Lecture Publication Series PCMI Math Forum Archive 2003 Program About the Program


IAS/Park City Mathematics Institute
Undergraduate Program 2003

The Undergraduate Program receives major funding from the National Security Agency.

The Undergraduate Program provides opportunities for talented undergraduate students to enhance their interest in mathematics. This program is open to undergraduates at all levels, from first-year students to those who have just completed their undergraduate education. Based on the backgrounds of the accepted students, they will be divided into two groups - introductory and advanced. There will be several activities organized for these groups, with some specifically intended for either the introductory or advanced group. There will be time for study groups and individual projects guided by advisors, as well as other activities.

2003 Course Descriptions:

1. Introductory Course: Fourier Analysis and its many uses
Thomas W. Körner, University of Cambridge (UK)

This will be a first introduction. I hope to consider Fourier transforms for the circle, the line and the group of $n$-th units and show how they are applied to such things as compact discs, telescopes and multiplying large numbers. Students who can read the expression \[\int_{0}^{2\pi}f(t)e^{int}\,dt\] without blanching have sufficient background for the course.

2. Advanced Course: Introduction to Wavelets
Lesley Ward, Harvey Mudd College.

This course will be a mathematical introduction to the study of wavelets. Wavelets provide beautiful connections between many fields, from (mathematical) harmonic analysis to (engineering) signal processing. In Fourier theory, we express functions as sums of sines and cosines. Wavelets give a new way to express functions as sums of shifted and compressed versions of a single function, the wavelet function.

 We will study how wavelets come about, their properties, different ways to write and compute them, and some of their many applications in mathematics and signal processing. We will be especially interested in the interplay between the discrete and continuous representations of wavelets.

 Specific topics will include:

1. The Discrete Wavelet Transform (DWT) and the Fast Wavelet Transform (FWT);
2. The pyramid algorithm and the matrix factorization formulation of these transforms;
3. The filter bank representation of wavelets (this is how they are thought of in signal processing);
4. The Dilation Equation: a bridge between the discrete and continuous manifestations of a given wavelet;
5. Proof that each finite set of filter coefficients corresponds to a unique, compactly supported wavelet function and scaling function;
6. Construction of these functions from the filter coefficients, using the Cascade Algorithm;
7. Applications such as the FBI Fingerprint Compression Standard.

Send questions and comments to C. Giesbrecht