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 y2=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
