MAT 401: Seminar in Mathematics (Conformal Mapping)

Fall 2009

Christopher Bishop

Professor, Mathematics
SUNY Stony Brook

Office: 4-112 Mathematics Building
Phone: (516)-632-8274
Dept. Phone: (516)-632-8290
FAX: (516)-632-7631


A conformal map between planar domains is one that preserves angles. The Riemann mapping theorem states that there is such a map between any two simply connected proper plane domains, and this is one of the most surprising and most important results in complex analysis. In this class we will introduce some basic ideas from complex analysis and hyperbolic geometry, give at least one proof of the Riemann mapping theorem, discuss its history and applications and study at least one algorithm for numerically computing it.

We will not use an assigned text, but I will hand out copies of book chapters, articles and my own lecture notes. I will give a few lectures to get the ball rolling, but within a week or two I will assign a few pages of reading to each student to prepare and lecture on. Near the end of the semester each student will prepare notes and a lecture on a topic of their choosing. This could be on a computer experiment you have performed, some research into the history or applications of conformal mappings, or an exposition on some theorem about conformal maps or their generalizations. There are numerous possibilities and I will help each student find an appropriate topic.

Grades will be based on homework problems I assign, the presentations you make in class, and the written version of your final presentation. I will start by giving a few lectures on the initial sections, but after a week or two will assign topics to students to present in class.


A rough draft of Chapters 1-3 of the notes The full set of notes can be found on the webpage for MAT 626 in Fall 2009

Some pictures

Pictures of power maps

History of Riemann mapping theorem by J. Gray

The Bieberbach conjecture by P. Zorn

article on the Bieberbach conjecture by J. Korevaar

a one page proof of the Bierbach conjecture

The CRDT algorithm by T. Driscoll and S. Vavasis

Function theory 1897-1932 by Bottazzini and Gray

The Riemann mapping problem by P. Ullrich

A history of the Riemann mapping problem by J. Walsh

T. Hales paper on a formal proof of the Jordan curve

T. Hales paper defending Jordan's proof of the Jordan curve theorem

Louis Howell's PhD thesis on SC mappings

paper of Banjai and Trefethen on Davis's method and mulitpole methods to handle large numbers of sides

paper of Banjai and Trefethen using numerical methods to examine the omitted area problem (what is the maximum area that can be omitted fromt the unit disk by a conformal map f so that f(0) =0 and f'(0) =1)

paper of Hale and Trefethen on quadrature methods motivated by conformal maps

paper of Trefethen on comparing Gauss quadrature to a simplier method that is as good in practice

paper of Trefethen on using SC maps to design electical resistors with desired resistance.

list of Trefethen's online papers

paper of mine on meshing using conformal maps

paper of mine about a quick approsimation to Schwarz-Christoffel parameters

SIAM article about Schwarz-Christoffel mapping for multiply connected domains

Introduction to Driscoll-Trefethen book; gives a history of the Schwarz-Christoffel mapping

Article by Feiszli and Mumford about using conformal maps in computer vision and shape recognition

Article by Oyma proving Hayman-Wu Theorem

Article by Oyma giving lower bound for Hayman-Wu constant

Survey article by Ken Stepenson on circle packing and conformal maps

webpage for Don Marshall's ZIPPER program

webpage for Toby Driscoll's SC toolbox for MATLAB

Survey of numerical conformal mapping by R.M.Porter

Hyperbolic metric and geometric function theory by Beardon and Minda

Geometric properties of hyperbolic geodesics by Ma and Minda

Hyperbolic-Type metric by Henri Linden

Introduction to QC mappings in n-space by Antti Rasila

Introduction to QC mappings in plane by Mercer and Stakewitz

International Workshop on QC mappings, Madras 2007 The last few papers were from a conference on QC mappings. The full list of downloadable papers from the converence is given here.

The uniformization theorem by William Abikoff

Fast transforms Thesis by Z. Tang

A Riemann mapping bibliography

Sinc Site Dates

We have reservations for the Math SINC site during classtime on the following dates: Oct. 2, 16, 30, Nov. 13, 25, Dec. 9, 11 from 9:30 - 10:30am. I will either present numerical examples in these sessions, or student my present experiments they have done.

Mathematica notebooks

If you click on one of these you will see the Mathematica code written in a Notebook format, which is rather verbose compared to the original form wrote it in. It would be nice if when you clicked onthelink, Mathematica automatically opened, but I don't known how to arrange this easily (.nb is not a recoginzed file type). Instead, right click on the link and save the file to your favorite directory, then in that directory type `mathematica filename'. Alernatively, you can start running Mathematica, and then open the file using the `open' option on the pull down menu.

Once you have the notebook open in Mathematica, select a cell with the mouse and then type `Shift-Enter'. Mathematica should evaluate the contents of that cell.

SquareGrid.nb draws a square grid and image of grid under power map

RectangleMap.nb Computes image of circle and rays in unit disk under conformal map to square or rectangle. THe map is a truncation of the power series of the map. The series is derived from the Schwarz-Christoffel map and using the binomial theorem to expand the factors in the SC formula.

PolarGrid.nb draws a square grid and image of grid under power map

Snowflake.nb draws the von Koch snowflake

Snowflake2.nb Alters the von Koch snowflake constuction to give Peano type curve

SCpowerseries.nb Computes the power series for a Schwarz-Christoffel functions and plots results.

Kakutani.nb estimates SC-parameters using random walks in polygon. You enter the vertices, the starting point, the number of random walks and the tolerence (distance from boundary when walk is said to have hit) and it returns plot of the polygon and the SC-image using the approximated harmonic measure.

A possible outline of topics:

Complex numbers
Linear fractional maps
Hyperbolic geometry
Holomorphic functions
Conformal maps
Brief history of Riemann's theorem
A proof of Riemann's theorem
The Schwarz-Christoffel formula
Davis's method for computing maps
Delaunay triangulations
The CRDT algorithm of Driscoll and Vavasis
The medial axis

Some possible topics for student presentations:

The quasi-hyperbolic metric
Why Riemann didn't prove Riemann's theorem
Osgoods' proof of Riemann's theorem
The uniformization theorem
Koebe's circle domain theorem
Applications to meshing
Applications to PDE's
Modulus and cross ratio
The iota map
Circle packing
Marshall's zipper program
Fornberg's method
Gauss-Jacobi quadrature
Carathodory's theorem
Kakutani's method
Symm's method
The Kertzman-Stein formula
Theodorson's method
Liouville's theorem in higher dimensions

Send the lecturer (C. Bishop) email at: bishop - at -

Send email to the whole class ((C. Bishop and students) class list

Week of Aug 31:
Monday - Introduction to the class, vague idea what conformal mappings are.
Wednesday - Introduction to complex numbers. Read chapter 1 of Churchill and Brown. We will discuss it in class and I will ask for students to work some problems on the board.
Friday - I reviewed the definiton of conformal map and 4 students made presentations.

Week of Sep 7
Monday - No Class (Labore Day)
Wednesday - We will discuss assignments and introduce some functions of a complex variable: powers, exponentials, logs,...

Week of Sep 14

Week of Sep 21

Week of Sep 28

Week of Oct 5

Week of Oct 12

Week of Oct 19

Week of Oct 26

Week of Nov 2

Week of Nov 9

Week of Nov 16

Week of Nov 23, Thanksgiving week

Week of Nov 30

Week of Dec 7, Last week of classes

Scheduled final: 11:15-1:45 Monday, Dec 14

Link to history of mathematics There are a lot of iteresting articles here. If you know of other math related sites I should link to, let me know.

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