Speaker: Carolyn S. Gordon, Dartmouth College

Date: Feb 18, 2025

Abstract: The Steklov spectrum of a bounded domain is the eigenvalue spectrum of the Dirichlet-to-Neumann operator. This operator maps the Dirichlet boundary values of each harmonic function on the domain to its Neumann boundary values. We will focus primarily on the inverse spectral problem for polygonal domains: If two polygonal domains have the same Steklov spectrum, must they be congruent?

We will begin with the question of whether the known examples of (non-convex) Laplace isospectral plane domains are Steklov isospectral. This is joint work with Peter Herbrich and David Webb.

We then address positive inverse spectral results for convex polygons, e.g., showing that generic convex n-gons are finitely determined among all convex n-gons by their Steklov spectra. Our main tools are very powerful Steklov spectral invariant found by Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich and David Sher. This work is joint with Emily Dryden, Javier Moreno, Julie Rowlett, and Carlos Villegas Blas.