Schedule

Abstract: For decades, number theory and ergodic theory have been treated as distinct mathematical continents - one dealing with discrete structures of integers, the other with continuous evolutions of dynamical systems. In 1977, Hillel Furstenberg bridged this gap by pioneering an approach to Szemerédi's Theorem - dense subsets of the natural numbers contain arbitrarily long arithmetic sequences. The first step in that proof was to convert a combinatorial statement into an ergodic theory one. Thus creating what is now known as Furstenberg's Correspondence Principle. In this talk, we explore the mechanics of this principle - specifically, how density translates into measure - and discuss the legacy it left on modern mathematics.

Abstract: t.b.a.

Abstract: In this talk, we present a general criterion for robust transitivity of partially hyperbolic flows and, as a consequence, obtain open sets of Riemannian metrics with conjugate points whose geodesic flows are transitive. Although topologically transitive geodesic flows for metrics with conjugate points are already known in dimension 2, through works of Donnay, Burns, and Pugh, our result applies to manifolds of dimension at least 4 and establishes this property in a robust way in the C^2 topology on Riemannian metrics. Furthermore, our result provides the first evidence that the celebrated theorem of Díaz, Pujals, and Ures—relating robust transitivity to the existence of a dominated splitting—may also hold in the context of geodesic flows.

Abstract: t.b.a.