Fine analysis of random surfaces and related processes

The field of random conformal geometry has been immensely successful in the last two decades, following the discovery of the Schramm-Loewner evolution (SLE).

A central object of study is the Gaussian free field (GFF) which is a random surface model that can be thought of as Brownian motion with two-dimensional time.

The GFF has deep connections with SLE via powerful couplings and is used to construct Liouville quantum gravity (LQG), a model of two-dimensional quantum gravity.LQG was introduced by Polyakov in the 1980’s in the context of string theory and one main aim of this project is to use the GFF and its level sets to construct a version of string theory that motivated Polyakov’s definition of LQG (but which is outside the typical range of parameters chosen for the model) and show that this is indeed the “correct” model (in the sense that it corresponds to LQG with the “right” parameter).

Another aim is to more deeply explore the rich structure of the different couplings and relations between SLE, GFF, LQG, etc., as well as develop new such couplings. Moreover, we further aim to illuminate and explain many untreated more analytic aspects of the couplings.

Finally, we aim to further analyse the geometry and regularity of the mentioned models, as well as use these techniques to better understand and analyse similar properties of deterministic sets, which is often harder as a bit of randomness has turned out to be helpful when tackling the questions for said models.