Symbolic Methods for Classification Problems In Lie Theory and Differential Geometry

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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.