SLE seminarOrganizer: Manjunath KrishnapurSchedule: Introductory lecure - May 6th, Thursday. 2:00-3:15 PM. |
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. |
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. |
Speaker | |
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 |