This course gives an introduction to the theory of Riemann surfaces. 

Riemann surfaces are certain geometric objects that can be approached from many different angles, and as such play an important role in different areas in mathematics. In this course we choose a complex geometric angle to introduce Riemann surfaces, using complex function theory as the basis. Topics we will discuss include, sheaves and their cohomologies, differential forms and residues. After that we come to the major results in the field, the Riemann-Roch theorem and Serre duality. We will also discuss covering spaces and the Riemann-Hurwitz formula. As an application of these results, we make contact with the algebraic point of view on Riemann surfaces (algebraic curves). If time permits, we will also treat the Jacobian and the Abel-Jacobi map.

Prerequisites

  • (Complex) analysis: holomorphic functions, Cauchy’s theorem, residues. Stokes’ theorem
  • Topology: basic notions such as open/closed/compact sets. Covering spaces and \pi_1
  • Algebra: fields and extensions.
  • A bachelor course on differentiable manifolds is useful, but not necessary.