In the second lecture, we will use integrability theorems from one variable complex analysis to show that matings of large classes of (anti-)holomorphic rational maps with Hecke groups can be realized in our correspondence framework. The interaction between correspondences and Schwarz reflection maps associated to quadrature domains will play a key role in the proof.

Based on joint works with Seung-Yeop Lee, Mikhail Lyubich, Nikolai Makarov, and Jacob Mazor.