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?
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12$\begingroup$ Here is the last thing I've seen: notes by bcnrd on the recent workshop at oxford on IUTT --- mathbabe.org/2015/12/15/… $\endgroup$– Vidit NandaCommented Feb 25, 2016 at 5:01
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6$\begingroup$ 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. $\endgroup$– David Roberts ♦Commented Feb 25, 2016 at 8:59
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1$\begingroup$ @DavidRoberts: is this upcoming workshop publicly announced yet, and if so, can you point us to the announcement? $\endgroup$– Peter LeFanu LumsdaineCommented Feb 25, 2016 at 9:04
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8$\begingroup$ maths.nottingham.ac.uk/personal/ibf/files/kyoto.iut.html $\endgroup$– Sylvain JULIENCommented Feb 25, 2016 at 9:24
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10$\begingroup$ 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. mainichi.jp/articles/20200403/k00/00m/040/093000c $\endgroup$– Daniel MoskovichCommented Apr 3, 2020 at 5:45
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7 Answers
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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:
It shows that people are understanding more of the proof.
It gives another avenue through which to check whether Mochizuki's work is invalid.
It makes Mochizuki's work that much more important.
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21$\begingroup$ 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! $\endgroup$– nfdc23Commented Feb 25, 2016 at 15:47
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9$\begingroup$ @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. $\endgroup$ Commented Feb 25, 2016 at 20:04
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8$\begingroup$ 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). $\endgroup$– nfdc23Commented Feb 26, 2016 at 5:39
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1$\begingroup$ 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"! $\endgroup$ Commented Feb 26, 2016 at 17:18
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2$\begingroup$ Does an effective abc conjecture give an effective Mordell conjecture? $\endgroup$– user19475Commented Dec 18, 2017 at 5:08
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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.
- ScSt report (August 2018 version, slightly updated from May 2018 version): http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf
- Mochizuki response:
- http://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-05.pdf (to StSc May 2018 version)
- http://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-08.pdf (update for StSc August 2018 version)
- Further discussion and links on Mochizuki's site: http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html
- Somewhat drama-ish Quanta magazine write-up by Erica Klarreich: https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/
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21$\begingroup$ The article by Erica Klarreich was really good! $\endgroup$ Commented Sep 21, 2018 at 0:26
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13$\begingroup$ The only criticism of M that I can understand that has real bearing on the ScSt report is that he claims (Comment (Lin) in 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. $\endgroup$– David Roberts ♦Commented Sep 21, 2018 at 5:26
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11$\begingroup$ A Corollary with a 9-page proof? Is that a record? $\endgroup$– bofCommented Sep 21, 2018 at 23:51
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28$\begingroup$ To a complete outsider, the thinly veiled insults Mochizuki addresses to Scholze and Stix in that response are surprising, to say the least. $\endgroup$ Commented Sep 22, 2018 at 15:17
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7$\begingroup$ See also "Comments on Mochizuki’s 2018 Report" by David Roberts : thehighergeometer.files.wordpress.com/2018/09/… $\endgroup$– jjcaleCommented Oct 17, 2018 at 22:32
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Today (3 April 2020) his papers have been accepted for publication on RIMS journal.
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38$\begingroup$ @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. $\endgroup$– LuciaCommented Apr 4, 2020 at 1:33
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40$\begingroup$ @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. $\endgroup$– LuciaCommented Apr 4, 2020 at 2:00
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28$\begingroup$ @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. $\endgroup$– LuciaCommented Apr 4, 2020 at 3:25
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27$\begingroup$ 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. $\endgroup$ Commented Apr 6, 2020 at 14:40
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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.
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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.
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8$\begingroup$ Has someone considered sharing this answer with Scholze/Stix? $\endgroup$ Commented Nov 14, 2018 at 20:11
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4$\begingroup$ I was hoping someone expert would point out quickly why it is wrong..! $\endgroup$ Commented Nov 15, 2018 at 22:50
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13$\begingroup$ 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). $\endgroup$ Commented Nov 19, 2018 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
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4$\begingroup$ plus.google.com/+johncbaez999/posts/DWtbKSG9BWD $\endgroup$ Commented Dec 17, 2017 at 19:42
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1$\begingroup$ Not to mention commentary by Peter Woit at Not Even Wrong, by Lieven le Bruyn on G+, and others. $\endgroup$– David Roberts ♦Commented Dec 17, 2017 at 22:29
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20$\begingroup$ Apparently the papers have not been accepted for publication. It's unclear how that claim originated. $\endgroup$– David Roberts ♦Commented Dec 24, 2017 at 22:47
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$\begingroup$ Back in December 2017, it was indeed an unsubstantiated rumor that the papers had been accepted for publication. However, in 2021, the papers were eventually published in a special issue of the Publications of the Research Institute for Mathematical Sciences, whose editor-in-chief was and is Shinichi Mochizuki himself. $\endgroup$ Commented Dec 30, 2023 at 13:59
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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).
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4$\begingroup$ No, not progress. As your update notes, those changes happened months ago. The concerns remain. $\endgroup$– nfdc23Commented Jan 8, 2018 at 4:38
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6$\begingroup$ galoisrepresentations.wordpress.com/2017/12/17/… $\endgroup$ Commented Jan 8, 2018 at 4:44
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4$\begingroup$ @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? $\endgroup$– ArchieCommented Jan 8, 2018 at 6:36
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8$\begingroup$ @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). $\endgroup$– nfdc23Commented Jan 8, 2018 at 21:47
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2$\begingroup$ @nfd23 Thank you for this clarification, what a puzzling situation. $\endgroup$– ArchieCommented Jan 9, 2018 at 8:15