Can one hear the shape of a lens space via Ehrhart theory?

Inverse spectral geometry studies in what extent the spectrum of the Laplace operator determines the geometry of a Riemannian manifold. The interest on this area increased a lot after M. Kac's article "Can one hear the shape of a drum?" in the 60's.

It has been recently discovered that the spectrum of the Laplace operator of a lens space (a quotient of a sphere by a cyclic group) can be encoded by the Ehrhart series of certain (very particular) polytope. It may be expected that some problems in spectral geometry can be solved by using Ehrhart theory.

In this talk, we will recall the mentioned connection in an elementary way. It will not be assumed any knowledge on spectral geometry.