Early last month Shinichi Mochizuki’s proof of a famous conjecture in number theory known as the abc conjecture appeared in Publications of the Research Institute for Mathematical Sciences (PRIMS). While the publication of the first proof of an important theorem is usually cause for celebration, the publication of Mochizuki’s proof has caused controversy because mathematicians do not agree that it is correct.
Mochizuki’s proof first appeared on his webpage as a preprint in 2012. It is incredibly complex, as he invented a whole new area of mathematics to try to prove the conjecture. It is also very long, with the proof split into four papers that total more than 500 pages in length. Many mathematicians found it difficult to understand the proof, making it hard for the community to come to a consensus about its correctness. Then in 2018, Peter Scholze and Jakob Stix, two highly-respected mathematicians, pointed to a problem at the heart of the proof. Mochizuki, however, maintained that there was no flaw in his reasoning and instead claimed that Scholze and Stix had misunderstood his argument. In the end, neither side could convince the other and an unhappy stalemate ensued.
The publication of Mochizuki’s proof seems to have done little to break this unfortunate stand-off. Physicist and mathematician Peter Woit writes in a comment on his blog that the editors of PRIMS “do not acknowledge the existence of the Scholze-Stix objection or the consensus of experts in the field that the proof is flawed” and “have refused to make public” how the editorial committee came to their decision to accept Mochizuki’s work. Mochizuki himself recently uploaded a 74 page document to his website defending his proof, but Woit comments that he “can’t find anywhere where he engages with the arguments Scholze has made.” To convince mathematicians that Mochizuki’s proof really is correct “a serious mathematical counter-argument to Scholze-Stix, from any source” is needed, says Woit. For now we have to wait to see if any is forthcoming.