Presidential Faculty Fellows

总统教职研究员

• 批准号: 9553134
• 负责人: Daniel Rockmore
• 金额: $ 50.49万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-02-01 至 2002-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9553134&HistoricalAwards=false
• 关键词: Presidential; Faculty; Fellows

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.

小行星9553134 调查员使用这个总统教师奖学金追求研究和教育创新的跨学科课程，由群论和计算的主题统一。 主要研究活动集中在广义快速傅立叶变换（FFT）的持续理论和实践发展上。 其中所追求的推广是有限和紧李群上的傅立叶变换和有效的多项式变换的有效计算。 目前解决这类问题的方法借鉴了群表示理论、算子代数理论和采样理论。 实际的考虑和目标的要求，实施方面，如数值稳定性和计算开销，以及解决，这样做需要的工具和技术的数值分析，以及一些知识的目标应用领域。 这种跨学科的范式也为教育方法提供了信息。 正在开发的是几个跨学科的课程，旨在通过突出数学与其他学科的联系来提高本科生对数学的兴趣和认识。 可以说，“经典”FFT是有史以来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，因此在各个方面影响着社会，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。 经典的FFT的效用表明，也许类似的成功，可以从应用这些新开发的广义FFT，并针对这一目标的建议的很大一部分。 到目前为止，这些新的计算方法的应用已经确定在气候建模，图像处理，医学成像，天文学和统计学等领域。 目前正在积极开展这些工作。 这项工作表明了纯研究和"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，研究人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，并在这样做的过程中努力重建国家科学基础设施。