Symbolic Methods for Classification Problems In Lie Theory and Differential Geometry

李理论和微分几何中分类问题的符号方法

• 批准号: 0410373
• 负责人: Ian Anderson
• 金额: --
• 依托单位: Utah State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0410373&HistoricalAwards=false
• 关键词: Symbolic; Methods; Classification; Problems; Lie

AndersonThe investigator develops and implements algorithms for a numberof classification problems in the areas of Lie algebras,differential geometry, and differential equations. For each ofthe problems of interest, the list of representative objects isvery large. Therefore he encodes these classification resultswithin a computer algebra system and develops software thatallows for the efficient but flexible manipulation of the largeamounts of data contained in these classification results. Thiswork provides researchers with powerful new tools to findexamples and test conjectures on a wide range of subjects. Thespecific classification problems that are addressed first dealwith the classification of low dimensional Lie algebras and theirsub-algebras, the classification of vector field systems in 2 and3 dimensions, and the classification of 4- and 5-dimensionalspace-times with symmetry.The classification of mathematical structures is a central themein many branches of mathematics. While these classificationproblems can be easy to state, the solutions are often very longand complex and therefore not accessible to the largermathematical and scientific community. A major component of thisproject explores new nontraditional methodologies (based oncomputer algebra systems and also data-based Java applets) forthe dissemination and generalization of these results. Applications to the automated solution of differential equationsare studied. The computer software that is being developed tosupport these specific goals has applications in other areas ofmathematical research and in theoretical physics. It also hasdemonstrated pedagogical value for both undergraduate andgraduate mathematics education.

Anderson研究人员为李代数、微分几何和微分方程领域中的一些分类问题开发和实现了算法。对于每个感兴趣的问题，具有代表性的对象的列表都非常大。因此，他在计算机代数系统中对这些分类结果进行编码，并开发了软件，允许高效而灵活地处理这些分类结果中包含的大量数据。这项工作为研究人员提供了强大的新工具，可以在广泛的主题上寻找样本和检验猜测。这些具体的分类问题首先涉及低维李代数及其子代数的分类，二维和三维向量场系统的分类，以及具有对称性的四维和五维时空的分类。数学结构的分类是许多数学分支的中心主题。虽然这些分类问题可以很容易地陈述，但解决方案往往非常长和复杂，因此对更大的数学和科学界来说是无法获得的。该项目的一个主要组成部分是探索传播和推广这些成果的新的非传统方法(基于计算机代数系统和基于数据库的Java小应用程序)。研究了它在微分方程组自动求解中的应用。正在开发的支持这些特定目标的计算机软件在数学研究的其他领域和理论物理中也有应用。它还展示了本科生和研究生数学教育的教学价值。