At a news conference in Tokyo today there evidently were various announcements made about IUT, the most dramatic of which was a 140 million yen (roughly one million dollar) prize for a paper showing a flaw in the claimed proof of the abc conjecture. It is generally accepted by experts in the field that the Scholze-Stix paper Why abc is still a conjecture conclusively shows that the claimed proof is flawed. For a detailed discussion with Scholze about the problems with the proof, see here. For extensive coverage of the IUT story on this blog, see here.
Between paywalls and the limitations of Google translate, I’m not sure exactly what the process is for Scholze and Stix to collect their million dollars. Perhaps they just need to publish their paper, but it seems that the decision may be up to the businessman who is contributing the funds, and it’s unclear what his process will be.
Update: Press release here. Rules for the million dollars are
Nobuo Kawakami makes his own judgment as an individual.
The review method will not be disclosed, but the papers to be reviewed must be papers in mathematics that have been published on MathSciNet and have published more than 10 papers on arithmetic geometry in the past 10 years. Only papers that have been peer-reviewed and published in journals.
Scholze and Stix may not want to take time to submit their paper to a journal (Scholze has a history of turning down large prizes…). It occurs to me that there are quite a few arithmetic geometers who understand well the problem with the proof, could write something up and possibly get it published. Maybe a collaboration could be formed to do this.
Update: New Scientist has a story here. The quotes from Fumiharu Kato aren’t especially encouraging for IUT, who “estimates that fewer than 10 people in the world comprehend the concept.”
Kato believes that the controversy stems from the fact that Mochizuki doesn’t want to promote his theory, talk to journalists or other mathematicians about it or present the idea in a more easily digestible format, believing his work speaks for itself. Kato says that his current and former students are also reticent to do the same because they see him “as a god” in mathematics and don’t want to go against his wishes.
Because of this, most mathematicians are “at a loss” for a way to understand IUT, says Kato, who concedes that, despite earlier optimism about the idea, it is possible that the theory will eventually be disproven.
Ivan Fesenko is much more of a believer:
He told New Scientist that there is no doubt about the correctness of IUT and that it all hinges on a deep understanding of an existing field called anabelian geometry.
“All negative public statements about the validity of IUT have been made by people who do not have proven expertise in anabelian geometry and who have zero research track record in anabelian geometry,” he says.
Update: Scientific American has a news story about this, which summarizes the situation with the proof as:
So despite Mochizuki’s latest publication, there is still doubt among experts about the state of the abc conjecture. Most number theorists cannot make up their own mind because they are unable to follow the proof. And because both Scholze and Mochizuki enjoy an excellent reputation in their field, it is unclear who is right.
This gets the story quite wrong, and misunderstands how mathematics works. The problem is that there is no proof that anyone (Mochizuki himself included) can explain to anyone else, this is not about “do I believe this guy or the other guy?”. Yes, most mathematicians don’t have the technical knowledge to evaluate this sort of proof, but there are plenty who do, and they are either saying there is no proof, or, for the few supporting the proof, unable to explain it to anyone else.