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Back in August 2012, Japanese mathematician Shinichi Mochizuki announced a proof of the abc conjecture using Inter-universal Teichmüller Theory. What has been the status of his proof? Has there been any progress made in verifying it?

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An update by Brian Conrad is here:

http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/

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There is no real update here. At the moment no experts outside a somewhat close circle have made any positive claims regarding the arguments, and if anything they are (on average) somewhat doubtful.

There is a basic difficulty that no-one knows where to really begin in order to understand what is going on in the argument, and Mochizuki more-or-less refuses to explain how the argument goes other than to say "read the papers".

The situation is peculiar; indeed, it may well be unprecedented. One possible outcome is that a group of researchers around Mochizuki accept his argument and techniques, and publish results building on it (say in relatively minor journals, where the fact that many people don't accept the results won't interfere with refereeing/publication), while the majority of researchers remain agnostic or skeptical. This would be pretty unfortunate!

Go Yamashita has been promising a write-up for a long time; perhaps when that appears the situation will become clearer.

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    $\begingroup$ Given the tone of Mochizuki's "On the Verification of Inter-universal Teichmüller Theory: A Progress Report", I think he would be none too pleased to hear people continuing to say that he has refused to explain his work. In his own words, he has personally given three talks and two seminars on the subject. Granted, he is not going on the "world tour" which might have been expected for such an important result, but I think this is a cultural difference: Mochizuki appears to me to be excessively cautious, to the point that others have considered obstinate. $\endgroup$ Jul 1, 2014 at 1:44
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    $\begingroup$ @EricStucky: In emails to experts he has more or less written "read the papers". It is not a question of being cautious; it is a deliberate choice to not attempt to summarize his new ideas and point of view. (The generous interpretation is that the ideas are so novel as to not summarizable.) This is why Go Yamashita's write-up will be so important; it will hopefully provide some insight into what is going on, insight which Mochizuki has basically refused to cooperate in providing. $\endgroup$
    – user160609
    Jul 1, 2014 at 2:22
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    $\begingroup$ I don't deny that it's a deliberate choice. I deny that it is uncooperative. I admit I have an inexpert grasp of Japanese culture. However, it seems reasonable to think that he interprets this action — possibly, of asking for (valid) insight into an unverified and possibly false result; or possibly, of asking for a useful summary of a decade of mental labor so soon after its completion, particularly after having ignored it for that time period — to be overstepping an obvious boundary. $\endgroup$ Jul 1, 2014 at 2:29
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    $\begingroup$ @EricStucky: I think people are making too much of the cultural aspects. Mochizuki spent a good part of his life (including graduate school) in the US, and some of the experts now trying to discuss this with him are people he went to graduate school with. Japanese mathematicians have been world leaders in number theory for 100 years now, and this kind of issue has never arisen before. (There was no problem when Takagi proved class field theory; there was no problem with people understanding Iwasawa theory; there was no problem with people learning about Kato's Euler system.) My sense ... $\endgroup$
    – user160609
    Jul 1, 2014 at 3:54
  • $\begingroup$ ... is that it's much more to do with the particular individual than anything cultural. But I am a westerner, viewing this from a western cultural viewpoint, so perhaps I'm wrong. $\endgroup$
    – user160609
    Jul 1, 2014 at 3:58
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2020 update: Quanta magazine published this article in September 2018 essentially saying that Fields medalist Peter Scholze had had doubts from the beginning, where after reading the entire work he could already say exactly where the heart of the proof was and this was exactly where he ceased to understand the argument.

He and Jakob Stix, an expert in the relevant field of anabelian geometry, visited Mochizuki in March 2018. In the end, they both remained convinced that there is an unfillable gap in the proof, while Mochizuki keeps claiming that it is true.

Here is a report written by Scholze and Stix where they explain their point.

In April 2020, the material is finally going to be published, even though the issue was not really settled.

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The persistent confusion over the status of the proof remains even in 2023, showing no sign of abating with one part of the mathematical community trying to build additional work over the method used and another part denying any value to the proof. Brief Report on the Current Situation Surrounding Inter-universal Teichmüller Theory (IUT)

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