Undergraduate research project

One of the most remarkable theorems of mathematics is the Riemann mapping theorem which says that any simply connected domain in the plane can be mapped conformally onto the unit disc. These maps are used in a variety of applications in physics and engineering. In a few special cases the map can be found explicitly, but in most applications one must compute a numerical approximation. The zipper algorithm constructs an approximation to the conformal map by composing a large number of simple maps. The goal of this project is to learn about conformal maps, the zipper algorithm and (for the ambitious) develop a much faster version of the zipper algorithm.

Depending on the student's interest, this project could last for a single semester or several semesters. The first semester would be spent learning about conformal maps, how the zipper algorithm works and applying this algorithm to some examples. There is a nice description of the zipper algorithm in a recent preprint by Stephen Rohde and Don Marshall.

Some pictures of conformal maps and Fortran code implementing the zipper algorithm may be downloaded from Don Marshall's web site. The code for the zipper algorithm can be downloaded and run relatively easily to produce some nice pictures of conformal maps.

Ambitious student(s) can continue after one semester to develop a faster zipper algorithm. The zipper algorithm is related to the Loewner differential equation which also produces conformal maps. In the last few years, there have been advances which greatly speed up finding solutions to the Loewner equation. These same ideas should lead to a much faster zipper algorith. The ultimate goal would be to produce code with a faster zipper algorithm that can be made available to the public (as is the current version of the zipper algorithm). The algorithm begins by approximating the boundary of the simply connected domain by a sequence of N points. The time the algorithm takes grows with N as N^2. The algorithm for the Loewner differential equation also takes a time that is of order N^2 to produce N points. However, the recent advances give an algorithm whose time is only of order N^1.4. It is expected that an equally dramatic improvement is possible for the zipper algorithm.

prerequisites: Familiarity with complex variables, e.g., Math 424 would be helpful, would is not essential. Conformal maps are not typically covered in our undergraduate courses, so it is expected that the student will begin by learning about these beautiful maps.