Georgia Institute of Technology College of Sciences

Can you hear the shape of a drum? A classical inverse problem in mathematical physics is to determine the shape of a membrane from the resonant frequencies at which it vibrates. This problem is very much still open for smooth, strictly convex planar domains and one tool in that is often used in this context is the wave trace, which contains information on the asymptotic distribution of eigenvalues of the Laplacian on a Riemannian manifold. It is well known that the singular support of the wave trace is contained in the length spectrum, which allows one to infer geometric information only under a length spectral simplicity or other nonresonance type condition. In a recent work together with Vadim Kaloshin and Illya Koval, we construct large families of domains for which there are multiple geodesics of a given length, having different Maslov indices, which interfere destructively and cancel arbitrarily many orders in the wave trace. This shows that there are potential obstacles in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different objects, at least insofar as the wave trace is concerned.