Eigenvalue ensembles of certain random matrix models can be viewed as two-dimensional Coulomb gases confined by an external potential. In the large-$N$ limit, their equilibrium distributions are supported on compact planar sets known as droplets. Remarkably, these droplets are closely related to quadrature domains, which admit Schwarz reflection maps – anti-meromorphic maps that extend continuously to the boundary as the identity.

We will discuss how this connection brings tools from conformal dynamics into the study of Coulomb gas droplets. In particular, Schwarz reflection dynamics and ideas from the Sullivan dictionary between rational dynamics and Kleinian groups can be used to study the topology and geometry of these droplets and to construct examples with prescribed configurations.