Davide Castelvecchi at Nature has talked to some of the mathematicians at the recent Kyoto workshop on Mochizuki’s proposed proof of the abc conjecture, and written up a summary under the appropriate title Monumental proof to torment mathematicians for years to come. Here’s the part that summarizes the opinions of some of the experts there:
Mochizuki is “less isolated than he was before the process got started”, says Kiran Kedlaya, a number theorist at the University of California, San Diego. Although at first Mochizuki’s papers, which stretch over more than 500 pages1–4, seemed like an impenetrable jungle of formulae, experts have slowly discerned a strategy in the proof that the papers describe, and have been able to zero in on particular passages that seem crucial, he says.
Jeffrey Lagarias, a number theorist at the University of Michigan in Ann Arbor, says that he got far enough to see that Mochizuki’s work is worth the effort. “It has some revolutionary new ideas,” he says.
Still, Kedlaya says that the more he delves into the proof, the longer he thinks it will take to reach a consensus on whether it is correct. He used to think that the issue would be resolved perhaps by 2017. “Now I’m thinking at least three years from now.”
Others are even less optimistic. “The constructions are generally clear, and many of the arguments could be followed to some extent, but the overarching strategy remains totally elusive for me,” says mathematician Vesselin Dimitrov of Yale University in New Haven, Connecticut. “Add to this the heavy, unprecedentedly indigestible notation: these papers are unlike anything that has ever appeared in the mathematical literature.”
Kedlaya’s opinion is the one likely to carry most weight in the math community, since he’s a prominent and well-respected expert in this field. Lagarias has a background in somewhat different areas, not in arithmetic algebraic geometry, and Dimitrov I believe is still a Ph.D. student (at Yale, with Goncharov as thesis advisor).
My impression based on this and from what I’ve heard elsewhere is that the Kyoto workshop was more successful than last year’s one at Oxford, perhaps largely because of Mochizuki’s direct participation. Unfortunately it seems that we’re still not at the point where others besides Mochizuki have enough understanding of his ideas to convincingly check them, with Kedlaya’s “at least three years” justifying well the title of the Nature piece.
Organizer Ivan Fesenko has a much more upbeat take here, although I wonder about the Vojta quote “now the theorem proved by someone in the audience” and whether that refers to Mochizuki’s IUT proof of the Vojta conjecture over number fields (which implies abc), or the Vojta conjecture over complex function fields (such as in Theorem 9 of the 2004 paper http://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1413.pdf), or something else. The reference to Dimitrov as discussing “applications of IUT” might be better worded as “would-be applications of IUT”.
There will be a conference at the University of Vermont in September, billed as “An introduction to concepts involved in Mochizuki’s work on the ABC conjecture, intended for non-experts.”