Mathematical Sciences: Geometric Applications of Exterior Differential Systems

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9626640 Landsberg The proposed research lies in the interface of differential geometry and algebraic geometry. Given a projective variety, its dual variety is defined to be the union of all hyperplanes tangent to it. In this project, the dimension and degeneracy of dual varieties will be investigated. Homogeneous varieties will also be studied using higher order contact invariants, thus from a local perspective. Algebraic geometry deals with solutions to polynomial equations, often emphasizing the global geometric aspect of the solution set; it is one of the most developed branches of modern mathematics with applications to control theory, robotics and other engineering disciplines. Differential geometry emphasizes local or infinitesimal properties of (hyper-) surfaces. One of the goals of this project is to recover various global and algebraic properties encoded in the local differential geometry of projective varieties.