Content: Riemann Surfaces arose naturally in the study of complex analytic functions. They are abstract objects, patched together from open domains of the complex plane according to a rigid set of patching data. The beauty of complex analysis carries over to this abstract setting: the apparently very general definition turns out to constrain the objects in a rather strong way. This leads to interesting geometric, analytic and topological theorems about Riemann surfaces, showing also their ubiquity in much of modern mathematics.
We will first review some of the important features of complex analysis in the plane, before moving on to defining Riemann surfaces as abstract objects modelled on planar domains, and give several examples such as the Riemann sphere, complex tori, and so on. We will explore how Riemann surfaces can be classified and uniformised, along the way taking in such results as the Monodromy theorem, the Riemann mapping theorem and introducing concepts such as universal covers and the covering group of deck transformations. The rest of the module will explore further topics: the degree of a mapping, triangulations and the Riemann-Hurwitz formula, the construction of holomorphic differentials and meromorphic functions on Riemann surfaces, metrics of constant curvature and the pants decompositions of Riemann surfaces, quasiconformal maps and the deformation of complex structures.
Aims: To motivate the idea of a Riemann surface along the lines of Riemann's original reasoning; to introduce the abstract concepts supported by examples; to explain the modern way of understanding Riemann surfaces and discuss their geometry and topology.
Objectives: Students at the end of the module should be able to define abstract Riemann surfaces with maps between them and give examples; use hyperbolic geometry and other geometries to construct Riemann surfaces; analyse topological and numerical properties of analytic mappings between Riemann surfaces; understand the classification of complex tori; and have an overall understanding of all Riemann surfaces as quotients of their universal covers using the statement of the Uniformisation Theorem.
L V Ahlfors, Complex Analysis: an introduction to the theory of analytic functions of one complex variable, McGraw-Hill.
A Beardon, A primer on Riemann surfaces, CUP.
O Forster, Lectures on Riemann Surfaces, Chapter I, Springer.