This course is taken in sequence, part 1 in the fall, and part 2 in the spring.

### COMPLEX ANALYSIS AND RIEMANN SURFACES I

Local theory of holomorphic functions

The Riemann surface of $w^2=z(z-1)(z-\lambda)$

Construction of holomorphic forms and meromorphic forms with prescribed poles

Abel’s theorem

The Jacobi inversion theorem

Function theory on tori:

- The point of view of Weierstrass
- The point of view of Jacobi: $\theta$ functions
- The point of view of partial differential equations

Line bundles, connections, metrics, curvature, first Chern class

The Riemann-Roch theorem, applications

The Riemann-Roch theorem as an index theorem

A heat equation proof of the Riemann-Roch theorem

Function theory on Riemann Surfaces of higher genus:

Theta functions, the theta divisor, the prime form

Theta characteristics and spinors

### COMPLEX ANALYSIS AND RIEMANN SURFACES II

Holomorphic vector bundles on complex manifolds

Connections and curvature, characteristic classes and Chern-Weil theory

The Uniformization theorem

The Ricci curvature and the Calabi conjecture:

Statement and complete proof; estimates for the complex Monge-Ampere equation

Hermitian-Einstein metrics and Mumford stability

The Donaldson-Uhlenbeck-Yau theorem

The proof by Uhlenbeck and Yau

Depending on the year, the last part of II (Hermitian-Einstein metrics, etc) may be replaced by a treatment of the $\bar\partial$ equation, as follows:

- Bochner-Kodaira formulas
- H\”ormander’s estimates for the $\bar\partial$ equation
- Singular metrics and multiplier ideal sheaves
- The Kodaira imbedding theorem
- The Ohsawa-Takegoshi extension theorem