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”.