MAT 670: Topics in Complex Analysis:
Dessins and Dynamics (introduction to quasiconformal folding)
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

Time and location

At present (June 2023) the course is scheduled to meet in person: TuTh 11:30-12:50pm in Physics P-117 (moved from original room Frey Hall 224). Over the summer I will try to guage whether there is interest in holding the class online, or at least recording class meetings.

First class is Tuesday, August 29, 2023.

Last class is Thursday, December 7, 2023.

No class on October 10 (Fall break) and November 23 (Thanksgiving break).

There is no final exam for this class.

Relevant sources

        "Introduction to Compact Riemann Surfaces and Dessins d'Enfants" by E. Girondo and G. Gonzalex-Diez, CUP, LMS Student Texts 79, 2012
        "Dessins d'Enfants on Riemann Surfaces", G.A. Jones and J. Wolfart, Springer Monographs in Mathematics, 2016.
        "The Grothendieck Theory of Dessins d'Enfants", edited by L. Schneps, LMS Lecture Note Series 200, CUP, 1994.
        Prescribing the Postsingular Dynamics of Meromorphic Functions, C. Bishop and Kirill Lazebnik, Math. Annalen., 365(3), 2019, 1761--1782. PREPRINT
        Constructing entire functions with quasiconformal folding, C,Bishop, Acta. Math. 214:1(2015) 1-60. PREPRINT
        True trees are dense, C.Bishop Inventiones Mat., vol 197, issue 2, 2014, pages 433-452. PREPRINT
        Models for the Eremenko-Lyubich class, C. Bishop, Journal of the London Math. Society., 92(2015), no 1, 202-221 PREPRINT
        Models for the Speiser class, C. Bishop, Procedings of the London Mathematical Society, (3) 114 (2017), no5, 765-797. PREPRINT
        A transcendental Julia set of dimension 1 , pdf , Inventiones Math., 212(2) 407--460, 2018. PAPER
        The Uniformization Theorem, by Don Marshall PREPRINT
        Structure theorems for Riemann and topological surfaces, V. Alvarez and J.M. Rodriguez, JLMS 69(2004), no 2, 153--168, PAPER
        "The geometry of discrete groups", Alan Beardon, 1983, Springer-Verlag Graduate Texts in Math 91.
        Entire functions arising from trees, Weiwei Cui, Science China, Oct 2021, vol 64, no. 10, 2231-2248, PAPER

Course description

Summary: We will start with an introduction to Grothendieck's theory of dessins d'enfants (children's drawings) and some connections to holomophic dynamics. A dessin is a finite graph (of a certain type) drawn on a compact topological surface and it induces a conformal structure, making the surface into a Riemann surface. This is based on Belyi's theorem: a Riemann surface is defined over the algebraic numbers iff it supports a meromorphic function branched over three points. Among the topics we might discuss are:
        - Compact surfaces and algebraic surfaces,
        - the uniformization theorem for Riemann surfaces,
        - Belyi's theorem and equilateral triangulations of Riemann surface
        - harmonic measure, conformal maps, Brownian motion,
        - quasiconformal mappings, measurable Riemann mapping theorem
        - finite "true trees" in the plane and Shabat polynomials, true trees are dense in all continua
        - extension to infinite planar trees (QC folding),
        - applications to dynamics: wandering domains, postcritical orbits, dimension of Julia sets
        - applications to polynomial and rational approximation,
        - an extension of Hilbert's lemniscate theorem
        - distribution of algebraic compact surfaces in moduli space

We will follow some recent papers and selected chapters of some textbooks. Prerequisites are the core courses in real and complex analysis; MAT 538 on Riemann surfaces will be helpful, but I will review facts about Riemann surfaces as we need them.

This is a topics class for PhD students. Others require permission of the instructor and graduate director to enroll, and must agree to do problems sets and exams on the material. It may be preferable to do this as an independent study course.

