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.