PhD comprehensive seminar | Mikhail Panine, Inverse Spectral Geometry

Inverse Spectral Geometry

Spectral Geometry studies the relationship between the shape of a Riemannian manifold and the spectra of geometrically meaningful operators, typically Laplacians, defined over it. Inverse Spectral Geometry studies the opposite direction: it strives to reconstruct the shape of a manifold from some spectra in a unique way. Informally, this problem is often referred to as "Can One Hear the Shape of a Drum?", since the spectrum of the Laplacian on scalar functions can be viewed as the resonant frequencies of a drum shaped like the manifold under study. In full generality, this question has been answered in the negative by providing a number of counterexamples (isospectral but not isometric manifolds). Still, the question of how often can the reconstruction succeed remains open. In this pre-comprehensive seminar, we review some of the known positive and negative results in Inverse Spectral Geometry, discuss our previous work in numerical explorations of inverse spectral geometry, make a few conjectures and state results that could plausibly be used to prove them.