Event Series
Event Type
Seminar
Friday, February 4, 2022 12:00 PM
Jeff Galkowski (University College London)

Abstract: We discuss the typical behavior of two important quantities on compact Riemannian manifolds: the number of primitive closed geodesics of a certain length and the error in the Weyl law. For Baire generic metrics, the qualitative behavior of both of these quantities has been well understood since the 1970's and 1980's. Nevertheless, their quantitative behavior for typical manifolds has remained mysterious. In fact, only recently, Contreras proved an exponential lower bound for the number of closed geodesics on a Baire generic manifold. Until now, this was the only quantitative estimate on either the number of geodesics for typical metrics, and no such estimate existed for the remainder in the Weyl law. In this talk, we give stretched exponential upper bounds on the number of primitive closed geodesics for typical metrics. Furthermore, using recent results on the remainder in the Weyl law, we will use our dynamical estimates to show that logarithmic improvements in the remainder in the Weyl law hold for typical manifolds. The notion of typicality used in this talk will be a new analog of full Lebesgue measure in infinite dimensions called predominance.

Based on joint work with Y. Canzani.