Let me first use the opportunity to thank in writing all the participants of the Paris conference. From my side these several days of science meant a lot and the great atmosphere of friendship moved me so deeply.

I did my best to make some comments after the talks. In general it is quite difficult to have some real interaction in such circumstances since one rarely keeps in touch with the speakers after the conference. But I believe one useful side of a blog, like this one, can be to give at least a place where such comments can be written and perhaps even discussed.

Here is one example. Thanks to the great work of S. Popa, one can now control in many interesting concrete cases the algebra H(N) of correspondences which are “finite” on both sides, for a II_1 factor N. This was illustrated quite succesfully by S. Vaes. One idea which emerges then is that, exactly as in the BC-system coming from the Hecke algebra associated to an almost normal subgroup of a discrete group, the ratio of the left and right dimensions of correspondences should define a natural time evolution on the algebra H(N). This algebra is in fact defined over the rational numbers and it is a natural problem, then, to classify the KMS states, and compute the range of the rational subalgebra under zero temperature states. From a more general perspective, on the one hand recent developments have shown that the type III theory provides a natural analogue of the Frobenius in characteristic zero with a sophisticated way to take the “points over the algebraic closure of F_1”. On the other hand, the theory of subfactors of Vaughan Jones is a striking extension of Galois theory to non-cocommutative group-like structures (like quantum groups, planar algebras etc…) and fits perfectly with the theory of correspondences. Time seems ripe now to merge the two sides (type III and subfactors), and in particular to explore possible relations with the other analogue of the Frobenius in characteristic zero coming from quantum groups at roots of unity.

Another very striking recent development was described in the talk of U. Haagerup on his joint work (I think it is with Magdalena Musat but am not sure, the paper is not out yet) on the classification of factors modulo isomorphism of the associated operator spaces. He gave an amazing necessary and sufficient condition for the class of the hyperfinite III_1 factor: that the flow of weights admits an invariant probability measure. (One knows that this holds for the von-Neumann algebra of a foliation with non-zero Godbillon-Vey class). This special case suggests that the general necessary and sufficient condition should be the “commensurability” of the flow of weights, and the idea of Mackey of viewing an ergodic flow as a “virtual subgroup” of the additive group R should be essential in developing the appropriate notion of “commensurability” for ergodic flows.

I was off at the beginning of the week for a short sobering trip in Sweeden (Atiyah’s “Witten” talk always has a sobering effect) and heard a really interesting talk by Nirenberg which suggests that the Holder exponent 1/3 which enters as the limit of regularity for the winding number formula of Kahane corresponds to the 3= 2 + 1 of the periodicity long exact sequence in cyclic cohomology.

There is yet another conference taking place the whole week in paris, organized by Vincent Rivasseau.