Anlaytic Geometry and Representation Theory

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Landsberg and Robles will continue their joint work in applying and advancing exterior differential systems (EDS). In particular they will complete their determination of the projective rigidity of rational homogeneous varieties, study questions regarding invariant differential operators (e.g., what is the image of the Killing operator on a Riemannian manifold?), and further the connections between Bernstein-Gelfand-Gelfand (BGG) resolutions and EDS. Robles will will identify necessary and sufficient conditions on curvature for the metric on a Riemannian 4-manifold to admit orthogonal coordinates, and determine geometric representatives of homology classes in Riemannian manifolds. Landsberg will work on problems originating in complexity theory and signal processing, in particular determining equations for secant varieties of homogeneous varieties, and establishing properties of the Mulmuley-Sohoni varieties associated to the determinant and permanant.Differential geometry may be thought of as the study of geometric objects "under a microscope" (i.e., infinitesimally). Landsberg and Robles are especially interested in geometric objects with symmetry. Over the last 40 years significant advances have been made in representation theory, which systematically studies symmetry (e.g., the BGG machinery), and these advances will be applied by the PIs to establish properties of systems of partial differential equations arising in differential geometry. Individually, the PIs will use also use differential geometry and representation theory to work on problems in topology (Robles), in theoretical computer science (Landsberg and Robles) and signal processing (Landsberg).