Carolyn Gordon

Abstract:   To what extent does spectral data associated with the Laplace operator of a Riemannian manifold determine the geometry of the manifold? In the case of compact manifolds, the spectral data we will consider are the eigenvalues of the Laplacian. For example, if the manifold is a plane domain, viewed as a vibrating membrane, the eigenvalue spectrum corresponds to the characteristic frequencies of vibration.

We will consider constructions of Riemannian manifolds with the same eigenvalue spectrum but with different local geometry. Through these examples, we will identify geometric invariants that are not spectrally determined.

For non-compact Riemannian manifolds, scattering resonances play a role similar to that of the Laplace eigenvalues for compact manifolds. We will construct continuous families of metrics on Rⁿ with the same resonances and scattering phase. (The latter construction is joint work with Peter Perry.)