Fractal Uncertainty Principle and Spectral Gaps

This lecture explores the concept of transfer operators and the Patterson-Sullivan measure, focusing on the Fractal Uncertainty Principle (FUP) and spectral gaps. It delves into the relationship between FUP, resonances, Selberg zeta function, and Schottky groups. The instructor discusses the uncertainty principle for Cantor sets, Ahlfors-David regular sets, and hyperbolic FUP. The lecture concludes with the application of FUP to hyperbolic surfaces and the connection between FUP and spectral gaps, providing insights into the finite number of resonances. Specialized results and new findings in the field are also presented.