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)
- Professor: Maryna Viazovska
- Teacher: Tanguy Vernet
