# Riemann Surfaces

### Instructor: Harry Tamvakis

Lectures: Tuesday and Thursday 2:00 - 3:15, Math 0401

Office: Math 4419
Office hours: By appointment
Telephone: (301)-405-5120
E-mail: harryt@math.umd.edu

### Course guide:

Main text: R. Narasimhan, "Compact Riemann Surfaces", Birkhauser, 1996.

Content:
This is a course on Riemann surface theory from the point of view of complex geometry, and can be taken as a second course after complex variables. We will only assume a rigorous background in real and one variable complex analysis, and cover any additional prerequisites during the lectures. The topics to be covered are a classic meeting ground of complex analysis and algebraic geometry, but the algebraic aspect of the theory will lie mostly in the background. In the spring semester, Math 607 will study complex manifolds in higher dimensions.

After discussing the basic facts and examples of Riemann surfaces, we will proceed to more advanced topics: the Riemann surface of an algebraic function, cohomology of line bundles, divisors and the Riemann-Roch theorem, the canonical bundle and Serre duality. Our proofs will be analytic, for example Serre duality will be proved using a regularity theorem for the d-bar operator. We will then discuss Abel's theorem and the Jacobian, and continue with a treatment of theta functions, the theta divisor, and Riemann's theorem about meromorphic functions and theta. If time permits we will prove Torelli's theorem and discuss the Dirichlet problem on Riemann surfaces and the uniformization theorem.

Homework:
I plan to distribute some homework problems during the course.

Other useful book references:

- O. Forster, "Lectures on Riemann Surfaces", Springer-Verlag 1999.

- R. Miranda, "Algebraic Curves and Riemann Surfaces", American Math. Society, 1995.

- Griffiths and Harris, "Principles of Algebraic Geometry", Wiley-Interscience, 1994.

## HOMEWORK

Assignment 1 (Due 9/13/18): ps, pdf

Assignment 2 (Due 10/4/18): tex, ps, pdf

Assignment 3 (Due 10/18/18): tex, ps, pdf

Assignment 4 (Due 11/8/18): tex, ps, pdf

Assignment 5 (Due 12/6/18): tex, ps, pdf