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

研究者发展了广义快速傅里叶变换（FFT）。最重要的是对有限群构造FFT的技术的进一步发展，以及对非abel紧群及其商上的带限函数的傅里叶变换的有效和数值稳定计算的算法工具的扩展。这种跨学科的研究从计算群理论、表示理论和复杂性理论等领域，以及那些探索应用的科学领域中汲取灵感。从历史上看，通过加速某些基础科学计算，“经典”FFT对社会产生了巨大的影响。它是数字信号处理中最重要的算法工具之一，在医学成像、国防、电信和高性能计算等领域发挥着关键作用。经典FFT的实用性表明，应用这些新开发的广义FFT可能会取得类似的成功。更有效地使用高性能计算网络（信息高速公路）的应用正在研究中。研究者和他的同事们继续在统计和数据分析中应用广义FFT。在气候模拟和结构化学方面的具体应用也已确定并正在进行中。

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.