Spectral geometry of Liouville quantum gravity

I will discuss Liouville Brownian motion, the canonical diffusion in the random geometry defined by Liouville quantum gravity (LQG). In particular I will present some recent results on the spectral geometry of LQG, showing that the eigenvalues satisfy a Weyl law. We will also discuss a number of striking conjectures which aim to relate LQG to a phenomenon known as "quantum chaos", which will also be explained.