I have some feeling of guilt for being late in reporting on the two recent conferences I (partially) attended: one in Vanderbilt, the other in Chicago, and for indulging in lighter stuff like reporting on the (very nice) conference on Pauli and Jung... Sorry, but that was easier.
As for the conference in Vanderbilt the slides are available on line, as well as the list of talks. I'll try to write something on that conference first and will report on the Chicago conference later. Unfortunately I was there only very briefly and it would be great if somebody who attended the whole meeting would volunteer to write a report! So far we got very little help, with the notable exception of David Goss, and we are badly in need for spontaneous contributions.
It is of course impossible to report on all of the talks, and one can at best give some "impressionist" summary of some of the talks that happened to trigger some interaction with one's mathematical preoccupations of the time. This generates a very personal view, and not mentioning a talk should, of course, not be taken as an offense.
Thus after these preliminaries I'll just begin. One of the "raison d'être" of the Vanderbilt school, this year, was to try and bridge the gap between quantum groups and NCG. Both the classes of Christian Kassel and the classes of Masoud Khalkhali and Atabey Kaygun gave the basics in the two subjects with as a possible goal the Hopf cyclic theory--the noncommutative analogue of the Chern-Weil theory-- in relation with quantum groups. There are plenty of open questions and problems to work on in that area (Hopf cyclic + quantum groups) and we hope that some of the students will be ready to work on these.
As it turned out an important puzzle in the interaction between quantum groups and NCG, that remained obscure for quite some time got completely resolved recently and opens up a new area of interaction. One natural set-up for the analogue of Riemannian geometry in NCG is the paradigm of "spectral triples" (A,H,D) where A is the algebra (of coordinates on the NC space) and D is the "Dirac operator", both concretely represented in the same Hilbert space H. The basic requirement is that the resolvent of D is compact, while the commutator [D,a] of D with elements of A is bounded. These two requirements create a tension which allows to measure distances in the corresponding NC space.
Now it has remained for long quite unclear whether this paradigm would work for the homogeneous spaces of compact quantum groups. The first guess is to take simply the q-analogue of the ordinary Dirac operator. This does not work because the eigenvalues of this q-analogue grow geometrically so that the space looks 0-dimensional, in contradiction with the positive dimension of the undeformed maximal torus in the q-group.
Together with Gianni Landi we proposed at some point, by analogy with a particular class of deformations, to replace the q-analogue by its isospectral counterpart, hoping that this would suffice to eliminate the above dimension obstruction. Then followed a rather epic story starting with a no-go theorem and then a first breakthrough by two Indian mathematicians, Partha Sarathi Chakraborty, Arupkumar Pal who managed to construct a spectral triple on the q-group SU(2)_q that had the hoped for boundedness and regularity. It was a very ingenious construction, all the more because it was not a deformation of the commutative spectral triple! In fact the obtained NC-geometry is sufficiently esoteric that I got cold sweat trying to compute what the local index formula was amounting to in that case. One extremely interesting feature is that whereas "locality" has the usual straightforward meaning in the commutative case, the above geometry of Chakraborty and Pal teaches us that in the NC-case "locality" can mean that you are able to strip all the formulas from the irrelevant "details" that only modify them by very small operators.
The next breakthrough was done in a paper by Ludwik Dabrowski, Giovanni Landi, Andrzej Sitarz, Walter van Suijlekom and Joseph C. Varilly who managed to show that there was a natural isospectral deformation of the classical Dirac operator with all the required properties (including the bi-invariance under the q-group action on itself!). Of course SU(2) is very important but the issue of finding the analogue for arbitrary compact q-groups (still in the Lie category) remained opened, and has now been beautifully settled by Sergey Neshveyev, and Lars Tuset who construct the isospectral Dirac in general, with the hoped for regularity properties. This opens up the question of computing the local index formula in that generality, the pseudo-differential calculus and the analogue of the cotangent space. There are also potential links with Hopf-cyclic and finally with the general theory of deformations.