Woit reports that a claimed proof of the abc conjecture, a major unproven conjecture in the sub-field of mathematics called number theory (the same sub-field of mathematics that includes Fermat's Last Theorem, which has been proven) is probably flawed:
James Douglas Boyd has recently spent a lot of time interacting with Mochizuki and others at RIMS working in anabelian geometry. Material from interviews he conducted are available here (Mochizuki on IUT) and here (on anabelian geometry at RIMS). He also has written a summary of IUT and of the basic problem with the abc proof. These include detailed comments on the issue pointed out by Scholze-Stix and why this is a significant problem for the proof. I’d be curious to hear from anyone who has looked at this closely about whether they agree with Boyd’s characterization of the situation.There’s also a lot of material [about] the IUT ideas, independent of the problematic abc proof, and about what Mochizuki and others are now trying to do with these ideas.
What is the abc conjecture?
The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers a,b and c (hence the name) that are relatively prime and satisfy a+b=c. The conjecture essentially states that the product of the distinct prime factors of abc cannot often be much smaller than c. A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Mathematician Dorian Goldfeld described the abc conjecture as "The most important unsolved problem in Diophantine analysis".The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves, which involves more geometric structures in its statement than the abc conjecture. The abc conjecture was shown to be equivalent to the modified Szpiro's conjecture.Various attempts to prove the abc conjecture have been made, but none have gained broad acceptance. Shinichi Mochizuki claimed to have a proof in 2012, but the conjecture is still regarded as unproven by the mainstream mathematical community.
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Woit, along with most western mathematicians, has taken that position since 2018 or so, when Scholze and Stix (Peter Scholze being widely considered the world's top number theorist) wrote up their claim that a particular step in the argument was flawed. They did this after a week of conversations with Mochizuki in Japan.
Their critique is directed at a step in Mochizuki's argument which they express as a hexagon of mappings between mathematical objects, which they argue must either be trivial or inconsistent. Mochizuki requires it to come out somewhere between those extremes, i.e. non-trivial in a way that has consequences.
Scholze and Stix use a simplified minimal form of the mappings to make their case. Mochizuki says the simplification is too simple and omits essential aspects of his theory that invalidate the Scholze-Stix argument. Neither side has changed the other's mind, and so the abc conjecture is proven in Japan, but still not proven elsewhere - this is the very rare aspect of the situation, that top mathematicians were unable to reach consensus on whether a proof had been achieved or not.
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