Let us start this report on this meeting in a light way with a picture, featuring the subject of this blog and David Goss...
Connes opened the meeting with a talk on some analogies between two grand challenges in mathematics and physics: On the one hand, the search for a geometric setting in which the methods of Weil's proof of the Riemann hypothesis (RH) for curves over finite fields could be applied to prove the original RH; and on the other hand, the search for a quantum theory of gravity starting from the NCG approach to the standard model of particle physics based on the spectral action principle of Connes and Chamseddine. If my memory from an earlier talk of Connes is correct, these analogies were discovered while Connes and Marcolli were finishing an early draft of their tome "Noncommutative geometry, quantum fields, and motives" (draft here), giving rise to the final part of the book that ties (conjecturally) the two major mathematics and physics strands mentioned above. Let me try to give a (somewhat disjointed) indication of the breadth of the analogies, while leaving the bigger picture completely in the fog.
First, Connes gave an overview of the Tomita-Takesaki theory (the "secret weapon" of operator algebraicists, in the words of Jack Morava), emphasizing its novelty and stark contrast with the commutative case: noncommutative operator algebras -- but not commutative ones! -- come endowed with a non-trivial, canonical (that is, up to alteration by an inner automorphism) time evolution. Surprisingly, the theory of Tomita-Takesaki also provided the correct framework for operator-algebraic quantum statistical mechanics. Now, these are old results from the 70's (and late 60's), but around 1992 in collaboration with C. Rovelli the two points of view were considered together in a novel way: Is there a thermodynamic basis for the origin of time? In particular, what should be the (noncommutative) algebra of observables of a quantum theory of gravity?
Connes also gave a rapid summary of his on-going project with Consani and Marcolli to build a geometric world in characterstic 0 hospitable to the methods of Weil's proof of the Riemann hypothesis for curves over finite fields. Apparently, there are fruitful analogies between the necessary ingredients for quantum gravity (QG) and aspects of the space of Q-lattices, the geometric space underlying the GL(2)-system of Connes-Marcolli and the Bost-Connes system (see the summary of Laca's talk below). For example, the moduli space of Dirac operators on the QG side, being described by a double quotient space of complex algebraic groups, is mirrored on the Q-lattice side by Shimura varieties (certain double quotient spaces of adelic algebraic groups). One will find a condensed dictionary of many more analogies in the last part of the book by Connes and Marcolli. It would be desirable if some knowledgeable reader of this blog could elaborate on this (perhaps even the authors themselves, the huge job of having written 700 or so pages notwithstanding).
Continuing in the bridge-building spirit of the meeting, van Suijlekom gave a talk on his recent work with S. Mahanta on their study of the noncommutative torus from the point of view of noncommutative algebraic geometry. This is a very natural undertaking: for while noncommutative tori have long been studied from a topological and differential perspective, classical tori can also be realized as 1-dimensional complex abelian varieties (a.k.a. elliptic curves) which have rich algebraic and arithmetic structures, so it is natural to try to examine noncommutative tori as noncommutative algebraic varieties of sorts. But whereas in the differential-topological approach pioneered by Connes and Rieffel a noncommutative space is a certain kind of noncommutative topological algebra, in the current algebraic-geometry approach, a noncommutative variety is regarded as a certain type of category. Indeed, from the work of A. Rosenberg, Bondal, and Orlov it is known that smooth (irreducible) projective varieties are characterized up to isomorphism by their bounded derived categories of quasi-coherent sheaves. The work of Mahanta-van Suijlekom is an attempt to connect these two worlds for NC tori. What they have done is to define a category that, roughly speaking, interpolates between the categories reflecting the differential and algebraic nature of the NC tori. Additionally, they've shown that this interpolating category is a Tannakian category equivalent to the category of representations of Z^2.
It would be interesting to see whether the categorical approach to noncommutative tori sheds any light on the conjectured relevance of noncommutative tori to an explicit class field theory for real quadratic fields (in analogy with the theory of complex multiplication, as suggested by Manin), or clarifies what it should mean for a noncommutative torus to be defined over Q or a number field (cf. the recent thesis of J. Plazas).
