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

Link to Moodle

Riemannsche Flächen

Link to RWTH Online

Riemannsche Flächen