The essential spectral gap question asks if linear waves on a given noncompact manifold have exponential local energy decay at high frequency. Spectral gaps for hyperbolic surfaces have important applications to partial differential equations, prime geodesic theorem (via their relation to the Selberg zeta function), and diophantine approximation.
I will explain the recent approach to spectral gaps based on fractal uncertainty principle, which states that no function can be localized near a fractal set in both position and frequency. I will describe recent progress on the fractal uncertainty principle and its applications to spectral gaps, in particular existence of an essential spectral gap for every convex co-compact hyperbolic surface. Time permitting, I will illustrate some of the ideas behind the proof of the fractal uncertainty principle in the case of Cantor sets.
This talk is based on joint works with Jean Bourgain, Long Jin, and Joshua Zahl.