Monday, April 3, 2023
04/03/2023 - 4:00pm
The Bowen-Ruelle conjecture predicts that geodesic flows on negatively curved manifolds are exponentially mixing with respect to all their equilibrium states. In a breakthrough in ‘98, Dolgopyat pioneered a method rooted in the thermodynamic formalism that settled the conjecture for flows satisfying certain strong regularity hypotheses. Soon after, Liverani introduced a more intrinsic refinement of Dolgopyat’s method which overcame these regularity limitations while simultaneously producing more precise rates of mixing, albeit at the price of being limited to smooth invariant measures. Despite these important developments, the conjecture remains open in general even for the measure of maximal entropy. In this talk, I will describe a new approach leveraging inverse theorems in additive combinatorics to overcome the limitations in Liverani’s approach in a concrete algebraic setting, namely the setting of geometrically finite quotients of rank one symmetric spaces.
04/03/2023 - 4:30pm
We study D. Kazhdan and G. Laumon's 1988 gluing construction for perverse sheaves on the basic affine space G/U and explore unexpected connections to other interesting objects in representation theory. We first define an analogue of Category O in the context of Kazhdan-Laumon categories and explicitly classify its simple objects, and then use this combinatorial data to discuss its connections to Braverman-Kazhdan's Schwartz space on G/U and perverse sheaves on the semi-infinite flag variety. Finally, we study the action of the braid group appearing in the definition of Kazhdan-Laumon categories and give a categorification of the "algebra of braids and ties" occuring in the context of knot theory.