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

Contact participants

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.

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:


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.

MSRI Workshops

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 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.

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.
        "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.

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.

Lecture Notes

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.

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.


Occasionally I will illustrate an idea using a program, usually in MATLAB. These are listed here. If you have a program you would like to share send me a copy to place here, or a link to post.

Lecture Recordings

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 here .

Lecture Schedule

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 Recording 1
        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.
        Due to Scott Sheffield's lecture 11-11:50 (NY time) at MSRI, class will run 12-12:50pm today.

Tuesday, Feb 1 Recording 2
        Finish introductory slides. How the DLA pictures are drawn. Some open problems about Brownian motion.

Thursday, Feb 3 Recording 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 Recording 5
        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 Recording 6
        Finish hypebolic metric, Gehring-Hayman theorem.

Thursday, Feb 17 Recording 7
        boundary continuity of conformal maps, Caratheodory-Torhorst theorem

Tuesday, Feb 22 Pre-recorded lecture, no live class meeting Recording 8
        Log capacity, definition, propeties

Thursday, Feb 24 Pre-recorded lecture, no live class meeting Recording 9
        Existence of equilibrium measure, Pfluger's theorem

Tuesday, March 1 Recording 10
        Harmonic measure, Beurling's estimate

Thursday, March 3 Recording 11
        Kesten's theorem on growth rate of DLA

Tuesday, March 8 Recording 12
        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 Recording 13
        Basic properties, nowhere differentiable

Tuesday, March 15
        No class -- Spring Break

Thursday, March 17
        No class -- Spring Break

Tuesday, March 22 Recording 14
        Dimension of graph, dimension of trace

Thursday, March 24 Recording 15
        Stopping times, Markov property, Wald's lemma

Tuesday, March 29 Recording 16
        Area of Brownian motion, Law of the iterated logarithm,

Thursday, March 31 Recording 17
        Strong law of large numbers, Dirichlet problem, recurrence in dimension 2

Tuesday, April 5 Recording 18
        Guest lecture by Amanda Turner on DLA
        Her slides

Thursday, April 7 No live class today. Lecture is pre-reocrded. Recording 19
        Conformal invariance of Brownian paths

Tuesday, April 12 Recording 20
        Start Part III: Martigales and Makarov's theorems
        Dyadic martingales

Thursday, April 14 Recording 21
        Limit theorems for martingales, Bloch harmonic functions and Bloch martingales

Tuesday, April 19 Recording 22
        Makarov's theorem: harmonic measure has dimension at most 1

Thursday, April 21 Recording 23
        Harmonic measure has dimension at least 1, weak version of LIL for dyadic martingales

Tuesday, April 26 Pre-recorded lecture Recording 24
        Makarov's LIL is sharp

Thursday, April 28 Pre-recorded lecture Recording 25
        From quasidisks to Jordan curves

Tuesday, May 3 Recording 26
        F. and M. Riesz Theorem, rectifiable domains

Thursday, May 5 Recording 27
        McMillan's twist point theorem, singularity of harmonic measure

Related readings

Some papers related to the topics of the class can be found here.


Technology Support: Student Technology Services. TLT provides academic technology support to all students. If you require assistance with Blackboard or other academic technologies, please contact TLT at:; Phone: 631.632.9602; Chat; or visit a SINC Site.
Students who need assistance with their personal devices can contact DoIT's service desk at: 631.632.9800, submit an online request, or visit the Walk In Center on the 5th floor of the Melville Library (West Campus), Room S-5410. For more information, visit:

The University Senate Undergraduate and Graduate Councils have authorized that the following required statements appear in all teaching syllabi (graduate and undergraduate courses) on the Stony Brook Campus.

DISABILITY SUPPORT SERVICES (DSS) STATEMENT: If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact Disability Support Services at (631) 632-6748 or They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.

ACADEMIC INTEGRITY STATEMENT: Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Facultyare required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology and Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website at

CRITICAL INCIDENT MANAGEMENT: Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures.

lso try to present some examples and applications not discussed in