Analytic Geometry and Representation Theory

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"Analytic Geometry and Representation Theory" by J.M. Landsberg & C. Robles. This project will make significant progress on problems originating in geometry and theoretical computer science. Each of the questions will be approached using techniques that lie on the interface of 3 areas of mathematics: differential geometry (submanifolds, exterior differential systems), algebraic geometry (projective varieties, geometric aspects of commutative algebra) and representation theory (orbit closures, Lie algebra cohomology, invariant theory). We emphasize that this synthesis yields a cross-pollination that significantly increases the richness, value an influence of the results. Landsberg and Robles will continue work related to (1) the Hwang-Mok program on Fano varieties in algebraic geometry, (2) the P = NP? problem in complexity theory, and (3) the geometric structure of orbits in representation theory. These seemingly disparate projects are unified by the Principle Investigators' approach. In each case there is an underlying differential geometry of lines on (and osculating to) a projective variety. A through understanding of this geometry will deepen our understanding of (and in some cases solve) these problems. This geometry is studied via techniques drawn from differential geometry, algebraic geometry and representation theory. While the geometric and representation theoretic tools used are classical, the implementation and resulting applications are innovative and originated with the principal investigators. Additional projects include: (1) A study of varieties in spaces of tensors. This work draws on geometry and representation theory, and will have applications to statistics, computer science and complexity theory. (2) Calibrated geometries. Not only an active area of research in mathematics, these geometries are of interest to physicists as components of string theory models. (3) The construction of compact G2-manifolds.