5/20/04 Paula Cohen, Texas A&M University Automorphic Pseudo-differential Operators Abstract: (Joint work with Y. Manin and D. Zagier) The theme of differential operators in the theory of classical modular forms has been important in recent work of Connes and Moscovici on the Hopf Algebra of Transverse Geometry. In this lecture we focus on the corresopndence between classical modular forms and pseudo-differential operators with some kind of automorphic behaviour. We shall describe a canonical lifting from modular forms to automorphic pseudo-differential operators, as well as certain families of non- commutative associative multiplications between modular forms that relate to Rankin-Cohen brackets. =========================================================================== Piotr Hajac, Polish Academy of Sciences Index Computations for New and Old Quantum Spheres Abstract: We compute the Chern-Connes pairing for line bundles associated with two types of noncommutative Hopf fibrations. These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U(1) with Galois-type extensions encoding the aforementioned families of Hopf fibrations. Taking advantage of the integrality of the pairing entailed by the noncommutative index formula, we show that the Chern numbers of these modules (indices of appropriate Fredholm operators) coincide with the winding numbers of representations defining them. =========================================================================== Xiaodong Hu, University of Toronto Transversally elliptic operators relative to compact Lie group action Abstract: We discuss the spectral triples for transversally elliptic operators relative to a compact Lie group action, and the Connes-Moscovici local index formula for such a spectral triple. =========================================================================== Hanfeng Li, University of Toronto Recent progress on Morita equivalence of noncommutative tori Abstract: I will survey recent progress on Morita equivalence and Picard groups of noncommutative tori, at both the C*-algebra level and the smooth algebra level. =========================================================================== Bertrand Monthubert, Universite Paul Sabatier (Toulouse) An Atiyah-Singer type index theorem for manifolds with corners We obtain an index theorem for the analytic index on the groupoid of a manifold with corners, using an embedding technique as in the work of Atiyah and Singer. =========================================================================== Ralf Meyer, University of Muenster The Baum-Connes conjecture via localization of categories Abstract: I am going to present my joint work with Ryszard Nest about a new approach towards the Baum-Connes conjecture. This approach uses the framework of triangulated categories and localization. The quickest way to summarize this approach is as follows. We call an object of the equivariant Kasparov category $KK^G$ weakly contractible if it is H-equivariantly KK-equivalent to 0 for all compact subgroups H in G. We describe the topological K-group Ktop(G,A) as the best possible approximation to the K-theory of the crossed product that vanishes for weakly contractible coefficients. The new approach is ideal for studying functorial properties of the Baum-Connes assembly map. In addition, it paves the way towards constructing analogues of the Baum-Connes assembly map for quantum group crossed products. =========================================================================== Marc Rieffel, UC Berkeley Distances between non-commutative spaces Abstract: I will discuss how to introduce metric data for C*-algebras, so that they become "non-commutative metric spaces". Then I will discuss how one defines the distance between non-commutative metric spaces, in analogy with Gromov-Hausdorff distance between ordinary metric spaces. I will give examples, and brief indications of further possibilities. =========================================================================== Andrzej Sitarz, Jagiellonian University Spectral triples and beyond Abstract: I shall present the construction of algebraic data of spectral triples on two nontrivial noncommutative spaces: first (briefly) on Podles sphere then I shall concentrate on the explicit construction of cyclic cocycle (using Connes-Moscovici formula) from the "Dirac operator" on the Heisenberg group algebra. =========================================================================== Erik van Erp, Penn State University The Atiyah-Singer Index Formula for Subelliptic Operators Abstract: We discuss the index theory of subelliptic differential operators on contact manifolds. We construct a symbol class in the K-theory of a noncommutative C*-algebra, which is naturally isomorphic to the K-theory of the cotangent bundle. By adapting the construction of Connes's tangent groupoid, we develop a proof of the Atiyah-Singer Index Theorem for such subelliptic operators. ======================================================================== Shmuel Weinberger, University of Chicago Rigidity of non-aspherical manifolds Abstract: The Baum-Connes conjecture is an analogue of the Borel conjecture which asserts the topological rigidity of aspherical manifolds. That conjecture, in turn, was suggested by the classical rigidity results for flat and hyperbolic manifolds. In this talk, I will try to explain some analogues of these results that apply to certain non-aspherical manifolds. (Some of this is joint work with Benson Farb and some is joint with Jonathan Block.) ===========================================================================