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, I will describe the emergence of strange attractors in a neighborhood of a Shilnikov-Hopf bifurcation. By identifying a broad class of dissipative return maps that capture the essential structure of this codimension-two global bifurcation, we provide a rigorous proof of the Bosch-Simó conjecture  (M. Bosch, C. Simó (1993), Physica D,  217--229). We show that, for a set of parameters with positive Lebesgue measure, the dynamics exhibit strange attractors winding around an annular region of phase space. These attractors are "large'' in the sense of Broer-Simó-Tatjer. The analysis relies on reducing the system to a one-dimensional model with strong transverse contraction, thereby placing the dynamics within the framework of rank-one strange attractors.

Abstract: t.b.a.