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In April 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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Here is the last thing I've seen: notes by bcnrd on the recent workshop at oxford on IUTT ---… – Vidit Nanda Feb 25 at 5:01
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 at 8:59
@DavidRoberts: is this upcoming workshop publicly announced yet, and if so, can you point us to the announcement? – Peter LeFanu Lumsdaine Feb 25 at 9:04
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"; – Gerry Myerson Jul 21 at 0:33
up vote 19 down vote accepted

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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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 at 15:47
@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 at 20:04
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 at 5:39
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 at 17:18

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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