Bass-Note Spectra of locally uniform geometries.

We formulate and report on the problem of the Bass-Note Spectrum of an invariant operator as one varies over locally uniform geometries. In the Euclidean setting this recasts classical problems of Mahler from the geometry of numbers in a new light. For certain operators homogeneous dynamics can be used decisively. In the non-Euclidea setting of hyperbolic manifolds we review some recent developments using the

conformal bootstrap method and of random covers to study the Bass-Note spectra.We highlight the theme and impact of rigidity.