The goal of this project is to explore geometric properties of certain random structures in
two dimensions which arise in an area of theoretical physics known as quantum field
theory, a basic building block in our understanding of the universe.
Concretely we will focus on several topics including Liouville conformal field theory, a two-
dimensional model of quantum gravity. Its construction and description requires new
methods at the interface between probability, analysis and geometry. We will in particular
focus on its spectral geometry (which describe for instance how waves propagate in this
random geometry) and aim to establish a conjectured connection to a phenomenon known
as "quantum chaos".
We will also aim to show how related ideas can be used to describe models of statistical
mechanics (which are microscopic models for the behaviour of a very large number of
particles) at a temperature which is close, but not exactly equal to the so-called critical point
where they undergo a phase transition. In particular in the case of the celebrated dimer
model, we will describe an associated boson-fermion correspondence.