Nathanael Berestycki: Weyl law in Liouville quantum gravity

Date: 2024-07-16

Time: 14:00 - 15:00

Speaker

Nathanael Berestycki, University of Vienna


Abstract

Can you hear the shape of Liouville quantum gravity (LQG)?

We obtain a Weyl law for the eigenvalues of Liouville Brownian motion:

the number of eigenvalues in an interval grows linearly with the length of the interval. The proportionality constant is given by the Liouville measure of the domain and a certain deterministic constant which is computed explicitly and is, surprisingly, strictly greater than its Riemannian counterpart. This result (joint work with Mo Dick Wong) comes from a fine analysis of the LQG heat trace, which homogenises despite overwhelming pointwise fluctuations.

 

We will also present a number of conjectures on the spectral geometry of LQG, notably suggesting a connection with “quantum chaos”.