This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves. We will cover the following topics:

  • Topology of Riemann surfaces. Fundamental group. Homology groups
  • Maps between Riemann surfaces. Degree of a map. Riemann–Hurwitz formula
  • Differential forms
  • De Rham cohomology
  • Hodge decomposition
  • Uniformization of Riemann surfaces
  • Holomorphic differentials
  • Periods of holomorphic differentials. Jacobian variety
  • Abel theorem
  • Riemann-Roch theorem
  • Embedding of Riemann surfaces into projective space
  • Riemann surfaces as algebraic curves
  • Jacobians, abelian varieties, and theta functions
  • Belyi maps (if time permits)