Presidential Faculty Fellows

9553134罗克莫尔研究员利用总统学院奖学金，通过群论和计算主题的统一，追求研究和教育创新的跨学科课程。主要的研究活动集中在广义快速傅立叶变换(FFT)的理论和实践的不断发展上。所追求的推广包括有限紧李群上的傅立叶变换的有效计算和有效的多项式变换。目前解决这类问题的方法包括群表示理论、算子代数理论和抽样理论。实际的考虑和目标要求还处理诸如数值稳定性和计算开销等实施方面的问题，在这样做时需要数值分析的工具和技术以及对目标应用领域的一些知识。这种跨学科的范式也为教育方法提供了信息。正在开发的是几门跨学科课程，旨在通过强调数学与其他学科的联系来提高本科生的数学兴趣和意识。可以说，“经典”的FFT是有史以来发展起来的最重要的计算技术之一。它是大多数数字信号处理背后的基本工具，因此影响着社会的方方面面，在医学成像、国防、电信、高性能计算和娱乐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可能会获得类似的成功，本提案的很大一部分是针对这一目标的。到目前为止，这些新的计算方法已经在气候建模、图像处理、医学成像、天文学和统计学等领域得到了应用。当局现正积极推行这些措施。这项工作揭示了纯研究和“现实世界”应用之间的联系，作为一种教育模式非常有用。通过关注这类主题，调查员致力于加强目前的数学课程，目标是提高数学入学率和认识，并在此过程中努力重建国家科学基础设施。

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.