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

### Complex Analysis and Riemann Surfaces I

I Holomorphic Functions

- Holomorphic functions, Cauchy-Riemann equations
- Conformal mappings
- Cauchy integral formula, residues

II Analytic Continuation

- Gamma and zeta functions
- Hypergeometric functions and monodromy
- Braid group representations
- Correlation functions in conformal field theory

III Riemann Surfaces

- The Riemann surface
*y*^{2}=*x*(*x*-1)(*x*-l) - Holomorphic and meromorphic differentials
- Homology, fundamental group, surface classification
- Weierstrass elliptic functions
- Theta functions
- The moduli space of tori
- Introduction to Riemann surfaces of arbitrary genera
- Fields of meromorphic functions, field extensions, Galois theory

### Complex Analysis and Riemann Surfaces II

I Theta Functions and Modular Forms

- Modular transformations and modular forms
- Eisenstein series, Dedekind eta-function, Kronecker limit formula
- Hecke operators
- Poisson summation, theta-functions of lattices
- Exact formulas for heat kernels

II Selected Topics, chosen from

- Integrable models, spectral curves, and solitons
- Modular forms and infinite-dimensional algebras
- Geometry of the moduli space of Riemann surfaces
- Solvable models in statistical mechanics or conformal field theory
- Introduction to
*L*-functions