Geometric Applications of Exterior Differential Systems

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PROPOSAL DMS-0305829PI: Joseph M. LandsbergTitle: Geometric applications of exterior differential systemsABSTRACTDr. Landsberg will work on two projects: the dual defect conjectureand the classification of Legendrian varieties and contact Fano manifolds.The second project originates with a question in differential geometry,the classification of compact quaternionic-Kahler manifolds, which, thanks to work of LeBrun and Salamon, is essentially equivalent to the classification of contact Fano manifolds. Inside the tangent space of a general point of a contact Fano manifold is an immersed Legendrian variety and knowledge about such varieties is important for the contact Fano problem. The Salamon conjecture is that all contact Fano manifoldsare homogeneous. The first project is to show that the dual variety of a smooth variety of codimension less than half its dimension is a hypersurface. This question is of importance in algebgraic geometry as part of the investigations motivated by Hartshornes's conjecture on complete intersections. The techniques that will be developed to solve this problem are as important as the problem iself- a combination of exterior differential systems and representation theory that will be useful for other problems in geometry and other areas of science such as computational complexity and algebraic statistics (Baysean networks).Dr. Landsberg will use techniques from partial differential equations(more precisely, exterior differential systems) and representationtheory (of simple Lie algebras) to study two problems in geometryrelated respectively to dual varieties and contact Fano manifolds. In addition to the importance of these questions to the mathematical community, the techniques that will be developed will be applicable to problems in other fields such as computational complexity and algebraic statistics (Baysean networks).