Riemann Surfaces
Vorlesung von O. Forster im SS 2016am Mathematischen Institut der LMU München
Wed, Fri 14-16, HS A027, Theresienstr. 39
Exercises Wed 16-18 (A027)
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). This course gives an introduction to the
theory of Riemann surfaces with special focus on
compact Riemann surfaces.
Some topics treated in this course: Definitions
and basic properties. Construction of Riemann surfaces
associated to algebraic functions and to algebraic
curves. Divisors, line bundles,
Theorem of Riemann-Roch. Periods of differential forms,
Abel's Theorem, Jacobi Inversion Problem.
für: Studierende der Mathematik und
Theoretischen Physik im Hauptstudium
mit Interesse in Funktionentheorie, Algebraischer Geometrie
oder Differentialgeometrie.
Vorkenntnisse:
Vorlesung Funktionentheorie I.
Nützlich sind auch
Grundkenntnisse aus Algebra, Topologie oder Differentialgeometrie.
Leistungsnachweis: Gilt für Masterstudiengang Mathematik (WP37 oder WP36, WP34), Masterstudiengang TMP
Contents:
- Definition of Riemann surfaces
- Elementary properties of holomorphic maps
- Branched and unbranched coverings
- Riemann surfaces of algebraic functions
- Sheaves
- Cohomology groups
- Theorem of Riemann-Roch
- The Serre Duality Theorem
- Harmonic Differential Forms
- Jacobi Variety and Abel's Theorem
Literature:
- S. Donaldson: Riemann surfaces. Oxford Univ. Press.
- Farkas/Kra: Riemann Surfaces. Springer
- O. Forster: Lectures on Riemann Surfaces. Springer
- Gunning: Lectures on Riemann Surfaces. Mathematical Notes. Princeton University Press
- J. Jost: Compact Riemann Surfaces. Springer
- K. Lamotke: Riemannsche Flächen. Springer
Vorlesungen vergangener Semester
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Otto Forster 2017-10-15