Mathematical Sciences: Harmonic Analysis Algorithms

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Six students will be chosen to participate in this Research Experiences for Undergraduates project. Seven faculty members engaged in five research programs will introduce the students to a variety of possible directions on which to focus their energies. They include work on the phi-transform, a new tool in harmonic analysis for decomposing general functions as sums of simple terms. It mixes time and frequency considerations, providing cancellation in powerful and subtle ways. Particular examples involving electromagnetic scattering, electrocardiogram analysis and NMR imaging will be studied. Work on Chern-Moser- Tanaka invariants in the theory of several complex variables has been hampered by the difficulty in calculating the invariants. Now it is possible to involve students in the use of symbol manipulation algorithms to make estimates and create numerical models to study these important mathematical structures. Students may also work on mathematical models of molecular evolution, (the distribution of nucleotides in the coding regions of individuals chosen from a random-mating population); on new high resolution methods for constructing magnetic resonance images from raw MR scanner data; on Fourier analysis over local fields; and on combinatorial number theory, specifically, on the randomness of the distribution of primes. Efforts will be made to encourage students to develop their work into publishable form or as the basis for senior honors theses.