Math 613 — Topics in Analysis: Conformal Mapping and Univalent Functions

Fall 2002

Instructor: Greg Lawler

Time: TR 2:55-4:10

Room: Malott 230

Prerequisite: Graduate real and complex analysis (Math 611-612 or the equivalent). Exposure to graduate probability will be useful but not required.

This will be a second course in complex variables focusing on the theory of conformal mappings of the plane. The starting point will be a review of the Riemann mapping theorem followed by: Koebe 1/4 Theorem, distortion theorems, extremal distance (extremal length), harmonic measure and boundary behavior (Beurling projection theorem, Makarov's theorem), Loewner differential equation and applications to Bieberbach conjecture. If there is time, we may discuss deBranges' proof of the Bieberbach conjecture.

Much of this material is classical. One reference text for this will be Duren, Univalent Functions. We will depart somewhat from classical treatments by using Brownian motion in some of our proofs. (Previous knowledge of Brownian motion is not required).

My interest in this material stems from work on conformally invariant limits of lattice models in statistical mechanics. I am scheduled to teach a course in the spring (Math 778) on the Stochastic Loewner evolution, which combines ideas from complex variables and probability.

I plan to prepare notes for the courses which will (I hope) turn into a book.