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Update: An ABC proof too tough even for mathematicians

Posted by Kevin Hartnett  December 21, 2013 12:09 PM

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Thirteen months ago, in November 2012, I wrote about new research that was unlike anything the world of professional mathematics had seen before. The work was by a Japanese mathematician named Shinichi Mochizuki, who claimed to have invented an entirely new branch of mathematics and in the process, to have found a proof for one of the most legendary unsolved problems in the field—a deceptively simple inequality known as the ABC conjecture.

Mochizuki’s work carried immense promise. It had the potential to fundamentally change the way mathematicians think about numbers—if it checked out. And that was the problem.

Mochizuki had been working almost entirely by himself for nearly twenty years—a rare solo performance in the normally collaborative world of higher mathematics—and during that time he had created thousands of pages of dense, complex mathematics that no one else in the world could understand. His proof of ABC ran 512 pages, and before anyone could hope to tackle that, they’d have to master 750 pages of foundational work in an incredibly complicated subfield called anabelian geometry that only about 50 people in the world fully understand. After taking a first look at Mochizuki’s proposed proof, mathematicians estimated that it could take years of study to determine if it really worked.

The question was, who, if anyone, would put in the effort, and how, more generally, was the mathematical community going to vet such a monumental body of work.

A year later, we have some answers, and many remaining questions. The accuracy of Mochizuki’s work is still up in the air, but an informal vetting process, as reported by Mochizuki himself, has started to take shape. In December he posted a six-page “progress report” to his website, which detailed the steps he’d taken over the last year to explain his work to other mathematicians.

In very detailed terms, he explained that since October 2012 he has conducted two intensive one-on-one seminars with two experts in anabelian geometry: Go Yamashita of Kyoto University, where Mochizuki works, and Mohamed Saıdi of the University of Exeter in England.

The seminars were an opportunity for Mochizuki to explain the branch of mathematics that he’s invented, which he calls Inter-Universal Teichmuller Theory or, IUTeich, to two of the people who had the best chance of understanding it. In the progress report, Mochizuki provided exact details about the meetings: how often he met with each mathematician, how much time they spent together overall, how many questions each posed to him. He wrote that the two seminars “exceed, by a substantial margin, the scope of a typical referee’s report for a mathematical journal”—meaning that they go above and beyond the typical vetting process in math (though Mochizuki also noted that he’s taken the traditional vetting route, too, having submitted his papers for publication and peer review). In understated language, Mochizuki explained that both Yamashita and Saidi left the seminars believing that Mochizuki’s theory is correct. He added that Yamashita is currently working on a “survey” that will eventually span 200-300 pages and which will aim to explain IUTeich in terms that other mathematicians can more readily grasp.

So where does this leave the mathematical community? At the end of the progress report, Mochizuki wrote that his experiences with Yamashita and Said have led him to believe that the amount of time it would take other mathematicians to study and understand his work is a “matter of months — i.e., roughly half a year or so — not a matter of years!” (a line which suggest he’s well-aware of the reservations his peers have expressed about approaching his work).

But even at six months of study, mathematicians are not lining up to read Mochizuki’s work. Last Thursday, December 19th, mathematician Peter Woit of Columbia University summarized Mochizuki’s update on his blog “Not Even Wrong.” In a comment on his own post, Woit wrote, “telling people they just need to sit down and devote six months of time to trying to puzzle out 1000 pages of disparate material is not especially helpful.” He was more optimistic, however, about the survey of IUTeich that Yamashita is working on, writing, “it’s sometimes the case that the originator of ideas is not the right person to explain them to others.”

This idea—that Mochizuki needs to develop ambassadors for his work—was expressed by other mathematicians as well. After reading Mochizuki’s progress report I emailed Jordan Ellenberg, a mathematician at the University of Wisconsin. Ellenberg had been among the first people to “break” the news of Mochizuki’s research back in 2012 and I’d interviewed him for my original story. Ellenberg wrote back, explaining that while the ultimate fate of Mochizuki’s proposed proof is far from clear, a path to resolution has begun to emerge:

I think the only way the community is going to be able to engage with this work is for someone other than Mochizuki to understand it all the way to the bottom. That seems to me what we're on our way towards. It sounds like we are still a ways from acceptance, but if Go and Mohammed have really understood it, after some serious work, that's a big advance towards figuring out what's going on with the proof.
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About brainiac Brainiac is the daily blog of the Globe's Sunday Ideas section, covering news and delights from the worlds of art, science, literature, history, design, and more. You can follow us on Twitter @GlobeIdeas.
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