The uncertainty principle goes fractal
The fractal uncertainty principle states that a function and its Fourier transform cannot both be localized in an appropriate sense on a fractal subset of the line. But why would anyone care about such an obscure statement? One reason is that it leads, unfortunately in a very roundabout way, to control of L^2-norms of eigenfunctions on compact hyperbolic surfaces, independent of the eigenvalue. In this talk I will introduce the statement of the FUP and give a proof when the fractal set is our best friend, the Cantor set. Time permitting, I will discuss how the FUP is used to prove control of eigenfunctions.