**Math 583, Conformal Invariance and Probability**

**Spring 2011**

Instructor: Steffen Rohde

Office Hours: by appointment in PDL-C337

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