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Mochizuki on ABC

[Update:  Lots of traffic coming in from Hacker News, much of it presumably from outside the usual pro number theory crowd that reads this blog.  If you’re not already familiar with the ABC conjecture, I recommend Barry Mazur’s beautiful expository paper “Questions about Number.”]

[Re-update:  Minhyong Kim’s discussion on Math Overflow is the most well-informed public discussion of Mochizuki’s strategy.  (Of course, it is still very sketchy indeed, as Minhyong hastens to emphasize.)   Both Kim’s writeup and discussions I’ve had with others suggest that the best place to start may be Mochizuki’s 2000 paper “A Survey of the Hodge-Arakelov Theory of Elliptic Curves I.”]

Shin Mochizuki has released his long-rumored proof of the ABC conjecture, in a paper called “Inter-universal Teichmuller theory IV:  log-volume computations and set-theoretic foundations.”

I just saw this an hour ago and so I have very little to say, beyond what I wrote on Google+ when rumors of this started circulating earlier this summer:

I hope it’s true:  my sense is that there’s a lot of very beautiful, very hard math going on in Shin’s work which almost no one in the community has really engaged with, and the resolution of a major conjecture would obviously create such engagement very quickly.

Well, now the time has come.  I have not even begun to understand Shin’s approach to the conjecture.  But it’s clear that it involves ideas which are completely outside the mainstream of the subject.  Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space.

Let me highlight one point which is clearly important, which I draw from pp.3–6 of the linked paper.

WARNING LABEL:  Of course my attempt to paraphrase is based on the barest of acquaintance with a very small section of the work and is placed here just to get people to look at Mochizuki’s paper — I may have it all wrong!

Mochizuki argues that it is too limiting to think about “the category of schemes over Spec Z,” as we are accustomed to do.  He makes the inarguable point that when X is a kind of thing, it can happen that the category of Xes, qua category, may not tell us very much about what Xes are like — for instance, if there is only one X and it has only one automorphism. Mochizuki argues that the category of schemes over a base is — if not quite this uninformative — insufficiently rich to handle certain problems in Diophantine geometry.  He wants us instead to think about what he calls the “species” of schemes over Spec Z, where a scheme in this sense is not an abstract object in a category, but something cut out by a formula.  In some sense this view is more classical than the conventional one, in which we tend to feel good about ourselves if we can “remove coordinates” and think about objects and arrows without implicitly applying a forgetful functor and referring to the object as a space with a Zariski topology or — ptui! — a set of points.

But Mochizuki’s point of view is not actually classical at all — because the point he wants to make is that formulas can be intepreted in any model of set theory, and each interpretation gives you a different category.  What is “inter-universal” about inter-universal Teichmuller theory is that it is important to keep track of all these categories, or at least many different ones.  What he is doing, he says, is simply outside the theory of schemes over Spec Z, even though it has consequences within that theory — just as (this part is my gloss) the theory of schemes itself is outside the classical theory of varieties, but provides us information about varieties that the classical theory could not have generated internally.

It’s tremendously exciting.  I very much look forward to commentary from people with a deeper knowledge than mine of Mochizuki’s past and present work.

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