Institut für Mathematik
This is a course of the Berlin Mathematical School held in English.
Riemann surfaces appear in complex analysis as the natural domains of holomorphic functions. Their theory provides powerful tools, examples, and inspiration for such diverse areas of pure and applied mathematics as number theory, algebraic geometry, topology, differential geometry, mathematical physics, and geometric analysis. Riemann surfaces appear in many different guises: as the result of analytic continuation, as algebraic curves, as quotients of a complex domains under discontinuous group actions, as smooth or polyhedral surfaces. This lecture course provides a first introduction to the theory of Riemann surfaces. Motivated by concrete examples and applications, the following topics will be treated: topology of Riemann surfaces, holomorphic maps, coverings and branched coverings, meromorphic functions on a Riemann surface, elliptic curves and elliptic functions, abelian differentials, theorems of Abel and Riemann/Roch, theta functions, discrete Riemann surfaces.
To get a certificate for the tutorial, you need to satisfactorily complete 50% of the homework assignments. Homeworks shall be prepared in groups of two or three students.
A growing file of Mini Lecture Notes: