Status: List C

Commitment: 30 one-hour lectures, and fortnightly example sheets.

Prerequisites: Complex Analysis and MA3F1 Introduction to Topology.

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. Today Riemann surfaces play a key role in geometry.

We will first define Riemann surfaces as abstract objects, and give examples from several sources: the Riemann sphere, complex tori, algebraic curves, solutions of differential equations and so on. The uniformisation and classification theorems for Riemann surfaces will be discussed (including the concept of the universal cover and the covering group of deck transformations). The rest of the module will explore the following topics: triangulations and the Riemann-Hurwitz formula, fundamental group and the first homology, tangent bundles and Euler classes, the construction of holomorphic differentials and meromorphic functions on Riemann surfaces, metrics of constant curvature and the pants decompositions of Riemann surfaces, the Moduli space 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.

Leads to: MA505 Algebraic Geometry, MA455 Manifolds.

Books:

L V Ahlfors, Complex Analysis, McGraw-Hill.

A Beardon, A primer on Riemann surfaces, CUP.

O Forster, Lectures on Riemann Surfaces, Chapter I, Springer.

Assessment: 15% credit for marked exercises fortnightly; 85% by a three-hour written exam.

Lecturer: Vladimir Markovic