Monday, October 16, 2023
10/16/2023 - 4:00pm
A trace on a group is a positive-definite conjugation-invariant function on it. These functions play an important role in harmonic analysis of discrete groups, and their study has found many exciting connections to rigidity, stability, and dynamics in the past couple of decades. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will then see the different behaviours of these spaces for free groups vs. higher-rank lattices. Finally, some open questions about free products and surface groups will be presented. This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.
10/16/2023 - 4:30pm
Recent work of Ben-Zvi, Sakellaridis, and Venkatesh (BZSV) proposes some conjectures about an analogue of the derived geometric Satake equivalence for spherical varieties, where the spectral side is related to Hamiltonian varieties for the dual group. If X is an affine spherical G-variety, this conjecture is concerned with describing the category of G[[t]]-equivariant sheaves of vector spaces over X((t)). In this talk, I will describe a homotopy-theoretic approach to this conjecture when X is an affine homogeneous spherical variety. When X is of rank 1, this leads to a derived Satake theorem as conjectured in BZSV. Along the way, we observe that the same techniques also allow a study of sheaves with coefficients in "complex connective K-theory"; this leads to a "grouplike deformation" of the derived Satake theorem. For instance, the adjoint quotient stack g^/G^ appearing on the spectral side of the usual derived geometric Satake equivalence is replaced by the canonical degeneration of the conjugation quotient G^/G^ into g^/G^.