Abstract: Let H be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra A. Baum, Dabrowski and Hajac conjectured that there does not exist an equivariant *-homomorphism from A to the equivariant noncommutative join C*-algebra A*H. When A is the C*-algebra of functions on a sphere, and H is the C*-algebra of functions on Z/2Z acting antipodally on the sphere, then the conjecture becomes the celebrated Borsuk-Ulam theorem. Recently, Chirvasitu and Passer proved the conjecture when H is commutative. The main goal of this talk is to show how to extend the Chirvasitu-Passer result to a far more general setting assuming only that H admits a character different from the counit. Also, assuming that our compact quantum group is a q-deformation of a compact connected semisimple Lie group, we prove that there exists a finite-dimensional representation of the compact quantum group such that, for any C*-algebra A admitting a character, the finitely generated projective module associated with A*H via this representation is not stably free. (Based on joint work with L. Dabrowski and S. Neshveyev.)

Abstract: We explain the structure of the reduced group C∗-algebras of complex semisimple quantum groups, and discuss a connection to the Baum-Connes assembly map for classical complex groups.

Abstract: This talk is based on my joint paper with Masoud Khalkhali and Ali Moatadelro (arXiv:1610.04740). First I will recall Gilkey’s theorem on asymptotic expansion of heat kernels for the special case of Laplacians. I will also introduce the noncommutatvie 3-torus (NCT3) and then I will conformally perturb the standard volume form on it. Then the corresponding perturbed Laplacian will be discussed, and using Connes’ pseudodifferential calculus, I will define the scalar curvature of NCT3. Finally, introducing a rearrangement lemma I will compute an explicit formula for the scalar curvature of the curved noncommutative 3-torus.