Laca gave a report on his recent work with N. Larsen and S. Neshveyev. This was an especially pleasing talk to attend as this work finally wraps up an analytic problem that has remained open for more than 10 years, namely the classification of KMS states for the Bost-Connes C*-dynamical system for number fields. Avoiding all details of what the Bost-Connes system is exactly -- an excellent summary is given in the book of Connes and Marcolli -- let me mention only that its most "fabulous" feature is that it admits an action of the abelianized absolute Galois group of Q on its so-called KMS infinity states, and upon evaluation of theses KMS states on a natural rational subalgebra, this Galois action coincides with the usual Galois action on the maximal cyclotomic extension of Q. (KMS-beta states were discussed by Connes in his talk and are surely discussed elsewhere on this blog as well. To describe them quickly, albeit in a rather cryptic manner: KMS-beta states are analogues of infinite-volume limits of Gibbs states in quantum statistical mechanics; beta, in the physical context, is inverse temperature.) A natural problem is to construct C*-dynamical systems with analogous Fabulous Features for general number fields. For the case of imaginary quadratic fields, this was accomplished about three years ago by Connes-Marcolli-Ramachandran. Paugam and one of the blog posters have defined a candidate analogue of Bost-Connes for general number fields, without, however, being able to show that it is fabulous. What Laca and his collaborators have done is overcome a key analytic obstacle towards establishing "fabulousness" of the Bost-Connes system for general number fields: namely, for all beta they have classified the KMS-beta states. The result is essentially the same as for the original Bost-Connes system, though the proof follows the ergodic-theoretic techniques developed by Neshveyev, later enhanced by Laca, Larsen, and Neshveyev to clean up the KMS states classification for the Connes-Marcolli GL(2)-system. To get truly "fabulous" systems in the general number field case capable of manifesting the Galois action, it remains to find an appropriate rational structure for such C*-systems. This is a problem of a different nature, which is not likely to fall without deep arithmetic insight, given that it has implications for Hilbert's 12th problem.
The second day started with two talks by Kreimer and Yeats discussing results obtained in (perturbative) quantum field theories, in particular on quantum electrodynamics. The recursive structures that appear are by now well-known to be captured by the structure of a Hopf algebra. On the analytical side, one can expand the probability amplitudes of interest in physics (such as the vacuum self-energy of the photon) as a series in certain functions gamma_k of the coupling constant. One then writes a recursive relation for the $\gamma_k$ and tries to (numerically) solve a differential equation for the $\gamma_1$. This involved only the computation of the amplitudes of primitive graphs, which was carried out up to fourth order in the loop number. Several vector flow diagrams were presented in the second talk, corresponding to the differential equation. Striking was the difference when moving to 4th loop order, where a separatrix
appeared. Although not yet completely understood, it was observed that the fine structure constant $\alpha = 1/137... lies on this separatrix!
In addition to the talks mentioned above there were also talks on the meeting by Ramachandran on computing Beilinson's ring of correspondences at the generic point of a smooth projective variety over a finite field; by Marcolli on her joint work with Manin on the pseudomeasure formalism for modular symbols (a manifestation of a "modular shadow" in their terminology); by Moscovici on twisted spectral triples (though, unfortunately, there wasn't enough time for him to go deeper into applications to the GL(2)-system of Connes-Marcolli-Moscovici); by Goss on Hecke operators and distributions (in the sense of probability theory) in characteristic p, and some work of Boeckle; and by Zhao on improving the Deligne-Goncharov upper bounds for the dimension of spaces of multiple zeta values (of a given weight). On the second day there were additional talks by Retakh on a construction of Lie algebras and Lie groups over noncommutative rings; by Gangl on Polygons and mixed Tate motives (with Brown and Levin); and finally Zhang on differential renormalization for multiple zeta values (joint with Guo).
Walter van Suijlekom