## Forster: Riemann Surfaces

Course (4 hours weekly + 2 hours Problem sessions) by
O. Forster

Winter Semester 2003/04,
Department of Mathematics, LMU München
**Time and Room:** Tue, Fri 11-13, E 27

Starts on Tuesday, October 21, 2003, at 11:15h

**Problem sessions:**
Tuesday 14-16, E 47

** What this course is about:**
Every serious study of analytic functions of one complex
variable will need Riemann surfaces. For example,
"multi-valued" functions like square root or logarithm
can be treated in a satisfactory way using Riemann surfaces
covering the complex plane. Abstractly speaking, a
Riemann surface is simply a complex 1-dimensional
manifold (which looks locally like an open set in the complex
plane). Some topics treated in this course: Definitions
and basic properties. Construction of Riemann surfaces
associated to algebraic functions (the square root is
the most elementary example). Quotients of Riemann
surfaces by discontinuous automorphism groups (this
allows an elegant treatment of modular functions
and forms). Divisors, line bundles, Theorem of
Riemann-Roch

** Prerequisites:**
A first course on the theory of analytic functions
of one complex variable (e.g. Funktionentheorie I).
Basic notions of algebra and topology.

** For:**
Studierende der Mathematik (und Theoretischen Physik)
im Hauptstudium.
Students of the International Master Program
in Mathematics

**Contents**

- 1. Definition of Riemann surfaces
- cf. [1], Sect. 1
- 2. Elementary properties of holomorphic mappings
- cf. [1], Sect. 2
- 3. Branched and unbranched coverings
- cf. [1], Sect. 4
- 4. Fundamental group, universal covering, covering transformations
- cf. [1], Sects 3 and 5
- 5. Algebraic functions
- cf. [1], Sect. 8
- 6. Sheaves, divisors, line bundles
- cf. [1], Sects 6, 29
- 7. Cohomology groups
- cf. [1], Sects 12-14
- 8. The Riemann-Roch theorem
- cf. [1], Sects 15, 16, 17

** Literature **

- O. Forster:
Lectures on Riemann Surfaces.
Springer Verlag
- Farkas/Kra: Riemann Surfaces. Springer Verlag
- Gunning: Lectures on Riemann Surfaces.
Mathematical Notes. Princeton University Press

Otto Forster (),
2003-07-07/2003-12-07