Teaching
Riemann surfaces – ETHZ – Spring 2024
Lecturer
Lectures. Thursdays 16:15–18:00, HG D 5.2. The first lecture is on Thursday, February 22, 2024.
Office hours. By appointment (office HG J 43). Absence periods: June 7–14, July 1–14, July 29–Aug 2.
Course description. The course will be a first introduction to Riemann surfaces. These are beautiful objects that sit at the intersection of algebra, geometry, and analysis. We will aim to cover the theorems of Riemann–Hurwitz and Riemann–Roch, as well as the basics of Hurwitz theory. Time permitting, we may delve into additional subjects such as abelian integrals and the Abel–Jacobi theorem.
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.
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.
A brief description of each lecture's content, together with some notes, will appear here.
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. Dates: August 6–7, 2024
The first question on your exam will be chosen randomly from this collection of questions.
A nice video hinting at some of the points explained in the lectures.
Linear Algebra (in Italian) – UTrieste – Fall 2016
Exercise classes