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I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such inclusion structures should be simpler if they are between categories , how that relates to F_1. It seems to me that his basic idea is that algebraic geometry has in general a kind of semantic feedback-loop, what sounds very beautifull, if it were true. His view of Grothendieck/Deligne's idea of using the section conjecture for indirect proving finiteness statements seems to me as if he would say "The first part of that is just the first jump into the feedback-loop".

Edit: A nice link was jut given in: Mochizuki's proof and Siegel zeros

Edit: A relatively new survey

Edit: Mochizuki' report on the current review status. Someone told me that there may be a seminar in Harvard on this next year.

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For those interested in Mochizuki's proof (or claimed proof) of ABC, it's worth noting that he has recently posted revised versions of IUTT I,II,III here. –  Daniel Miller Mar 19 '13 at 12:53
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In a research statement, he says:

"The essence of arithmetic geometry lies not in the various specific schemes that occur in a specific arithmetic-geometric setting, but rather in the abstract combinatorial patterns, along with the combinatorial algorithms that describe these patterns, that govern the dynamics of such specific schemes."

Regarding this, he then talks about how his main motivations are monoids, Galois categories, and dual graphs of degenerate stable curves, which leads him to talking about his geometry of categories stuff, and then to "absolute anabelian geometry." He then links to a bunch of papers that I would assume elaborate a bit on it. He then goes on to talk about extending Teichmuller Theory.

Generally, his research statement is fairly readable (and consider that I'm very much a nonspecialist in arithmetic anything) and seems to link to things with more details.

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I only browsed his articles and his remarks about a structure of selfrefferentiality are unclear to me. The research statement you link to indicates that he regards such a thing as a pretty fundamental issue. –  Thomas Riepe Jan 1 '10 at 10:41
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