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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?

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    $\begingroup$ 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 Nanda Feb 25 '16 at 5:01
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    $\begingroup$ 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 Feb 25 '16 at 8:59
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    $\begingroup$ @DavidRoberts: is this upcoming workshop publicly announced yet, and if so, can you point us to the announcement? $\endgroup$ – Peter LeFanu Lumsdaine Feb 25 '16 at 9:04
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    $\begingroup$ maths.nottingham.ac.uk/personal/ibf/files/kyoto.iut.html $\endgroup$ – Sylvain JULIEN Feb 25 '16 at 9:24
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    $\begingroup$ 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 uvm.edu/~tdupuy/anabelian.html." $\endgroup$ – Gerry Myerson Jul 21 '16 at 0:33
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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.

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    $\begingroup$ 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! $\endgroup$ – nfdc23 Feb 25 '16 at 15:47
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    $\begingroup$ @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. $\endgroup$ – Pace Nielsen Feb 25 '16 at 20:04
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    $\begingroup$ 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). $\endgroup$ – nfdc23 Feb 26 '16 at 5:39
  • $\begingroup$ 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"! $\endgroup$ – Pace Nielsen Feb 26 '16 at 17:18
  • $\begingroup$ Does an effective abc conjecture give an effective Mordell conjecture? $\endgroup$ – TKe Dec 18 '17 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.

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    $\begingroup$ The article by Erica Klarreich was really good! $\endgroup$ – Bjørn Kjos-Hanssen Sep 21 '18 at 0:26
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    $\begingroup$ 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 Sep 21 '18 at 5:26
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    $\begingroup$ A Corollary with a 9-page proof? Is that a record? $\endgroup$ – bof Sep 21 '18 at 23:51
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    $\begingroup$ To a complete outsider, the thinly veiled insults Mochizuki addresses to Scholze and Stix in that response are surprising, to say the least. $\endgroup$ – PseudoNeo Sep 22 '18 at 15:17
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    $\begingroup$ See also "Comments on Mochizuki’s 2018 Report" by David Roberts : thehighergeometer.files.wordpress.com/2018/09/… $\endgroup$ – jjcale Oct 17 '18 at 22:32
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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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    $\begingroup$ Has someone considered sharing this answer with Scholze/Stix? $\endgroup$ – Sam Hopkins Nov 14 '18 at 20:11
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    $\begingroup$ I was hoping someone expert would point out quickly why it is wrong..! $\endgroup$ – user36212 Nov 15 '18 at 22:50
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    $\begingroup$ 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). $\endgroup$ – 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

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    $\begingroup$ plus.google.com/+johncbaez999/posts/DWtbKSG9BWD $\endgroup$ – Sylvain JULIEN Dec 17 '17 at 19:42
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    $\begingroup$ Not to mention commentary by Peter Woit at Not Even Wrong, by Lieven le Bruyn on G+, and others. $\endgroup$ – David Roberts Dec 17 '17 at 22:29
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    $\begingroup$ Apparently the papers have not been accepted for publication. It's unclear how that claim originated. $\endgroup$ – David Roberts Dec 24 '17 at 22:47

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