Singularities and Complexity in CR Geometry

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The PI proposes to study the theory of singularities and complexity in CR geometry. In particular the PI proposes to study the connections between his previous work on Levi-flat hypersurfaces, properties of nowhere minimal submanifolds of complex euclidean space, and the complexity of proper maps between balls in different dimensions. The common link can be described as studying the set where two squared norms of holomorphic maps are equal, and relating the CR geometric information of the solution set of this equation to the complexity of the maps. The PI proposes to prove that the singular set of a Levi-flat hypervariety is Levi-flat, to classify algebraic Levi-flat hypervarieties in complex projective space, to study the regularity of Levi-flat hypersurfaces with boundary, to extend the work on complexity of proper maps between balls, and finally, to further develop computational methods to help in gaining insight into the combinatorial aspects of the proper maps of balls and related problems.Study of several complex variables, of which CR geometry is part, is central to the understanding of modern mathematics, physics and other applied sciences. For example, to understand behavior of differential equations, one must understand the geometry of the space where the equation lives. The theory of singularities and complexity in CR geometry is not well understood currently, and there is great interest in the CR geometry community in building proper foundations in this area. Furthermore, there is fertile ground to build connections with other areas of mathematics. The study of proper maps of balls has already yielded unexpected connections with number theory, combinatorics, linear algebra, and has computational aspects that may perhaps yield advances in symbolic and numerical computation. Many of the methods applied in this research are easily accessible to beginning graduate and even advanced undergraduate students. The project will therefore not only advance the understanding of this new area in complex analysis, but may serve to involve young researchers.