9553134 Rockmore The investigator uses this Presidential Faculty Fellowship to pursue an interdisciplinary course of research and educational innovation, unified by the themes of group theory and computation. The primary research activity centers upon the continued theoretical and practical development of generalized fast Fourier transforms (FFTs). Among the generalizations pursued are the efficient computation of Fourier transforms on finite and compact Lie groups and efficient polynomial transforms. Current approaches to these sorts of problems draw on the areas of group representation theory, the theory of operator algebras and sampling theory. Practical considerations and goals demand that implementational aspects such as numerical stability and computational overhead be addressed as well, and in so doing require the tools and techniques of numerical analysis, as well as some knowledge of the targeted applications areas. This sort of interdisciplinary paradigm also informs the educational approach. Under development are several interdisciplinary courses, intended to increase mathematics interest and awareness among undergraduates by highlighting the connections of mathematics to other disciplines. Arguably, the ``classical'' FFT is one of the most important computational techniques ever developed. It is the fundamental tool behind most of digital signal processing and as such impacts society in all aspects, playing a key role in areas such as medical imaging, defense, telecommunications, high performance computing, and entertainment. The utility of the classical FFT indicates that perhaps similar success may be obtained from applying these newly developed generalized FFTs, and a large part of this proposal is directed towards this end. To date, applications of these new computational methods have been identified in areas such as climate modeling, image processing, medical imaging, astronomy and statistics. These are being pursued actively. The connection that this work evinces between pure research and ``real-world'' application is tremendously useful as an educational model. By focusing on this type of theme, the investigator works towards enhancing the current mathematics curriculum with the goal of increasing mathematics enrollments and awareness and in so doing working towards rebuilding the national scientific infrastructure.
