In 1966, Marc Kac asked whether it is possible to hear the shape of a drum. Specifically, he wondered about the degree to which a planar domain $\Omega$ is uniquely characterized by the eigenvalues of its associated Laplace operator (i.e., the spectrum of the domain). This question has sparked decades of ongoing research regarding the relationship between the spectrum of a Riemannian manifold and its underlying geometry. And, while there are numerous examples of geometrically distinct manifolds that have identical spectra, it is interesting to explore whether certain special classes of manifolds can be characterized by their spectra (or “sound”).

The positive resolution of the Geometrization conjecture tells us that, in dimension three, locally homogeneous spaces are the fundamental building blocks of space: in a precise sense, every (closed) three-dimensional manifold can be constructed by joining together a collection of locally homogeneous three-manifolds. Partially inspired by this special role, we initiate the study of the extent to which compact locally homogeneous three-manifolds are characterized by their spectra and present strong evidence that (up to universal Riemannian cover) the shape of such spaces is “audible". 

This is joint work with Samuel Lin (Oklahoma) & Benjamin Schmidt (Michigan State).