## WS19/20

## I welcome everyone to:

### Advanced Topics in Stochastic Analysis - Introduction to Schramm-Loewner evolution

**Mondays 12–14 and Thursdays 8:30–10 **in Endenicher Allee 60 - SemR 1.008

**NB:** Monday lectures begin **12:15** and Thursday lectures begin **8:30** !

Schramm-Loewner Evolution (SLE) is a key concept when studying the geometry of random structures. It lies at the interface between probability, geometry, and analysis, combining beautiful theory of conformal mappings to stochastic analysis and properties of Brownian motion. The SLE is a random fractal curve in the plane, whose most famous applications include the understanding of geometric properties of statistical models in two dimensions (random walks, percolation, Ising model, Gaussian free field, ...).

The goal of the course is to provide an introduction to the definition, properties, and applications of the SLE.

We will cover some background material on complex analysis and stochastic analysis, when needed (the precise plan will depend on the participants’ background knowledge and wishes). Along the way, we develop Loewner’s theory for growth processes encoded in conformal maps. Then we define the SLE processes and prove their basic properties. Time permitting, we discuss further properties and applications of the SLE.

## Course Log:

Here appear rough descriptions of the past (and following) lectures.

- 07.10: Overview presentation, practicalities (for the slides, please contact me by email)
- 10.10: Refreshing some basics in complex analysis: holomorphic/analytic maps, their properties, maximum modulus principle, Schwarz lemma
- 14.10: Complex analysis continues: conformal maps, Möbius transformations, Riemann mapping thm, SLE(0)
- 17.10: Area theorem, Koebe 1/4 theorem, conformal invariance of 2D Brownian motion, Kakutani's formula

## Exercises:

Each week an exercise set is published here. The exercises are not mandatory but strongly advisable!

We will discuss the problems once in a while, tentatively on Thursday's lecture every 3 weeks.

If you have any questions, please don't hesitate to contact me:

*firstname.lastname @ hcm.uni-bonn.de*