Introduction to the Theory of Riemann Surfaces
Permission of the instructors. You will need a basic knowledge
of complex function theory at the level of Complex Analysis by
Lars Ahlfors or
Functions of One Complex Variable, Volume I, by
Riemann's analysis of finite genus one dimensional complex manifolds
is a mathematical gem. This course will be an introduction to these
manifolds. The topics are
- Definitions and Examples
- Topology of Riemann Surfaces
- Differential Forms
- Integration Formulae
- Hodge Decomposition
- Harmonic Differentials
- Meromorphic Functions and Differentials
- Compact Riemann Surfaces
- Bilinear Relations
- The Riemann--Roch Theorem
- Hyperelliptic Riemann Surfaces
- Torelli's Theorem
- Additional topics as time permits
- Automorphisms of Compact Riemann Surfaces
- Theta Functions
H. M. Farkas and I. Kra, Riemann Surfaces,
Springer-Verlag, 2nd Edition, 1992.
Other possible references include
All handouts, problem sets, etc. will be posted on the web
- A. Beardon, Riemann Surfaces - A Primer.
- C. H. Clemens, A Scrapbook of Complex Curve Theory.
- R. Miranda, Algebraic Curves and Riemann Surfaces.
- G. Springer, Introduction to Riemann Surfaces.