Mathematical Sciences: The Differential Geometry of PartialDifferential Equations

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9505125 Bryant The proposed research lies in the area of exterior differential systems. The proposer seeks to classify systems of various types which admit non-trivial conservation laws, and to develop effective algorithms for computing integrable extensions of a given system. The proposer also wishes to understand the geometric invariants of a control system and to investigate control algorithms. The main tools to be used are Cartan's method of equivalence and Cartan-Kahler theory. Exterior differential systems generalize systems of partial differential equations; their study is heavily geometric and heavily computational at the same time. Control theory is used extensively in such areas as manufacturing and robotics; the proposed research may help to create more efficient design algorithms based upon a deeper understanding of the geometric invariants involved.