Description

Loewner's theorem is a correspondence between curves in the plane (satisfying certain natural conditions) and real-valued functions of time. Roughly speaking, the behaviour of these "driving functions" encode how much and in what direction the tip of the curve moves. The aim of the project would be to study this relationship, and then look more deeply at further topics, for example: regularity of the curves; conformal welding problems; Schramm-Loewner evolutions.

Prerequisites

Complex analysis II is essential

Resources

To get an idea of the topic, search the internet for "Riemann mapping theorem" and "Schramm-Loewner evolution".

Further reference material:

• G. Lawler's book Conformally invariant processes in the plane , see a short version here
• N. Berestycki and J.Norris, Lectures on Schramm-Loewner evolutions .

Get in touch by email if you have questions.

Note: the supervisor will be travelling in Epiphany term, so meetings in this term will be online .