MAT 627, Topics in complex analysis: conformal mappings and hyperbolic geometry

Christopher Bishop

Professor, Mathematics
SUNY Stony Brook

Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631

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Time and place: M,F 11:20-12:40, Math 5-127

In this class we will read Greg Lawler's book ``Conformally invariant processes in the plane'' which deals with SLE (Schram-Loewner Evolutions). These are basically the classical Loewner's equation for conformal mapping onto slit domains, but with Brownian motion as a forcing term., The result is a conformally invariant class of simple random curves. Such objects have long been sought as the limits of well known, but poorly understood, discrete models for producing simple random walks (the key word is `simple'; Brownian motion provides the model for random paths with self-intersections allowed). This is a `hot' topic right now and Lawler's 2005 book provides up-to-date survey of what is known so far. We will attempt to cover enough of the book to get to the sections on Brownian intersection exponents which allow one to compute the Hausdorff dimension of various interesting subsets of the Brownian trace, such as the set of cut points and the boundary of the complementary components.

postscript code for drawing an approximate Brownian path

Mathematica code for drawing percolation on a square lattice

Conformally invariant scaling limits by Oded Schramm

SLE short course in Poland by Michel Zinsmeister