Singularities and Complexity in CR Geometry

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI建议研究CR几何中的奇点和复杂性理论。 特别是PI建议研究之间的联系，他以前的工作列维平坦超曲面，性质无处极小子流形的复杂欧氏空间，和复杂性适当的地图之间的球在不同的层面。 其共同环节可以描述为研究两个全纯映射的平方范数相等的集合，并将该方程的解集的CR几何信息与映射的复杂性联系起来。 PI提出证明Levi-flat超簇的奇异集是Levi-flat的，对复射影空间中的代数Levi-flat超簇进行分类，研究带边界的Levi-flat超曲面的正则性，推广球间真映射复杂性的工作，进一步发展计算方法，以帮助深入了解球的适当映射和相关问题的组合方面。研究几个复杂的变量，其中CR几何的一部分，是中央的现代数学，物理学和其他应用科学的理解。 例如，要理解微分方程的行为，必须理解方程所在空间的几何。 CR几何中的奇异性和复杂性理论目前还没有得到很好的理解，在CR几何社区中，人们对在这一领域建立适当的基础有很大的兴趣。 此外，还有肥沃的土壤与其他数学领域建立联系。 球的真映射的研究已经与数论、组合学、线性代数产生了意想不到的联系，并且在计算方面可能会在符号计算和数值计算方面取得进展。 本研究中应用的许多方法对于刚开始的研究生甚至是高年级的本科生来说都很容易使用。 因此，该项目不仅将促进对复杂分析这一新领域的理解，而且可能有助于吸引年轻研究人员的参与。

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.