Office: 4-112 Mathematics Building
Dept. Phone: (516)-632-8290
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
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. TangA Riemann mapping bibliography
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.
Linear fractional maps
Brief history of Riemann's theorem
A proof of Riemann's theorem
The Schwarz-Christoffel formula
Davis's method for computing maps
The CRDT algorithm of Driscoll and Vavasis
The medial axis
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
Marshall's zipper program
Demo of SCTOOLBOX
The Kertzman-Stein formula
Liouville's theorem in higher dimensions
Send the lecturer (C. Bishop) email 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.