Math 583, Conformal Invariance and Probability

Spring 2011


Instructor: Steffen Rohde

Office Hours: by appointment in PDL-C337

 

Syllabus

Exercise 1

Exercise 2

Exercise 3

Exercise 4

 

Exercise 5 (Autumn 2011)

Exercise 6

Exercise 7

 

Topics covered include

 

·         The self-avoiding walk  (Following Hugo Duminil Copin-Stas Smirnov: The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$, see also Hugo’s talk, and see the recent paper by Martin Klazar for some details, particularly concerning the winding)

·         Basic theory of conformal maps: Riemann mapping theorem, distortion theorems, (partly following Michel Zinsmeister’s notes),  the zipper algorithm (partly following Michel Zinsmeister’s notes)

·         The Loewner differential equation 

·         Basic theory of Brownian Motion

·         SLE, Schramm's principle (here is the link to the Java program of Joan Lind and her students, and here the Mathematica notebook discussed in class)

·         Basic Stochastic Calculus (Ito Integral, diffusions, Dynkin's formula; applications: Conformal invariance of BM, recurrence vs transience)

·         Path properties of the deterministic and the stochastic LE (Continuity, Phases, Transience, Dimensions; overview, few proofs)

·         SLE_6 (restriction property; conformally invariant measures) and Cardy's formula

·         Smirnov's Theorem (convergence of critical percolation interfaces to SLE_6)

·         SLE_{8/3} (restriction property; SAW)

·         Intersection exponents for BM, Mandelbrot conjecture, and the work of Lawler, Schramm and Werner