Riemann Surfaces

Abstract

Riemann Surfaces are important objects in the study of Algebraic Curves. In this thesis, we will begin by extending various concepts and results from Com- plex Analysis to Riemann surfaces. We will go on to describe the consequences of these concepts and results on a particular class of Riemann surfaces, the complex torus. We will then define various objects related to the Riemann surfaces,such as differential forms, divisors and spaces related to divisors. Finally, we shall discuss the Riemann-Roch theorem, Serre Duality and Abel’s theorem. Above all, we will look at the relation between the geometric/topological structure (genus, homology) and the analytic structure (holomorphic maps, meromorphic functions and related spaces, differential forms) of the Riemann Surface.