Topics in Riemann Surfaces

Abstract

This thesis consists of a study of some topics in Riemann surfaces. Starting with a quick overview of definitions and properties of maps between Riemann surfaces, we study a proof of the Uniformization theorem for Riemann surfaces. We give a brief overview of the properties of Green’s functions on Riemann surfaces needed to prove the Uniformization theorem. We then describe the correspondence between divisors and line bundles on Riemann surfaces. The final section deals with some consequences of the Behnke-Stein Runge theorem.