Instead of thinking of complex numbers as lying on a plane, it is much better use stereographic projection to put them on a sphere, called the Riemann sphere. The Möbius group acts on this sphere, giving a fascinating set of transformations, crucially important in many branches of mathematics. A natural development is to consider multi-valued complex functions, such as the logarithm, leading to the amazing concept of a Riemann surface. The more advanced ideas here are at the level of a masters project.
- Wikipedia Riemann sphere
- Needham, T 1997 Visual Complex Analysis Oxford University Press.
- Jones, G and Singerman, D 1987 Complex Functions Cambridge University Press.
Supervisor: Professor Stephen Huggett