Speaker
Description
Introduced by Polyakov in the 1980s, Liouville quantum gravity ($LQG$) is in some sense the canonical model of a random fractal Riemannian surface. $LQG$ can be defined as a path integral over fields corresponding to the Liouville action, or equivalently as a random metric measure space that turns out to describe the scaling limit of a host of two-dimensional discrete objects. In particular, certain discrete conformal embeddings of random planar maps converge to canonical (up to conformal reparametrization) embeddings of $LQG$ surfaces into 2D Euclidean space. Though one might expect these metric embeddings to retain some vestige of conformality, in fact no embedding of an LQG surface into $\mathbb{R}^n$ can be quasisymmetric. This generalizes a result of Troscheit in the special case of $\sqrt 8/3-LQG$ (corresponding to uniform random planar maps). Time permitting, I will also discuss future directions in the study of metric embeddability for $LQG$.