Weyl laws and closed geodesics on typical manifolds

The HADES seminar on Tuesday, March 15 will be at 3:30 pm in Room 740.

Speaker: Jeffrey Galkowski

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.
Given recent results of myself and Canzani on the Weyl law, all of these estimates are reduced to a study of the closed geodesics on a typical manifold. We will recall these results on the Weyl law and discuss the ideas used to understand closed geodesics on typical manifolds.
Based on joint work with Y. Canzani.

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