21:47
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    @nfd23 Thank you for this clarification, what a puzzling situation. – Archie Jan 09 '18 at 08:15
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    Four more months have passed, any news since? – Raphael J.F. Berger May 07 '18 at 16:27
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    @R_Berger Yes, there is a recent rumor that a rebuttal may appear publicly in the next few months, see this blog post https://totallydisconnected.wordpress.com/2018/05/09/the-latest-hot-abc-news/ – Archie May 13 '18 at 11:37
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    @user2802238, and now that rumor seems to have disappeared...?!? – Yaakov Baruch Jul 13 '18 at 13:13