Analysis and Probability seminar

Abstract: Detecting geometric and analytic features via the spectrum of the Laplacian has a long history in Spectral Geometry. For example, we can detect the dimension, the volume, the perimeter, the topological type of surfaces, and many other spectral invariants.

In a variation on this theme, we show that generically, the Steklov spectrum detects the regularity of the boundary of a surface. This is obtained via precise upper and lower bounds on the remainder in Weyl's law for the Steklov problem. In this talk, I will discuss how to obtain this result via an analysis of operators reminescent of Toeplitz operators.