Riemann Surfaces

Lecturer

Jørgen Ellegaard Andersen (Aarhus).

Contents

Riemann surfaces are important in many areas of mathematics, in topology, analysis and also in algebraic geometry. The aim of the course is to give an introduction to the elementary properties of such surfaces.

A Riemann surface is a surface with a complex structure; thus sufficiently small neighbourhoods of its points may be treated as open subsets of the complex plane. We will extend the usual concepts of complex function theory to Riemann surfaces, defining holomorphic and meromorphic functions, and proving, for example, the identity theorem, the maximum principle, and the residue theorem.

We will, in particular, study compact Riemann surfaces. We will prove the celebrated Riemann-Roch theorem, relating the functions on a surface to its "shape", and Abel's theorem on finding meromorphic functions with prescribed zeroes and poles. If time permits it, we will also treat Hurwitz' theorem on the automorphisms of a compact Riemann surface of high genus (i.e. "complicated shape").

The course can be continued in several directions.

Literature

O. Forster, Lectures on Riemann Surfaces, Springer, GTM 81.

H.M. Farkas, I. Kra, Riemann Surfaces, Springer, GTM 71.

Schedule