Dessins and Dynamics (introduction to quasiconformal folding)

Spring 2022

Dept of Mathematics

Stony Brook University

Office: 4-112 Mathematics Building

Dept. Phone: (631)-632-8290

FAX: (631)-632-7631

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.

"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

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.

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 Matemàtica, Barcelona, April 19-23, 2021,

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.)

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.

Introduction to the course

Introduction to dessins and dynamics (recap Barcelona lectures)

Introduction to dessisn and dynamics (recap Barcelona lectures)

Introduction to dessisn and dynamics (recap Barcelona lectures)

Introduction to dessisn and dynamics (recap Barcelona lectures)

Introduction to dessisn and dynamics (recap Barcelona lectures)

Introduction to dessisn and dynamics (recap Barcelona lectures)

Class canceled

Introduction to Riemann surfaces, state uniformization theorem

Start proof of uniformization theorem

Finish uniformization theorem, Dipole Green's functions

Riemann surfaces, fundamental domains, Y-pieces

No class (Fall break)

equilateral triangulations, equivalence of compact Riemann surfaces and algebraic varieties

Belyi's theorem, Part I, algebraic implies Belyi function

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

Quasiconformal mappings II, Equicontinuity

Quasiconformal mappings III, 1-QC = conformal;

No class

Quasiconformal mappings IV, Geometric definition implies analystic definition

Quasiconformal Mappings V, Proof of MRMT, Reverse Holder inequality

QC Folding I

QC folding II

QC Folding III (class will take place as usual)

No class (Thanksgiving)

No class Wandering domains

Guest lecture: Lasse Rempe on equilateral triangulations of Riemann surfaces

Guest lecture: Kirill Lazebnik on prescribing post-signular orbits

Gues lecture: Kirill Lazebnik on approximation via folding

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: https://it.stonybrook.edu/students

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