On the spectral geometry of Liouville quantum gravity

We give a concise presentation of the construction of the Liouville quantum gravity (LQG) eigenvalues and eigenfunctions, i.e., the spectrum associated to the infinitesimal generator of Liouville Brownian motion, the canonical diffusion in the geometry of LQG. We describe the recently obtained Weyl law in this context (giving a short summary of its proof) and report on some work in progress concerning the associated heat trace. Finally, we summarise and propose some new key open problems in this direction.