I’ve stolen the title of this posting from Michael Harris, see his posting for a discussion of the same topic.

A big topic of discussion among mathematicians this week is the ongoing workshop at Oxford devoted to Mochizuki’s claimed proof of the abc conjecture. For some background, see here. I first wrote about this when news arrived more than three years ago, with a comment that has turned out to be more accurate than I expected “it may take a very long time to see if this is really a proof.”

While waiting for news from Oxford, I thought it might be a good idea to explain a bit how this looks to mathematicians, since I think few people outside the field really understand what goes on when a new breakthrough happens in mathematics. It should be made clear from the beginning that I am extremely far from expert in any of this mathematics. These are very general comments, informed a bit by some conversations with those much more expert.

What I’m very sure is not going to happen this week is “white smoke” in the sense of the gathered experts there announcing that Mochizuki’s proof is correct. Before this can happen a laborious process of experts going through the proof looking for subtle problems in the details needs to take place, and that won’t be quick.

The problem so far has been that experts in this area haven’t been able to get off the ground, taking the first step needed. Given a paper claiming a proof of some well-known conjecture that no one has been able to prove, an expert is not going to carefully read from the beginning, checking each step, but instead will skim the paper looking for something new. If no new idea is visible, the tentative conclusion is likely to be that the proof is unlikely to work (in which case, depending on circumstances, spending more time on the paper may or may not be worthwhile). If there is a new idea, the next step is to try and understand its implications, how it fits in with everything else known about the subject, and how it may change our best understanding of the subject. After going through this process it generally becomes clear whether a proof will likely be possible or not, and how to approach the laborious process of checking a proof (i.e. which parts will be routine, which parts much harder).

Mochizuki’s papers have presented a very unusual challenge. They take up a large number of pages, and develop an argument using very different techniques than people are used to. Experts who try and skim them end up quickly unable to see their way through a huge forest of unrecognizable features. There definitely are new ideas there, but the problem is connecting them to known mathematics to see if they say something new about that. The worry is that what Mochizuki has done is create a new formalism with all sorts of new internal features, but no connection to the rest of mathematics deep enough and powerful enough to tell us something new about that.

Part of the problem has been Mochizuki’s own choices about how to explain his work to the outside world. He feels that he has created a new and different way of looking at the subject, and that those who want to understand it need to start from the beginning and work their way through the details. But experts who try this have generally given up, frustrated at not being able to identify a new idea powerful enough in its implications for what they know about to make the effort worthwhile. Mochizuki hasn’t made things easier, with his decision not to travel to talk to other experts, and with most of the activity of others talking to him and trying to understand his work taking place locally in Japan in Japanese, with little coming out of this in a form accessible to others.

It’s hard to emphasize how incredibly complex, abstract and difficult this subject is. The number of experts is very small and most mathematicians have no hope of doing anything useful here. What’s happening in Oxford now is that a significant number of experts are devoting the week to their best effort to jointly see if they can understand Mochizuki’s work well enough to identify a new idea, and together start to explore its implications. The thing to look for when this is over is not a consensus that there’s a proof, but a consensus that there’s a new idea that people have now understood, one potentially powerful enough to solve the problem.

About this, I’m hearing mixed reports, but I can say that some of what I’m hearing is unexpectedly positive. It now seems quite possible that what will emerge will be some significant understanding among experts of a new idea. And that will be the moment of a real breakthrough in the subject.

Update: Turns out the “unexpectedly positive” was a reaction to day 3, which covered pre-IUT material. Today, when things turned to the IUT stuff, it did not go well at all. See the link in the comments from lieven le bruyn to a report from Felipe Voloch. Unfortunately it now looks quite possible that the end result of this workshop will be a consensus that the IUT part of this story is just hopelessly impenetrable.

Update: Brian Conrad has posted here a long and extremely valuable discussion of the Oxford workshop and the state of attempts to understand Mochizuki’s work. He makes clear where the fundamental problem has been with communication to other mathematicians, and why this problem still remains even after the workshop. The challenge going forward is to find a way to address it.