Riemann surfaces are one of the most fundamental objects in mathematics and physics. They arise as zeros of equations in two complex variables, the complex 1-dimensional manifolds (complex curves). They also arise from conformal geometry, namely 2-dimensional oriented real manifolds with a conformal class of Riemannian metrics. That these two view points are equivalent, is already a non-trivial result. Thus, any surface you see in 3-space is in fact an example of a Riemann surface. Special such surfaces, obeying variational constraints, arise in geometry as minimal surfaces and in physics as string propagations. If the Riemann surface is compact, a deep results says that it can be described by complex algebraic equations and we are in the realm of algebraic geometry. At which point we could also contemplate Riemann surfaces over any field, e.g. Diophantine equations, Langlands program etc.
This course will explore these different facets of Riemann surfaces. The necessary concepts such as vector bundles and holomorphic structures will be developed as the course progresses. The main results proven in the course include the integrability of almost complex structures in complex dimension one, Riemann-Roch for vector bundles, Serre's GAGA principle, and the Abel-Jacobi theorem describing the moduli space of holomorphic line bundles. Possible applications (or final project topics) include the classification of holomorphic vector bundles over the Riemann sphere and elliptic curves, algebro-geometric integrable systems, moduli spaces of higher rank holomorphic vector bundles, and non-Abelian Hodge theory.
good knowledge of abstract (linear) algebra, complex variables, basic topology, perhaps some exposure to basic concepts of manifold theory (e.g. Spivak's book on Calculus on manifolds or Hitchin's notes on Differentiable Manifolds).
Hermann Weyl: The Concept of a Riemann Surface (good historical reading).
Griffiths and Harris: Principles of Algebraic Geometry (selected chapters).
Knoeppel, Pedit, and Pinkall: Riemann Surfaces from a Differential Geometric View Point (pdf lecture notes).
Nigel Hitchin: pdf lecture notes on Geometry of Surfaces, Differentiable Manifolds, and Algebraic Curves.
Simon Donaldson: Riemann Surfaces, Oxford Graduate Texts in Mathematics.
R. Narasimhan: Compact Riemann Surfaces, Lectures in Mathematics. ETH Zuerich.
Otto Forster: Lectures on Riemann Surfaces, Springer-Verlag 1981.