Wolff Memorial Lectures

In this talk, I will describe how Schramm-Loewner evolutions (SLE) are related to Liouville quantum gravity (LQG) surfaces as random conformal weldings and some complex analytic questions which arise. I will also describe how this representation can be used to link random planar maps with LQG. Finally, I will explain how this perspective has led to a number of new results about random planar maps, SLE, and two-dimensional statistical mechanics models.