Visions in Arithmetic and Beyond: Celebrating Peter Sarnak's Work and Impact

I'll discuss spectral gaps in the following contexts:
- d-regular graphs
- locally symmetric spaces e.g. hyperbolic manifolds
- finite dimensional unitary representations of discrete groups e.g. free groups, surface groups
 

Of particular interest are spectral gaps that are in some a priori sense optimal or almost optimal.
The discussion will center on recent developments giving new examples of (almost) optimal spectral gaps.