MAT 627: Topics in Complex Analysis: Conformal Fractals, Spring 2022

Prof. Christopher Bishop
Dept of Mathematics
Stony Brook University

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

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Times and places

Course description

The course is motivated by topics in two MSRI programs this semester. One is on geometry of random sets and the other is on holomorphic dynamics. My plan is to cover some basic elements of geometric function theory and then apply these ideas to prove some well known results involving conformal maps, harmonic measure and fractals. Each is of these is associated to famous problems that remain open.
        (1) Kesten's theorem on the growth rate of DLA (diffusion limited aggregation). The sharp rate of growth is unknown and no non-trivial lower bound is known at all.
        (2) A brief introduction to Brownian motion. Our main goals are to construct Brownian, motion and learn enough about it to derive the law of the iterated logarithm (LIL) and prove the conformal invariance of Browian paths. All details are given in my book with Peres (link below), so some parts may only be sketched in class.
        (3) Makarov's theorem that the dimension of harmonic measure on simply connected plane domains is 1. Here we will make use of the LIL for martigales, which can be deduced from the one for Brownian motion. There have been some recent breakthrough's in higher dimensions related to harmonic measure, but the extension of Makarov's bound to dimensions larger than 2 is still open. If time permits, I may describe the LIL for Kleinian groups by Peter Jones and myself, or the Jones-Wolff theorem that harmonic measure has dimension at most 1 for all planar domains.

I am planning to run the course online so that participants at MSRI can also attend if they wish (although 11:30 here is 8:30 in Berkeley). Perhaps we can also view and discuss some of the lectures at MSRI as part of this class. As a topics class, the grade is based on participation. There will not exams or problem sets for advanced doctoral students, although undergraduates or first year graduate students who are attending for credit will be expected to do some problem sets; talk to me if you are in this situation.

The course will generally follow some lecture notes of mine, which I will update throughout the semester.

        Conformal Fractals (Chapters on DLA and Makarov's theorems)

        Fractals in probability and analysis

For improved readablility during class, I will convert the material in these preprints into a landscape format with larger print. I will follow these:

        SLIDES ON FUNCTION THEORY AND DLA

MSRI Colloquia and Seminars

MSRI Workshops

Jan 19-21 MSRI Connections Workshop on Random Sets

Jan 24-28 MSRI Introductory Workshop on Random Sets

Feb 2-4, MSRI Connections Workshop on Holomorphic Dynamics

Feb 8-17, MSRI Introductory Workshop on Holomorphic Dynamics

Lecture Notes

Code

Lecture Recordings

Lecture Schedule

Related readings

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