Holomorphic and harmonic approximation on Riemann surfaces

The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.