Quadrature domains are planar domains admitting 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 instances of "matings'" of rational maps and groups.

We will focus on one-parameter families of Schwarz reflection maps, and describe how typical maps in these families arise as conformal matings of anti-holomorphic rational maps and the modular group.

Joint work with Seung-Yeop Lee, Mikhail Lyubich, and Nikolai Makarov.