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Complex Analysis and Riemann Surfaces

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
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