Singularities and Complexity in CR Geometry

CR 几何中的奇点和复杂性

• 批准号: 0900885
• 负责人: Jiri Lebl
• 金额: $ 9.99万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900885&HistoricalAwards=false
• 关键词: Singularities; Complexity; CR; Geometry

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.

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。 PI 提议研究 CR 几何中的奇点和复杂性理论。 特别是，PI 提议研究他之前关于列维平坦超曲面的工作、复杂欧几里得空间的无处最小子流形的性质以及不同维度的球之间的固有映射的复杂性之间的联系。 共同的联系可以描述为研究全纯​​映射的两个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度相关联。 PI 提议证明 Levi 平超变性的奇异集是 Levi 平，对复杂射影空间中的代数 Levi 平超变性进行分类，研究带边界的 Levi 平超曲面的正则性，扩展球之间真映射复杂性的工作，最后，进一步开发计算方法以帮助深入了解组合方面 球的固有映射和相关问题。对几个复杂变量的研究（其中 CR 几何是其中的一部分）是理解现代数学、物理学和其他应用科学的核心。 例如，要理解微分方程的行为，必须了解方程所在空间的几何形状。 目前，CR 几何中的奇点和复杂性理论尚未得到很好的理解，CR 几何界对在该领域建立适当的基础抱有极大的兴趣。 此外，还有与其他数学领域建立联系的肥沃土壤。 对球的正确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带来符号和数值计算方面的进步。 这项研究中应用的许多方法对于研究生甚至高年级本科生来说都很容易掌握。 因此，该项目不仅将增进对复杂分析这一新领域的理解，而且可能有助于年轻研究人员的参与。