EVENT DETAILS AND ABSTRACT

Title: Inverse spectral problems for the Dirichlet-to-Neumann map
Speaker: David Sher
Speaker Info: University of Michigan
Brief Description:
Special Note:
Abstract:

The Dirichlet-to-Neumann operator on a compact Riemannian manifold with boundary is the map which takes the boundary value of any harmonic function to the boundary value of its normal derivative. It appears in many physical settings, such as electric impedance tomography, and has been extensively studied over the last thirty years. The inverse spectral problem for the Dirichlet-to-Neumann map is the following: given knowledge of the spectrum of the map, as an operator acting on functions on the boundary, what can be said about the geometry of the manifold? In this talk, I will first give an introduction to the subject (requiring no specialized knowledge) and then briefly discuss two recent specific results. The first, joint with I. Polterovich (Montreal), uses heat equation techniques to show that any compact three-dimensional Euclidean domain with connected boundary which has the same Dirichlet-to-Neumann spectrum as a ball must in fact be a ball. The second, joint with A. Girouard (U. Laval), L. Parnovski (UCL), and I. Polterovich, shows how to determine the set of lengths of the boundary components of any compact surface from the Dirichlet-to-Neumann spectrum.