Given two von Neumann algebras A,B with an action by a locally compact (quantum) group G, one can consider its associated equivariant correspondences, which are usual A-B-correspondences (in the sense of Connes) with a compatible unitary G-representation. We show how the category of such equivariant A-B-correspondences carries an analogue of the Fell topology, which is preserved under natural operations (such as crossed products or equivariant Morita equivalence). If time permits, we will discuss one particular interesting example of such a category of equivariant correspondences, which quantizes the representation category of SL(2,R). This is based on joint works with Joeri De Ro and Joel Dzokou Talla. 

The geodesic cycles (resp. Eisenstein classes) for SL(2,Z) are special classes in the homology (resp. cohomology) of modular curve (for SL(2,Z)) defined by the closed geodesics (resp. Eisenstein series). It is known that the pairing between these geodesic cycles and Eisenstein classes gives the special values of partial zeta functions of real quadratic fields, and this has many applications. In this talk, I would like to report on some recent observations on the size of the homology subgroup generated by geodesic cycles and their applications. This is a joint work with Ryotaro Sakamoto.

A trace on a group is a positive-definite conjugation-invariant function on it. These traces correspond to tracial states on the group's maximal  C*-algebra. In the past couple of decades, the study of traces has led to exciting connections to the rigidity, stability, and dynamics of groups. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will see the different behaviours of these spaces for free groups vs. higher-rank lattices, and how our strategy for the free group can be used to answer a question of Musat and Rørdam regarding free products of matrix algebras. This is based on joint works with Arie Levit, Joav Orovitz, and Itamar Vigdorovich.