Institute for Applied Mathematics  / Probability Theory and Stochastic Analysis  / Eveliina Peltola
University of Bonn Institute for Applied Mathematics
 
NoLinebreak

V5F3 - Advanced Topics in Stochastic Analysis:

Introduction to Schramm-Loewner Evolution 

Course announcement

WS19/20: Mondays 12–14 and Thursdays 8–10.  Endenicher Allee 60 - SemR 1.008

---

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, Koebe distortion theorem, ...
  • 21.10: Boundary properties of conformal maps, half-plane capacity, ...
  • 24.10: Recap of some notions from stochastic analysis. Conformal invariance of 2D Brownian motion, Kakutani's formula and harmonic functions, Beurling estimate
  • 28.10: Continuity and positivity of half-plane capacity, first steps towards Loewner theory of growth processes
  • 31.10: Discussion on Exercise Sets 1,2, and 3 (about complex analysis)
  • 04.11: Loewner's theorem, idea of the proof, examples, and consequences
  • 07.11: proof of Loewner's theorem
  • 11.11: finishing the proof of Loewner's theorem, definition of SLE and first properties
  • 14.11: Schramm's theorem, ideas from discrete models
  • 18.11: Bessel flow associated to SLE, SLE hitting the real line, phases of SLE
  • 21.11: Discussion on Exercise Sets 4,5, and 6
  • 25.11: transience of the SLE curve when 0 ≤ kappa < 8
  • 28.11: SLE swallowing points in the interior, transience of the space-filling SLE curve with kappa ≥ 8
  • 02.12: Schramm's formula, SLE Green's function (one-point function), sketch why SLE is a curve
  • 05.12: Brownian excursions, comparing Loewner evolution in different domains
  • 09.12: SLE in other domains, locality property (kappa = 6), Cardy's formula
  • 12.12: change of target point for SLE, SLE(kappa, rho), tilting SLE by (local) martingales involving Brownian loop measure
  • 16.12: probability of SLE avoiding a hull, conformal restriction property (kappa = 8/3)
  • 17.12: Discussion on Exercise Sets 7,8, and 9
  • 19.12: conformal restriction measures and related topics
  • 13.01: Definition and main properties of the Gaussian free field (GFF): discrete case
  • 16.01: GFF on planar domains as a random distribution, level lines of the GFF as SLE(4) curves
  • 20.01: proof of the GFF - SLE(4) coupling, Step 1: the coupling
  • 23.01: proof of the GFF - SLE(4) coupling, Step 2: SLE(4) is determined by the GFF
  • 27.01: further relations of the GFF and SLE
  • 30:01: Discussion on Exercise Sets 9,10, and 11

 

Exercises:

  1. Exercise Set 1
  2. Exercise Set 2
  3. Exercise Set 3
  4. Exercise Set 4
  5. Exercise Set 5
  6. Exercise Set 6
  7. Exercise Set 7
  8. Exercise Set 8
  9. Exercise Set 9
  10. Exercise Set 10
  11. Exercise Set 11

 

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

firstname.lastname @ hcm.uni-bonn.de

 

Literary Remarks:

We are not following any source directly. Below appear some pointers to find most of the material covered so far. (In the lectures, some additional results might have been discussed and references were given there.)

 

[Ahl]: Lars Ahlfors. Complex Analysis.

[Con]: John Conway. Functions of One Complex Variable, Volumes 1-2

[BN]: Nathanael Berestycki and James Norris. Lecture notes on SLE

[Kem]: Antti Kemppainen. Schramm-Loewner evolution

[Law]: Gregory Lawler. Conformally Invariant Processes in the Plane

[Wer]: Wendelin Werner. Random planar curves and Schramm-Loewner evolutions.