The course is mainly devoted to the classical theory of Riemann surfaces and complex algebraic curves, founded by Riemann, Abel, Jacobi, Weierstrass, Hurwitz and others. This fundamental theory is the basis of algebraic geometry, the theory of complex manifolds and other important parts of contemporary mathematics. Lately, the theory of Riemann surfaces has played a remarkable role in mathematical physics. Some of these applications will also be considered in the course.

**Prerequisites:** introductory course to Complex Analysis

### Program of the course

- Riemann surfaces and Fuchsian groups.
- Meromorphic functions and the Riemann-Hurwitz formula.
- Holomorphic differentials and bilinear Riemann relations.
- Meromorphic differentials.
- Riemann-Roch theorem.
- Weierstrass points.
- Canonical embedding.
- Elliptic curves and functions.
- Jacobi manifolds.
- Abel’s theorem.
- Riemann surfaces and algebraic curves.
- Θ-functions.
- Addition theorems for Θ-functions.
- Abel’s map.
- The Riemann theorem on zeros.
- Θ-divisor.
- Θ-functions and integrable systems.

### Textbooks

- G. Springer, Introduction to Riemann surfaces, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, USA, 1957.
- D. Mumford, Tata Lectures on Theta I, Progress in Mathematics, vol. 28, Birkhauser, 1983.