As discussed here a couple months ago, Peter Scholze and Jakob Stix believe they have found a serious problem with Mochizuki’s claimed proof of the abc conjecture, and traveled to Kyoto in March to discuss it with him. Their write-up is now available here. Mochizuki has made public his response to this, creating a web-page available here. There’s also an updated version of Ivan Fesenko’s take on the story, as well as a possibly relevant FAQ on IUTeich from Go Yamashita.
Erica Klarreich has an excellent long and detailed article about this story at Quanta.
Update: Looking through these Scholze/Stix/Mochizuki documents, my non-expert opinion is that Mochizuki does not seem to effectively address the Scholze-Stix objections, which are aimed at a very specific piece of his argument. Unfortunately, he also does his own credibility a huge amount of damage by including over-the-top attacks on the competence of Scholze and Stix, in typefaces that make him look unserious. For instance, there’s
I can only say that it is a very challenging task to document the depth of my astonishment when I first read this Remark! This Remark may be described as a breath-takingly (melo?)dramatic self-declaration, on the part of SS, of their profound ignorance of the elementary theory of heights, at the advanced undergraduate/beginning graduate level.
or the last couple pages of his report.
Update: More of the same about IUT from Fesenko available here. His argument is that the overwhelming majority of leading experts in arithmetic geometry who are skeptical of the purported abc proof should be ignored, since they haven’t put in the two years of continuous study of IUT necessary. I don’t think this collection of ad hominem arguments will do anything to change anyone’s mind. I also don’t see why he doesn’t instead produce what could change minds: a clear and convincing technical refutation of the Scholze-Stix argument.