Mikhail Panine, Department of Applied Mathematics, University of Waterloo

A Numerical Exploration of Infinitesimal Inverse Spectral Geometry
The discipline of spectral geometry studies the relationship between the shape of a Riemannian manifold and the spectra of natural differential operators, mainly Laplacians, defined on it. Of particular interest in that field is whether one can uniquely reconstruct a manifold solely from knowledge of such spectra. This is commonly stated as “can one hear the shape of a drum?”, after Mark Kac's paper of the same title. Famously, the answer to this question is negative. Indeed, it has been shown that one can construct pairs of isospectral, yet non-isometric manifolds. However, it remains unknown how frequent such counterexamples are. It is expected that they are rare in some suitable sense. We explore this possibility by linearizing this otherwise highly nonlinear problem. This corresponds to asking a simpler question: can small changes of shape be uniquely reconstructed from the corresponding small changes in spectrum? We apply this strategy numerically to a particular class of planar domains and find local reconstruction to be possible if enough eigenvalues are considered. Moreover, we find evidence that pairs of isospectral non-isometric shapes are exceedingly rare.