Have there been any updates on Mochizuki's proposed proof of the abc conjecture?

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.

  • ScSt report (August 2018 version, slightly updated from May 2018 version): http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf
  • Mochizuki response:
    • http://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-05.pdf (to StSc May 2018 version)
    • http://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-08.pdf (update for StSc August 2018 version)
  • Further discussion and links on Mochizuki's site: http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html
  • Somewhat drama-ish Quanta magazine write-up by Erica Klarreich: https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/

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.


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.

  • On the verification of Inter-Universal Teichmüller theory: a progress report (December 2013)

  • On the verification of Inter-Universal Teichmüller theory: a progress report (December 2014)

Tags:

Ag.Algebraic Geometry

Nt.Number Theory

Arithmetic Geometry

Abc Conjecture

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