As usual, Arkani-Hamed makes wildly enthusiastic claims. In this case, he claims to have finally found a remarkable new understanding of the subject, based on some combinatorial objects and dramatic new mathematical ideas. He does note that this still hasn’t been written up, and that it’s the third time in the past year that he has thought he had things understood, with the last two times not working out.
14 years ago
Well, when I personally told Arkani-Hamed in 2009 that he should learn some combinatorial category theory (ie. Grothendieck's mathematics) he said, "oh, yuck, no". Guess he may still have a mental block, heh.
ReplyDeleteHe also said, "oh, yuck, no topos theory here" at the mention of Grothendieck's ideas. Then Penrose mentioned something related to topos theory, so I had to remind him that we weren't allowed to talk about topos theory.
ReplyDeleteIndeed, the twistor-string program is just the beginning. Much of the "Grothendieck mathematics" was noted in your thesis, as I recall. A fertile testing ground, which seems to generalize the twistor context, involves topological strings and automorphic black holes. One challenge would be the proper tiling of the Shimura varieties that serve as moduli spaces for such black holes. This would give rise to a whole family of exceptional polytopes.
ReplyDeleteHi, kneemo. And the Batanin (et al) polytopes do come in suitably infinite varieties. We would of course start with $1$, $2$ and $3$ ordinals, taking us to the octonions, as we were discussing in 2006, and include some special higher cases, such as in $6$ dimensions.
ReplyDeleteLink: Shimura variety.
Personally I was never particularly interested in the Langlands thing, or the Riemann hypothesis, but if these things happen to need polytopes too ...
ReplyDeleteYes, the relation comes from the motivic L-functions attached to the moduli spaces. Moreover, in the full quantum theory U-duality groups are broken to discrete subgroups so the constructions are naturally arithmetical, even from the physical viewpoint.
ReplyDeleteWell, as you know, the physical viewpoint must be arithmetical. Arithmetic is the best motivation for weak versions of a topos, going back to Grothendieck, or even to Galois himself, who of course defined adjunctions.
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