Aidan Backus will give a talk at 1/31/2023.

An open manifold $M$ is said to have an essential spectral gap if the Schrödinger equation on $M$ has only finitely many resonant states with resonance close to the real line. If $M$ is hyperbolic, then one can prove the existence of an essential spectral gap by showing that one cannot localize a function and its Fourier transform to a neighborhood of the boundary at infinity of $\pi_1(M)$.

We shall see that the above considerations are one manifestation of a general principle called the fractal uncertainty principle, which says that one cannot localize a function and a Fourier transform to the neighborhood of a fractal. Time permitting we will also sketch some of the ideas used in the proofs of the various cases of the fractal uncertainty principle.

Lecture note is available. See link