Riemannsche Flächen
Prof. Dr. Umberto Hryniewicz
Schedule
The meetings take place on Tuesdays, from 10:30 to 12:00, in room klPhys(1090|334).
Plan of Lectures
Lecture 1 (October 11):
- Quick review of basic topological facts about manifolds.
- Definition of Riemann surfaces.
Reference: [Ahlfors-Sario]
Lecture 2 (October 18):
- Analytic continuation
- Covering spaces
- Some examples of Riemann surfaces of algebraic functions
Reference: Chapter IX from [Jänich], subsection I.2.9 and section I.3 from [Ahlfors-Sario], sections 1 and 2 of chapter 8 from [Ahlfors]
Lecture 3:
- Riemann surfaces of general algebraic functions
Reference: Chapter IX from [Jänich], subsection I.2.9 and section I.3 from [Ahlfors-Sario], sections 1 and 2 of chapter 8 from [Ahlfors].
Lecture 4:
- Covering transformations
- Fundamental group
Reference: Chapter IX from [Jänich], sections I.3 and I.7 from [Ahlfors-Sario].
Lecture 5:
- Existence of harmonic functions
- Applications of Dirichlet's principle
Reference: sections II.2 and II.3 from [Ahlfors-Sario].
Lecture 6:
- Normal operators
Reference: section III.1 from [Ahlfors-Sario].
Lecture 7:
- Principal operators
Reference: section III.2 from [Ahlfors-Sario].
Lecture 8:
- Principal functions
Reference: section III.3 from [Ahlfors-Sario].
Lecture 9:
- Planar surfaces and the uniformization theorem
Reference: section III.4 from [Ahlfors-Sario].
Bibliography
- AHLFORS, L.V. - Complex Analysis
McGraw-Hill, New York 1978 - AHLFORS, L.V. and SARIO, L. - Riemann Sufraces
Princeton Mathematical Series 26, Princeton University Press, Princeton 1960 - JÄNICH, K. - Topology
Springer-Verlag, Berlin 2005