In case you haven’t been following this story, “abc” refers to a famous conjecture in number theory, for which Shin Mochizuki claimed last year (see here) to have found a proof. His argument for abc involves a new set of ideas he has developed that he calls “Inter-Universal Teichmuller Theory” (IUTeich). These are explained in a set of four papers with a total length over 500 pages. The papers are available here, and he has written a 45 page overview here. One can characterize the reaction to date of most experts to these papers as bafflement: what Mochizuki is doing is just so far removed from what is known and understood by the experts that they have no way of evaluating whether or not he has a new idea that solves the abc problem.
In principle one should just be able to go line by line through the four papers and check the arguments, but if one tries this, one runs into the problem that they depend on a long list of “preparatory papers”, which run to yet another set of more than 500 pages. So, one is faced with an intricate argument of over 1000 pages, involving all sorts of unfamiliar material. That people have thrown up their hands after struggling with this for a while, deciding that it would take years to figure out, is not surprising.
Mochizuki has just released a new document “concerning activities devoted to the verification of IUTeich”. It explains the state of his efforts to get other mathematicians to check his work, a project that has been going on since last year, leading to many ongoing updates to the papers making up the proof. He explains that he submitted the four IUTeich papers to a journal last August, but will not have anything to say about the journal or the state of the submission process. This is the way mathematics is supposed to work: the papers should be refereed by experts who have agreed to go through and check them carefully (and confidentially). Given the unusual character of the series of papers, finding willing and able referees may be very difficult. It would of course be most satisfying if such referees can be found and can either identify holes in his argument, or vouch for correctness of the whole thing.
In the meantime, he has been working since October 2012 with Go Yamashita, who has carefully gone through the papers and is now writing a 200-300 page survey of what is in them. Yamashita may also give a course on the topic at Kyushu University sometime after next April. As part of this process, three other mathematicians participated in a seminar in which Yamashita lectured on the papers.
Another mathematician working on this is Mohamed Saïdi, who devoted about six months to studying the papers, then spent three months visiting Kyoto and discussing them with Mochizuki. According to Mochizuki, he has said that he believes the theory to be correct. Mochizuki summarizes the current situation as
the issue of whether or not one should regard the verification of IUTeich as being, for all practical purposes, complete, i.e., as a result of the activities of Yamashita and Saïdi, is by no means clear, and any sort of “final conclusion” on this topic must be regarded as a matter that lies beyond the scope of the present report.
Mochizuki goes on to claim that, based on what he has heard from Yamashita and Saïdi, researchers trying to read his papers should find it possible to understand the theory if they work on it for roughly half a year. He warns that they do need to be aware though that an attempt to make sense of what he is doing by expecting “a similar pattern of argument to existing mathematical theories is likely to end in failure.” They also need to keep in mind that he’s not particularly focused on proving abc, that for him it is the IUTeich theory itself that is the object of interest.
This is a remarkable story, with little precedent. After more than a year, I haven’t heard anyone willing to bet either way on how it will turn out. Mochizuki is a talented mathematician and maybe he has a proof. Or maybe he has a complicated set of ideas which don’t do what he hopes. Perhaps someday one of these alternatives will start to emerge, but it doesn’t now look like this will be anytime soon.
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