A domain in the complex plane is called a quadrature domain if it admits a global Schwarz reflection map. Topology of quadrature domains has important applications to physics, and is intimately related to iteration of Schwarz reflection maps. As dynamical systems, Schwarz reflection maps produce various examples of ''matings" of rational maps and groups.
We will look at a specific one-parameter family of Schwarz reflection maps, and show that "typical" maps in this family arise as unique conformal matings of a quadratic anti-holomorphic polynomial and the ideal triangle group. Time permitting, we will also describe a combinatorial model for the "connectedness locus'' of this family.
Joint work with Mikhail Lyubich and Nikolai Makarov.