Riemann Surfaces

A first introduction to the theory of Riemann surfaces. These are beautiful objects that sit at the intersection of algebra, geometry, and analysis. We covered the theorems of Riemann–Hurwitz, Riemann–Roch, and Abeli–Jacobi, as well as the basics of Hurwitz theory.

Course Details

Schedule. Thursdays 16:15–18:00, HG D 5.2.

Office hours. By appointment (office HG J 16.2).

Prerequisites. Theory of functions of one complex variable, basics of topology. Familiarity with the theory of smooth manifolds and algebraic topology would be useful, but not necessary.

References

• Notes of the course
• E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris. Geometry of Algebraic Curves, Volume 1. Springer–Verlag, 1985
• R. Cavalieri, E. Miles. Riemann Surfaces and Algebraic Curves. Cambridge University Press, 2016
• O. Forster. Lectures on Riemann Surfaces. Springer–Verlag, 1981

Course log

• 22 Feb 2024. Presentation of the course. Real smooth vs. analytic. Holomorphic functions.
  Exercise sheet 1, solutions
• 29 Feb 2024. The problem of multi-valued functions. Manifolds. Definition of Riemann surfaces.
  Exercise sheet 2, solutions
• 7 Mar 2024. Maps between manifolds. Classification of topological compact surfaces.
  Exercise sheet 3, solutions
• 14 Mar 2024. Classification of complex tori, plane affine and projective curves.
  Exercise sheet 4, solutions
• 21 Mar 2024. Elliptic curves. Multiplicity of holomorphic maps at a point.
• 28 Mar 2024. Riemann–Hurwitz formula, genus-degree formula, meromorphic functions.
  Exercise sheet 5, solutions
• 11 Apr 2024. Meromorphic functions on the Riemann sphere and the tori.
  Exercise sheet 6, solutions
• 18 Apr 2024. Divisors, principal divisors, Picard group, linear spaces of meromorphic functions.
  Exercise sheet 7, solutions
• 25 Apr 2024. Finiteness theorem. Meromorphic forms, canonical divisors, Serre duality.
  Exercise sheet 8, solutions
• 2 May 2024. Residue theorem. Riemann–Roch theorem.
  Exercise sheet 9 (solutions can be found in the notes)
• 16 May, 2024. Abel–Jacobi theory.
• 23 May, 2024. Abel–Jacobi theory (conclusion), Hurwitz numbers, monodromy representation.
• 30 May, 2024. Hurwitz numbers and permutations.
  Exercise sheet 10, solutions

Exam

The exam is a 20 minute oral exam. The first question on your exam will be chosen randomly from this collection of questions.