Mathematical Sciences: Applications of Lie Theory to Control, Computer Science and Physics

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This research will focus on four areas: 1) integrability theory for linear differential systems and perturbation theory for the feedback linearization equations of nolinear control theory. This work will support the program of engineering research of nonlinear control of air-space-craft at NASA-Ames Research Center directed by Dr. George Meyer. This research was cited in the recent NSF Report on Control Theory as a prime example of the practical success of the application of geometric methodology. 2) Geometric and sheaf-theoretic methodology in the knowledge-representation and computation theory side of computer science. This will provide mathematical support for the Discrete Event System and Artificial Intelligence research. 3) Lie- theoretic methods in numerical analysis developing new geometric and Lie-theoretic methodology for the numerical algorithms of linear algebra and differential equations. 4) Infinite pseudogroups in elementary particle physics looking for the ntural mathematical synthesis of String and Standard Model gauge theories, utilizing the Cartan-Matshushima-Guillemin-Sternberg- Conn theory of structure of infinite Lie pseudogroups.