Laura Monk

What does the spectral gap of a hyperbolic surface tell us?

The spectral gap of a hyperbolic surface is the size of the smallest non-zero eigenvalue of its Laplacian. The aim of this talk is to motivate the study of this quantity and show how much information is captured in this number. I will overview results connecting the spectral gap to different geometric features of the surface, as well as its dynamical properties and some counting problems, notably in number theory. I will conclude the talk by announcing new results on the spectral gap of random surfaces, joint with Nalini Anantharaman.