Conformal mapping of multiply connected domains: theory and applications

The theory of conformal mappings plays an important role in many applications and, recently, has been shown to arise in integrable systems theory. This talk will focus on the mathematical construction of conformal mappings to multiply connected domains. For example, one of the few general constructive techniques of conformal mapping theory is based on the Schwarz-Christoffel mapping formula. In recent years, powerful new software has been developed to construct such mappings, based on this classical formula, in the simply connected case. In this talk, a new general formula for the Schwarz-Christoffel mapping to multiply connected polygonal domains will be derived. The idea of the construction is to perform the analysis on a compact Riemann surface known as the Schottky double of a conformally equivalent circular domain and to make use of an associated prime function. Applications and numerical issues will be discussed.