MAT 627: Topics in Complex Analysis: Conformal Fractals,
Office: 4-112 Mathematics Building
Dept. Phone: (631)-632-8290
There is a contact list on the page obtained by
following this page's address by "email.html". There
are three email links: email me (Bishop), email enrolled
students (mostly for me to communcate to the students) and email
all interested parties (for me to send general announcements or
for participants to share comments, papers, links,...)
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.
Times and places
First class is Thursday, January 27, 2022. (No class Tuesday due to MSRI lectures.)
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.
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.
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
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
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
SLIDES ON FUNCTION THEORY AND DLA
MSRI Colloquia and Seminars
Link to upcoming seminars and colloquia at MSRI
Most of these require registration to get email link, and this,
inturn, requires you create a MSRI account. Some seminars are
listed as for students and postdocs only; no faculty.
As mentioned above the topics in this class are paratly
inspried by two programs at MSRI this semester:
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
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.
Jan 19-21 MSRI Connections Workshop on Random Sets
The workshop schedule is posted at
Jan 19-21 schedule.
The most relvant talk to our class are
"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.
Jan 24-28 MSRI Introductory Workshop on Random Sets
The schedule for this week is posted at
Jan 24-28 schedule.
There numerous talks that are interesting and relevant to us.
A few highlights are
(times are in Berkeley; add 3 hours for NY):
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.
Nikolai Makarov, 8am (11am NY) Tue Jan 25.
"Conformal Welding in Liouville Quantum Gravity",
Nina Holden, 9am (noon NY) Tue Jan 25.
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.
Feb 2-4, MSRI Connections Workshop on Holomorphic Dynamics
The schedule for this week is posted at
Feb 2-4 schedule.
Feb 8-17, MSRI Introductory Workshop on Holomorphic Dynamics
The schedule for this two week workshop is posted at
Feb 8-17 schedule.
Here are some
notes on extremal length,
harmonic measure and DLA. We will follow these (with some additions and
deviations) for the first few weeks.
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.
Occasionally I will illustrate an idea using a program, usually in MATLAB. These
here. If you have a program you would like to share
send me a copy to place here, or a link to post.
Recording links in the Zoom cloud are posted next to each date below.
The passcodes will all be
Kesten<2/3. These will expire 180 days after the recording was made.
A list of recordings stored more permanently at Stony Brook as MP4 files is posted
Below I list a tentative schedule of lecture topics. This will be
extended as the semester proceeds (and I hope that our meetings
will be more like discussions than lectures).
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.
Tuesday, Jan 25
Due to Makarov's and Holden's lectures
(11-11:50 and 12:00-12:50
NY time) at MSRI, we will not meet today.
Students are requested to watch one or both of these talks.
Thursday, Jan 27
First class: introduction to DLA and some other random sets.
Here are the
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.
Due to Scott Sheffield's lecture 11-11:50 (NY time) at MSRI, class
will run 12-12:50pm today.
Tuesday, Feb 1
Finish introductory slides. How the DLA pictures
are drawn. Some open problems about Brownian motion.
Thursday, Feb 3
Start Part I: Function theory and DLA
Modulus and Extremal length
Tuesday, Feb 8
Recording 4 (contains short gap due to internet problem)
Symmetry and Koebe's (1/4)-theorem,
Thursday, Feb 10
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.
Tuesday, Feb 15
Finish hypebolic metric, Gehring-Hayman theorem.
Thursday, Feb 17
boundary continuity of conformal maps, Caratheodory-Torhorst theorem
Tuesday, Feb 22 Pre-recorded lecture, no live class meeting
Log capacity, definition, propeties
Thursday, Feb 24 Pre-recorded lecture, no live class meeting
Existence of equilibrium measure, Pfluger's theorem
Tuesday, March 1
Harmonic measure, Beurling's estimate
Thursday, March 3
Kesten's theorem on growth rate of DLA
Tuesday, March 8
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
Thursday, March 10
Basic properties, nowhere differentiable
Tuesday, March 15
No class -- Spring Break
Thursday, March 17
No class -- Spring Break
Tuesday, March 22
Dimension of graph, dimension of trace
Thursday, March 24
Stopping times, Markov property, Wald's lemma
Tuesday, March 29
Area of Brownian motion,
Law of the iterated logarithm,
Thursday, March 31
Strong law of large numbers,
Dirichlet problem, recurrence in dimension 2
Tuesday, April 5
Guest lecture by Amanda Turner on DLA
Thursday, April 7 No live class today. Lecture is
Conformal invariance of Brownian paths
Tuesday, April 12
Start Part III: Martigales and Makarov's theorems
Thursday, April 14
Limit theorems for martingales, Bloch harmonic functions and Bloch martingales
Tuesday, April 19
Makarov's theorem: harmonic measure has dimension at most 1
Thursday, April 21
Harmonic measure has dimension at least 1, weak version
of LIL for dyadic martingales
Tuesday, April 26 Pre-recorded lecture
Makarov's LIL is sharp
Thursday, April 28 Pre-recorded lecture
From quasidisks to Jordan curves
Tuesday, May 3
F. and M. Riesz Theorem, rectifiable domains
Thursday, May 5
McMillan's twist point theorem, singularity of harmonic measure
Some papers related to the topics of the class can be found
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lso try to present some examples and applications not discussed