VIGRE Seminar, Spring 2003: Frames, Group Representations and Wavelets

Instructors: Ken Dykema, Keri Kornelson, Nico Spronk and with the help of Dave Larson

Students Enrolled: Ila Cobbs, Brady McCrary, Nathaniel Strawn, Ryan Westbrook (undergrad math majors) and Victor Ginting, Troy Henderson, Quynh Nguyen (math grad students).
Description: Group representation theory blends group theory, matrix algebra and geometrical analysis to study groups and their actions. It has plenty of applications to almost every area of mathematics, including the theory of wavelets and frames. We will introduce the topic of tight frames in finite dimensions, which is already accessible to students with good linear algebra skills. Many easily stated problems in frame theory, however, prove highly challenging and are of interest to the research community. Finite frames with equal norm components, for example, are the subject of research at places like Bell Laboratories due to their applications in signal processing. There is also a new formulation of the Kadison-Singer problem in terms of tight frames. This is a longstanding problem in operator theory, and is likely to be difficult to solve completely. However, the reformulation suggests the opportunity for research and partial results. We will present group representations and lead into a discussion of wandering vectors for unitary systems and frame representations. We will also look at positive definite functions on finite groups and introduce noncommutative Fourier analysis. This can serve as an access point to advanced topics in representation theory for infinite groups. These topics will serve as an excellent introduction for advanced courses in real analysis, operator theory, representation theory and harmonic analysis.