Symbolic Methods for Classification Problems In Lie Theory and Differential Geometry

安德森研究员开发和实现算法的一些分类问题的领域李代数，微分几何和微分方程。 对于每个感兴趣的问题，代表对象的列表非常大。 因此，他编码这些分类结果内的计算机代数系统和开发软件，允许有效而灵活地操纵大量的数据包含在这些分类结果. 这项工作为研究人员提供了强有力的新工具，以寻找样本和测试广泛的主题。 本文首先讨论了低维李代数及其子代数的分类问题，二维和三维向量场系统的分类问题，以及四维和五维对称时空的分类问题。 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漫长和复杂，因此不适合更大的数学和科学界。 该项目的一个主要组成部分是探索新的非传统方法（基于计算机代数系统和基于数据的Java小程序），以传播和推广这些结果。研究了其在微分方程自动求解中的应用。 正在开发的支持这些特定目标的计算机软件在数学研究和理论物理的其他领域也有应用。 它对本科数学教育和研究生数学教育都具有一定的教学价值。

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.