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# 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.