Riemann surfaces

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)