Prerequisites
Basic knowledge of complex function theory is assumed, even though the main ingredients from that field (in particular the Riemann mapping theorem and related results) will be briefly presented during the course. It will also be assumed that the student is familiar with the basic properties of Brownian motion, and familiarity with martingales is desirable (the Mastermath course "Stochastic Processes" would be a good preparation w.r.t. these subjects). Some knowledge of stochastic calculus (Ito's formula and related techniques) is very helpful but not a crucial prerequisite: a short introduction to that topic will be included in the course.
Finally, maybe the most important is a good level of mathematical maturity, curiosity and intuition.
Aim of the course
Introduction
Stochastic Loewner Evolution (SLE), introduced by Schramm around 1999, and developed further by him with Lawler and Werner (Fields medal 2006), is a random fractal curve in the plane. It has been extremely useful for analysing probabilistic models of important physical processes with critical behaviour in the plane. Examples of such processes are percolation, loop-erased random walks, and systems of magnetic particles.
A main ingredient in SLE is the deterministic classical work by Loewner. That work provides an interesting technique to encode a growing curve in the upper half-plane, in terms of a real-valued 'driving function' (which may be interpreted as a moving particle on the real line). In this encoding, conformal maps on the complex plane are central. This work received new widespread interest, beyond pure mathematics, when Schramm initiated the study of such curves with a Brownian motion as driving function.
Course plan
The first few lectures will present and explain several results from complex function theory (in particular conformal mapping) which will be used later. Next, the notions of 'hulls', and capacity of hulls, will be introduced, followed by the deterministic Loewner evolution (and differential equation). Then some necessary results from stochastic calculus will be introduced, after which the Stochastic Loewner Evolution is studied. The SLE has a parameter, and it will be explained that changing the parameter can give rise to a drastic and surprising change of the qualitative behaviour of the process. Finally, one or two applications of SLE will be highlighted.
Lecturers
Rob van den Berg (VU and CWI), and Henk Don (Radboud Universiteit Nijmegen)
- Docent: Henk Don
- Docent: Rob van den Berg