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 Hugoís 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 Zinsmeisterís notes),the zipper algorithm (partly following Michel Zinsmeisterís 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