I don’t really have time to write seriously about this, and there’s a very good argument that this is a topic anyone with any sense should be ignoring, but I just can’t resist linking to the latest in the abc saga, the REPORT ON THE RECENT SERIES OF PREPRINTS BY K. JOSHI posted yesterday by Mochizuki.

To summarize the situation before yesterday, virtually all experts in this subject have long ago given up on the idea that Mochizuki’s IUT theory has any hope of proving the abc conjecture. Back in 2018, after a trip to Kyoto to discuss in depth with Mochizuki, Scholze and Stix wrote up a document explaining why the IUT proof strategy was flawed. Scholze later defended this argument in detail and as far as I know has not changed his mind. Taking a look at these two documents and at Mochizuki’s continually updated attempt to refute them, anyone who wants to try and decide for themselves can make up their own minds. All experts I’ve talked to agree that Scholze/Stix are making a credible argument, Mochizuki’s seriously lacks credibility.

The one hope for an IUT-based proof of abc has been the ongoing work of Kirti Joshi, who recently posted the last in a series of preprints purporting to give a proof of abc, starting off with “This paper completes (in Theorem 7.1.1) the remarkable proof of the abc-conjecture announced by Shinichi Mochizuki…”. My understanding is that Scholze and other experts are so far unconvinced by the new Joshi proof, although I don’t know of anyone who has gone through it carefully in detail. Given this situation, an IUT optimist might hope that the Joshi proof might work and vindicate IUT.

Mochizuki’s new report destroys any such hope, simultaneously taking a blow-torch to his own credibility. He starts off with

.. it is

conspicuously obviousto any reader of these preprints who is equipped with a solid, rigorous understanding of the actual mathematical content of inter-universal Teichmüller theory that the author of this series of preprints isprofoundly ignorantof the actual mathematical content of inter-universal Teichmüller theory, and, in particular, that this series of preprintsdoes not contain, at least from the point of view of the mathematics surrounding inter-universal Teichmüller theory,any meaningful mathematical content whatsoever.

and it gets worse from there.

**
Update**: A commenter points to a response from Joshi here.

**Update**: Scholze has a comment on MathOverflow indicating precisely where Joshi’s attempted proof runs into trouble.

**Update**: Mochizuki and those around him award themselves \$100,000 (this is the IUT Innovator Prize described here).

@Unfollow Joshi has very slightly updated his response on May 2: see https://www.math.arizona.edu/%7Ekirti/local-global-issue.pdf ; pages 3 and 6 inserted a line or two that arguably can be taken to be a response to the concerns about locality raised by Scholze and Sawin.