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.”
Update: Fesenko has updated his report on the conference (see here) to include a more accurate characterization of talks by Vojta and Dimitrov (you can see changes to that report here). Between this and the Nature quotes, there seems to be a consensus among the experts quoted (Kedlaya, Dimitrov, Vojta, Lagarias) that they still don’t understand the IUT material well enough to judge whether it will provide a proof of abc or not. Unfortunately it still seems that Mochizuki is the one person with a detailed grasp of the proof and how it works. I hope people will continue to encourage him to write this up in a way that will help these experts follow the details and see if they can come to a conclusion about the proof, in less than Kedlaya’s “at least three years”.
Update: New Scientist has a piece about this which, as in its typical physics coverage, distinguishes itself from Nature by throwing caution to the wind. It quotes Fesenko as follows:
I expect that at least 100 of the most important open problems in number theory will be solved using Mochizuki’s theory and further development.
Fesenko also claims that “At least 10 people now understand the theory in detail”, although no word who they are (besides Mochizuki) and why if they understand the theory in detail they are having such trouble explaining it to others, such as the experts quoted in the Nature article. He also claims that
the IUT papers have almost passed peer review so should be officially published in a journal in the next year or so. That will likely change the attitude of people who have previously been hostile towards Mochizuki’s work, says Fesenko. “Mathematicians are very conservative people, and they follow the traditions. When papers are published, that’s it.”
I think Fesenko here seriously misrepresents the way mathematics works. It’s not that mathematicians are very conservative and devoted to following tradition. The ethos of the field is that it’s not a proof until it’s written down (or presented in a talk or less formal discussion) in such a way that, if you have the proper background, you can read it for yourself, follow the argument, and understand why the claim is true. Unfortunately this is not yet the case, as experts have not been able to completely follow the argument.
If it is true that a Japanese journal will publish the IUT papers as is, with Mochizuki and Fesenko then demanding that the math community must accept that this is a correct argument, even though experts don’t understand it, that will create a truly unfortunate situation. Refereeing is usually conducted anonymously, shielding that process from any examination. Lagarias gives some indication of the problem:
It is likely that the IUT papers will be published in a Japanese journal, says Fesenko, as Mochizuki’s previous work has been. That may affect its reception by the wider community. “Certainly which journal they are published in will have something to do with how the math community reacts,” says Lagarias.
While refereeing of typical math papers can be rather slipshod, standards have traditionally been higher for results of great importance like this one. A good example is the Wiles proof of Fermat, which was submitted to Annals of Mathematics, after which a team of experts went to work on it. One of these experts, Nick Katz, finally identified a subtle flaw in the argument (the proof was later completed with the help of Richard Taylor). Is the refereeing by the Japanese journal being done at this level of competence, one that would identify the sort of flaw that Katz found? That’s the question people will be asking.
In some sense the refereeing process for these papers has already been problematic. A paper is supposed to be not just free of mistakes, but also written in a way that others can understand. Arguably any referee of these papers should have begun by insisting that the author rewrite them first to address the expository problems experts have identified.
Update: Fesenko is not happy with the Nature article, see his comment here.