The investigator develops and implements algorithms for a number of classification problems in the areas of Lie algebras, differential geometry, and differential equations. For each of the problems of interest, the list of representative objects is very large. Therefore he encodes these classification results within a computer algebra system and develops software that allows for the efficient but flexible manipulation of the large amounts of data contained in these classification results. This work provides researchers with powerful new tools to find examples and test conjectures on a wide range of subjects. The specific classification problems that are addressed first deal with the classification of low dimensional Lie algebras and their sub-algebras, the classification of vector field systems in 2 and 3 dimensions, and the classification of 4- and 5-dimensional space-times with symmetry.
The classification of mathematical structures is a central theme in many branches of mathematics. While these classification problems can be easy to state, the solutions are often very long and complex and therefore not accessible to the larger mathematical and scientific community. A major component of this project explores new nontraditional methodologies (based on computer algebra systems and also data-based Java applets) for the dissemination and generalization of these results. Applications to the automated solution of differential equations are studied. The computer software that is being developed to support these specific goals has applications in other areas of mathematical research and in theoretical physics. It also has demonstrated pedagogical value for both undergraduate and graduate mathematics education.
