CMSA Colloquium: Liouville quantum gravity and distances in random planar maps

I will give an introduction to Liouville quantum gravity, a theory of random surfaces. Such random surfaces describe the large-scale behavior of random planar maps (e.g., uniformly sampled triangulations of the sphere), equipped with the graph distance, in the sense of Gromov-Hausdorff convergence. I will also discuss some recent work with J. Borga and others concerning directed distances in random planar maps equipped with an orientation on their edges, which are related to directed metrics on LQG. I will assume no background knowledge beyond basic probability and analysis.