Prerequisites

  • Linear Algebra
  • Basic Algebra (Group Theory, Rings and Fields.)
  • Basic Point set Topology, including a first introduction to the fundamental group.
  • Familiarity with the notion of a manifold.
  • (Complex) Analysis: holomorphic and meromorphic functions in 1 variable, Cauchy's theorem, residues, Stokes's theorem.

Aim of the course

In this course we study Riemann Surfaces, which are 1-dimensional complex analytic varieties. Riemann Surfaces are very closely related to algebraic curves, but we shall mostly take an analytic approach.

The main goal of the course is to arrive at a good understanding of the geometry of Riemann Surfaces. For this we will have to develop quite a bit of theory. In particular, we will study holomorphic and meromorphic functions, differential forms, divisors, ramification theory and linear systems.

Among the important theorems that we will cover are the Riemann-Roch theorem and the Riemann-Hurwitz formula. These will enable us to study projective embeddings of Riemann Surfaces and maps between them. We will illustrate the theory with many concrete examples.

Rules about Homework / Exam

During the semester, some homework assignments will be given. At the end, there will be a written exam.

    Lecture Notes / Literature

    We mostly follow the book 'Algebraic Curves and Riemann Surfaces' by Rick Miranda. (Graduate Studies in Mathematics, Volume 5, AMS.) Where necessary we may complement this with excerpts from other books.

    Lecturer

    Ben Moonen (Radboud University)