MSc Pure Mathematics and Mathematical Logic / Course details

Riemann surfaces are surfaces (two-dimensional shapes) that are orientable (they have an inside and an outside, like a sphere but unlike a Möbius strip) and, crucially, on which there exists a notion of angles. Two Riemann surfaces are conformally isomorphic (considered the same) if one can be mapped to the other without changing angles. Such surfaces appear naturally in many areas of mathematics, and in many different guises, such as smooth surfaces in 3-space, complex algebraic curves (solutions of an algebraic equation), and domains of definition of differentiable functions of one complex variable.  

This module introduces the theory of Riemann surfaces and discusses their properties as well as key examples such as the Riemann sphere and the space of complex tori. Although the theory of Riemann surfaces has its origins in the 19th century, they play an important role in many modern developments of mathematics. By the end of the module we will be able to touch upon some recent and ongoing research results in the area.  

The theory of Riemann surfaces connects to many other branches of mathematics. As such, a number of other units are related to the material covered here. We assume familiarity with the theory of metric spaces (particularly the notions of open, closed and compact sets and of continuous functions and homeomorphisms), as covered in MATH21111. The theory of Riemann surfaces is closely related to the subject of complex analysis; familiarity with complex analysis as covered by the first half of MATH34011 will be helpful but is not required, as we will consider the properties of complex-differentiable functions from a more geometric point of view than in MATH34011. Some further topics in which there are some connections with other units include: Möbius transformations and the Riemann sphere (MATH21120 Groups and Geometry), geometry of surfaces (MATH31072 Differential Geometry of Curves and Surfaces), hyperbolic geometry (MATH32052) and the fundamental group (MATH31010 Topology and Analysis). No familiarity with any of these units is required.  

The unit aims to: introduce students to the theory of Riemann surfaces as orientable surfaces (two-dimensional shapes) on which there is a notion of angles; study the geometric properties of holomorphic (complex-differentiable) functions and their connection to Riemann surfaces; discuss the classification of Riemann surfaces in terms of their geometry (elliptic, parabolic and hyperbolic), and discuss how Riemann surfaces arise in different areas of mathematics. 

On the successful completion of the course, students will be able to:

1. Define Riemann surfaces and conformal maps, and determine whether a given function is a conformal isomorphism between Riemann surfaces
2. Use stereographic projection and Möbius transformations to transform the Riemann sphere and its subsets
3. Determine whether a given Riemann surface is elliptic, parabolic or hyperbolic.
4. Use a variety of methods to determine whether two Riemann surfaces are conformally isomorphic.
5. Use results from complex analysis and conformal geometry to prove results about Riemann surfaces.

Syllabus:

Conformal maps and Riemann surfaces; stereographic projection and the Riemann sphere [3 lectures]

Conformality and complex analysis; geometric properties of holomorphic functions [3 lectures]

The Riemann surfaces of the square root and the logarithm [2 lectures]

Möbius transformations of the sphere, plane and disc [2 lectures]

Quotients of Riemann surfaces; complex tori as quotients of the plane [2 lectures]

Covering maps and the universal covering surface [3 lectures]

Content will be delivered in two lectures a week; these lectures will include the use of instant feedback systems to identify gaps in understanding and topics that will require further discussion in lectures or tutorials. Tutorials will provide an opportunity for active learning activities to help students consolidate their, as well as discussions of the homework sheets. Further feedback can be obtained directly from the lecturer during office hours. 

Written exam - 100% weighting.

Biweekly problem sheets - 0% weighting, but feedback will be provided on returned scripts within two weeks of submission and in tutorials.

1. Forster, Lectures on Riemann Surfaces
2. Girondo and González-Diez, Introduction to Compact Riemann Surfaces and Dessins d’Enfants
3. Miranda, Algebraic Curves and Riemann Surfaces