Symbolic Methods for Classification Problems In Lie Theory and Differential Geometry

研究者开发并实现了李代数、微分几何和微分方程领域的许多分类问题的算法。对于每个感兴趣的问题，具有代表性的对象列表都非常大。因此，他在计算机代数系统中对这些分类结果进行编码，并开发软件，允许对这些分类结果中包含的大量数据进行有效而灵活的操作。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找到例子和测试猜想。本文首先讨论了低维李代数及其子代数的分类，二维和三维矢量场系统的分类，以及四维和五维对称时空的分类。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说明，但解决方案往往很长很复杂，因此无法被更大的数学和科学界所接受。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 applet），以传播和推广这些结果。研究了微分方程自动解的应用。为支持这些具体目标而开发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。对本科和研究生数学教育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.