Dept of Mathematics

Stony Brook University

Office: 4-112 Mathematics Building

Dept. Phone: (631)-632-8290

FAX: (631)-632-7631

In you are not enrolled in MAT 627, but would like to be on the "interested parties" list and get the class Zoom link, send me an email (even if you intend to only attend a few times). I will send out the link on Wed Jan 26, to the names I have collected by that point, and as needed thereafter.

Last class is Thursday, May 5, 2022.

No class on March 15 and 17 (Spring break).

Class time -- 11:30am-12:50pm (eastern time) Tu and Th (time may change)

Location -- online; enrolled may join via the link in Blackboard. Otherwise, contact me for the Zoom coordinates.

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

Analysis and Geometry of Random Shapes

Complex Dynamics: from special families to natural generalizations in one and several variables

Online participation is open to anyone that registers; registration requires giving an ORCID id (which is also needed is some other contexts; NSF may require it for grant proposals, I think). An ORCID ID number is free and easy to obtain at ORCID webpage MSRI lectures will generally be recorded and put in the MSRI video archive at MSRI video dashboard I will post links to talks that seem relevant to the class.

"Sobolev Spaces via Upper Gradients in Non-Smooth Setting", Nageswari Shanmugalingam, Wed Jan 19 at 9am (noon in NY).

"Scaling Limits of Laplacian Random Growth Models", Amanda Turner on Thur Jan 20 8:20am (11:30am in NY).

Turner's talk is on random growth models. The first goal of our class is study the best known example of such a growth model, DLA. The talk by Nageswari Shanmugalingam should also be accessible to any interested in analysis on fractals.

Short course "An Elementary Introduction to Multiplicative Chaos", Eero Saksman, 8am (11am NY) Mon Jan 24 and 11:40am (2:40pm NY) Wed Jan 26.

"Talk TBA", Nikolai Makarov, 8am (11am NY) Tue Jan 25.

"Conformal Welding in Liouville Quantum Gravity", Nina Holden, 9am (noon NY) Tue Jan 25.

"Talk TBA", Scott Sheffield, 8am (11am NY) Thur Jan 27.

Short course "Removability of Planar Sets", Malik Younsi, 10:20am (1:20pm NY) Thur Jan 27 and 9:50am (12:50pm NY) Fri Jan 28.

Introductory slides on DLA and some other random sets. Later, we will only treat DLA in detail by proving Kesten's theorem on the growth rate of DLA.

This will be adjusted if we fall behind, or substitute watching a lecture from MSRI for a regular class meeting. I expect that several times we will omit a "class meeting" in favor of watching a talk from MSRI live, or invite a MSRI participant to discuss their work with us.

Students are requested to watch one or both of these talks.

First class: introduction to DLA and some other random sets.

Here are the SLIDES I will use today.

Please watch Amanda Turner's Jan 20 lecture before this class (wideo link is on workshop schedule page).

She dicusses classic DLA, but also many other variants that are easier to work with, and explains the physical motivations for studying these random processes.

Here is an EXPOSITORY ARTICLE (pages 14-19) by Amanda Turner on DLA and DLA-like processes.

How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth by Denis S. Grebenkov and Dmitry Beliaev, Phys. Rev. E 96, 042159 – Published 30 October 2017. This shows numercially that DLA on a lattice forms a "fractal cross" at large scales. This won't happen for DLA based on Brownian motion in the plane, which must be roationally invariant.

Finish introductory slides. How the DLA pictures are drawn. Some open problems about Brownian motion.

Slides

Start Part I: Function theory and DLA

Modulus and Extremal length

Slides

Symmetry and Koebe's (1/4)-theorem,

Slides

hyperbolic metric, uniformization

Following the lecture, Martin Chuaqui, pointed out the paper of Ludwig Bieberbach "Eine singularitätenfreie Fläche konstanter negativer Krümmung in Hilbertschen Raum", Commentarii Mathematici Helvetici volume 4, pages 248–255 (1932). This constructs an embedding of the hyperbolic disk into Hilbert space, so that the hyperbolic isometries are just restrictions to the image of ambient isometries. The idea is to find a holomorphic mapping F=(f_1,f_2,...) of infinitely many components so that F*(euclid)=hyp, which happens if |f'_1|^2+|f_2'|^2+... = 1/(1-|z|^2)^2. I do not have a link to an online version the paper.

Slides

Finish hypebolic metric, Gehring-Hayman theorem.

Slides

boundary continuity of conformal maps, Caratheodory-Torhorst theorem

Slides

Log capacity, definition, propeties

Slides

Existence of equilibrium measure, Pfluger's theorem

Slides

Harmonic measure, Beurling's estimate

Slides

Kesten's theorem on growth rate of DLA

Slides

Start Part II: Introduction to Brownian Motion

Introduction to Brownian motion, Levy's construction

Alternate presentation of Levy's construction by Steven Lalley

Fractals in probability and analysis Slides are condensed version of Chapters 6 and 7 of this book

Brownian Motion by Morters and Peres; extended treatment of Brownian motion; more details

Slides

Basic properties, nowhere differentiable

No class -- Spring Break

No class -- Spring Break

Slides

Dimension of graph, dimension of trace

Slides

Stopping times, Markov property, Wald's lemma

Slides

Area of Brownian motion, Law of the iterated logarithm,

Slides

Strong law of large numbers, Dirichlet problem, recurrence in dimension 2

Guest lecture by Amanda Turner on DLA

Her slides

Slides

Conformal invariance of Brownian paths

Slides

Start Part III: Martigales and Makarov's theorems

Dyadic martingales

Slides

Limit theorems for martingales, Bloch harmonic functions and Bloch martingales

Slides

Makarov's theorem: harmonic measure has dimension at most 1

Slides

Harmonic measure has dimension at least 1, weak version of LIL for dyadic martingales

Slides

Makarov's LIL is sharp

Slides

From quasidisks to Jordan curves

Slides

F. and M. Riesz Theorem, rectifiable domains

Slides

McMillan's twist point theorem, singularity of harmonic measure

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