Barcelona Lectures

Most of the topics we will discuss in the course were touched on in a series of two lectures I gave at a conference in Barcelona during April 2021. The slides and a video recording are posted below. We will go through the slides in the first few lectures of the course to give an overview, and later return, with greater detail and background, to topics of interest to the participants.
        Slides for Trees, Triangles and Tracts, Part I
        Slides for Trees, Triangles and Tracts, Part II .
        Link to Video Recording . (one YouTube video covers both talks).
        Link to conference website: Transcendental Dynamics and Beyond: topics in complex dynamics 2021, (Centre de Recerca Matem├ática, Barcelona, April 19-23, 2021,

Course Lecture Slides

A short set of SLIDES on true form of truncated, 3-regular tree. The infinite 3-regular tree does not have a true form in the plane (but it does in the disk).

Steffen Rohde's SLIDES from the August 2023 Quasiworld Workshop in Helsinki. These discuss the true form of truncations of the infinite 3-reguler tree, and outline an alternate proof of my "true trees are dense" theorem.

Preliminary SLIDES on Riemann surfaces, uniformization and Belyi's theorem.

Preliminary SLIDES on Extremal length and quasiconformal mappings. Sketch of proof of Measurable Riemann Mapping Theorem. Estimates for QC maps, removability.

Preliminary SLIDES on Quasicoformal Folding and some applications.

Dimensions of transcendental Julia sets, last day lecture (Dec 7, 2023). (I did not give this lecture; Kirill Lazebnik gave a guest lecture on rational approximation instead.)

Course Lecture Recordings

Class meets in person at Stony Brook, but I am using slides projected on a screen n the class room and streaming/recording lectures via Zoom. Email me for the Zoom Address/Passcode.

Zoom meeting ID is 993 5153 2637. Email me if you need the passcode.

Click HERE for links to Zoom recordings of previous lectures.

Lecture Schedule

Below I list a tentative schedule of lecture topics. This will very likely change as the semester proceeds (and I hope that our meetings will be more like discussions than lectures).

Tuesday, Aug 29
        Introduction to the course

Thursday, Aug 31
        Introduction to dessins and dynamics (recap Barcelona lectures)

Tuesday, Sept 5
        Introduction to dessisn and dynamics (recap Barcelona lectures)

Thursday, Sept 7
        Introduction to dessisn and dynamics (recap Barcelona lectures)

Tuesday, Sept 12
        Introduction to dessisn and dynamics (recap Barcelona lectures)

Thursday, Sept 14
        Introduction to dessisn and dynamics (recap Barcelona lectures)

Tuesday, Sept 19
        Introduction to dessisn and dynamics (recap Barcelona lectures)

Thursday, Sept 21
        Class canceled

Tuesday, Sept 26
        Introduction to Riemann surfaces, state uniformization theorem

Thursday, Sept 28
        Start proof of uniformization theorem

Tuesday, Oct 3
        Finish uniformization theorem, Dipole Green's functions

Tuesday, Oct 5
        Riemann surfaces, fundamental domains, Y-pieces

Tuesday, Oct 10
        No class (Fall break)

Thursday, Oct 12
        equilateral triangulations, equivalence of compact Riemann surfaces and algebraic varieties

Tuesday, Oct 17
        Belyi's theorem, Part I, algebraic implies Belyi function

Thursday, Oct 19
        Belyi's theorem Part II, Belyi function implies algebraic (sketch)

Tuesday, Oct 24 Quasiconformal mappings I, defininitions

Thursday, Oct 26
        Quasiconformal mappings II, Equicontinuity

Tuesday, Oct 31
        Quasiconformal mappings III, 1-QC = conformal;

Thursday, Nov 2
        No class

Tuesday, Nov 7
        Quasiconformal mappings IV, Geometric definition implies analystic definition

Thursday, Nov 9
        Quasiconformal Mappings V, Proof of MRMT, Reverse Holder inequality

Tuesday, Nov 14
        QC Folding I

Thursday, Nov 16
        QC folding II

Tuesday, Nov 21
        QC Folding III (class will take place as usual)

Thursday, Nov 23
        No class (Thanksgiving)

Tuesday, Nov 28
        No class Wandering domains

Thursday, Nov 30
        Guest lecture: Lasse Rempe on equilateral triangulations of Riemann surfaces

Tuesday, Dec 5
        Guest lecture: Kirill Lazebnik on prescribing post-signular orbits

Thursday, Dec 7
        Gues lecture: Kirill Lazebnik on approximation via folding


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