SLE seminar

Organizer: Manjunath Krishnapur

Schedule: Introductory lecure - May 6th, Thursday. 2:00-3:15 PM.

June meetings: 14, 16, 18, 21, 23, 25. From 11 AM to 12:30 PM.
Sept-Nov meetings: Yet to be decided.

Objective: To give an introduction to Schramm-Löwner evolution with fairly complete details. We start with the complex analysis background needed beyond a first course in complex analysis. Derive Löwner equation for slit mappings. Define SLE. Derive basic properties of SLE - transience, locality, restriction, and the associated martingales. Make a link with discrete models of statistical mechanics (Loop-erased random walk, Percolation, Self-avoiding walks, Ising model). If time permits, talk in depth about Smirnov's proof of conformal invariance of site percolation on triangular lattice or talk about restriction measures and Brownian loop soup.

Topics for June lectures [NOTES]
Overview of conformal invariance in statistical mechanical models
Boundary behaviour of conformal maps.
Harmonic measure, Beurling's projection theorem.
Half-plane capacity.
Löwner's differential equation - chordal version.

Topics for Sep-Nov lectures (tentative)
Definition of chordal SLE(κ). Statement about generation by a curve. Manjunath Krishnapur
Basic properties of SLE. Recurrence, Transience, Phase transition at κ=4. Siva Athreya
Bessel processes. Computation of hitting probabilities. Siva Athreya
Locality. Siva Athreya
SLE in a triangle. Cardy's formulas for SLE. Manjunath Krishnapur
Percolation on the hexagonal lattice. RSW theorem. Lecture notes (same notes, smaller file) Manjunath Krishnapur
Smirnov's proof of conformal invariance of critical percolation (on hexagonal lattice) Rajesh Sundaresan
SLE(6) as the limit of percolation interfaces. Sreekar Vadlamani
Description of ful scaling limit of percolation.Manjunath Krishnapur