Mathematical Sciences: Generalized FFT's & Computational Methods in Group Representation Theory

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The investigator develops generalized fast Fourier transforms (FFT's). Of principal importance is the further development of techniques for constructing FFT's for finite groups, as well as the extension of algorithmic tools for the efficient and numerically stable computation of Fourier transforms for band-limited functions on nonabelian compact groups and their quotients. This interdisciplinary research draws from the areas of computational group theory, representation theory, and complexity theory, as well as those areas of science in which applications are explored. Historically, by speeding certain fundamental scientific computations, the ``classical'' FFT has had a tremendous impact on society. It is one of the most important algorithmic tools in digital signal processing and as such plays a key role in such areas as medical imaging, defense, telecommunications, and high performance computing. The utility of the classical FFT is an indicator that perhaps similar success may be obtained from applying these newly developed generalized FFT's. Applications towards more efficient use of high performance computing networks (the information superhighway) are under investigation. The investigator and his colleagues continue to apply generalized FFT's in statistics and data analysis. Specific applications in climate modelling and structural chemistry have also been identified and are being pursued.