Discrete conformal welding

Classically the process of conformal welding played a key role in the development of Teichmüller theory and quasiconformal maps. More recently, it has gained attention from physicists as the mechanism for switching between two models of the Universal Teichmüller Space, which is becoming increasingly important in string theory. We develop a geometric method for constructing conformal weldings using circle packings. We consider weldings created by bilipschitz, quasisymmetric, and locally quasisymmetric identification maps and exhibit several computer-generated examples. This provides the first method for computing these weldings.