Wolff Memorial Lectures

Over the past several decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity (LQG), which has roots in string theory and conformal field theory from the 1980s and 1990s, and which comes with a parameter $\gamma \in [0,2]$ that indicates how "rough" the surface is. LQG with the special parameter $\gamma=\sqrt{8/3}$ is sometimes called "pure Liouville quantity gravity." The second is the Brownian map (TBM), which has its roots in planar map combinatorics from the 1960s. In this talk, I will describe work from the probability community starting around 2000 aimed at rigorously defining LQG, TBM, the scaling limits of other models of random planar maps, and conjectures from the 1980s about their equivalence.