Riemann surfaces are spaces on which one can naturally define the notion of holomorphic functions. These objects lie at the crossroad of many different fields such as differential geometry (hyperbolic metrics), number theory (modular forms), dynamical systems (Teichmüller spaces) or algebraic geometry (projective curves).

The general aim of the course is to give an introduction to several geometric aspects of Riemann surfaces. We shall also introduce the notion of covering spaces and the fundamental group, and we shall more specifically discuss the case of compact Riemann surfaces.

Plan of the course:

  • Refresh on the theory of holomorphic functions;
  • Riemann surfaces : definition and first examples;
  • Covering theory and Galois correspondence;
  • The fundamental group;
  • Van Kampen theorem;
  • The topology of compact Riemann surfaces.

Allen Hatcher: Algebraic topology.

Eric Reyssat: Quelques aspects des surfaces de Riemann.

Langue du cours : Français

Credits ECTS : 5