The motivation for studying Riemann surfaces: Branches of the logarithm and the square root. Analytic

continuation and the Monodromy theorem. Definition and basic examples of Riemann surfaces.

Complex tori. The Riemann surface of an algebraic function. Riemann’s existence theorem.

Calculus on Riemann surfaces: Differential forms. deRham cohomology. Poincare duality. Laplace

operator and harmonic functions. The Dirichlet norm.

The Euler Characteristic: Meromorphic forms. The Riemann--Hurwitz formula.

Some major theorems on Riemann surfaces: Existence of meromorphic and holomorphic functions on

Riemann surfaces. The Riemann--Roch theorem. The uniformisation theorem.