The physical problems of QG and Yang-Mills and precise mass values are closely related to the Riemann hypothesis, which is naturally what Connes and Marcolli are trying to solve, as is well known. One important ingredient in this program is the notion of Grothendieck-Teichmuller group, as discussed in this lecture by Schneps. Note the pretty tree diagram on page 11. Many people, such as Kontsevich and Cartier, have thought about this structure from different angles. Kontsevich said we should think of the GT group as the quotient of the motivic Galois group by its action on the spectrum of an algebra generated by (2 pi i), its formal inverse and all the MZVs. Apparently there are some problems with this idea. But the question is, do we really need to define this GT group? Perhaps we can get physical parameters much more directly.
In AQFT one prefers to think about Tomita-Takesaki modular theory. I went to an interesting NCG seminar on this yesterday by Paolo Bertozzini. He has been trying to understand the basic NCG geometry/algebra correspondence from a more categorical point of view. In fact, he made it clear that the categorical duality is far from understood. The correspondence usually works only on the level of objects: take a spin manifold and get a spectral triple, or take a spectral triple and find its spectrum. But to describe the correspondence properly the adjunction natural transformations need to be fully described. The categories need morphisms. Now one can do this, but it leads to the question that Paolo is thinking about: the spectral triples appear to be approximations to something higher categorical. They are recovered as endofunctors of some kind in a richer structure. What is this structure? Paolo was thinking along the lines of a quantum topos theory. Funny thing was that just after I was introduced to Paolo we realised that we had met on the internet, in a discussion on the fqxi funds on Woit's blog. Paolo was one of those rejected for his over enthusiastic category theoretic proposal.
It was a nice day yesterday. There was a Feynman film night run by the Physics group. I was talking to a quantum optics guy and he told me they are getting six new staff in Quantum Information next year. Given the small size of the department at present, this is just flabbergasting. I stood there like a stunned mullet and he pointed out that there was an awful lot of money in this game. None of these guys seemed to have the least interest in what the Category Theory group are doing, even though they haunt the same corridors. One of the new guys will be Terno who has been at Perimeter, so I'm looking forward to meeting him in a few weeks when he arrives.
The pizza was yummy, too.