Project

A remarkable theory of random conformal geometry has emerged at the interface between probability, geometry and analysis. The theory, which is strongly motivated by mathematical physics and especially Liouville quantum gravity (LQG), offers a fascinating new and rigorous window into questions which, until recently, still seemed far away from the realm of mathematics. In this project we follow two interlinked main objectives: on the one hand, explore fundamental geometric properties of LQG; and on the other hand, make progress on a number of theories to which LQG is thought to be related, including integrable quantum field theory as well as the question of holographic projection.

Concretely, we will work on questions related to the spectral geometry of LQG, following the blueprint put forward in this work by the PI, including potential intriguing connections to so-called quantum chaos. We will also investigate massive scaling limits of models of planar statistical mechanics such as the dimer model, which arise when we consider these models slightly away from their critical points. Here we conjecture that, from a geometric perspective the associated curves are related to massive SLEs (introduced by Makarov and Smirnov), while from a field-theoretic perspectives the corresponding height functions should be related to the sine-Gordon model, an integrable (yet non-conformal) quantum field theory. 

The project starts on 01.09.2025.