Hermitian Forms and CR Geometry

埃尔米特形式和 CR 几何

• 批准号: 1066177
• 负责人: John D'Angelo
• 金额: $ 22.16万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1066177&HistoricalAwards=false
• 关键词: Hermitian; Forms; CR; Geometry

John D'Angelo will continue his study of Hermitian forms and their applications to several complex variables and CR geometry. This work lies at the foundation of a developing area in mathematics, complexity theory in CR Geometry, while also enabling connections to other branches of mathematics. The starting point concerns CR mappings between spheres and hyperquadrics; it goes on to study various notions of complexity for CR mappings, the signature pair of a Hermitian symmetric function, and representation theory for finite unitary groups. Applications include Hermitian analogues of Hilbert's 17-th problem, revolving around new matrix positivity conditions for Hermitian polynomials on algebraic sets. This work is closely related to a non-linear form of the Cauchy-Schwartz inequality and to the notion of Hermitian nullity of a set with respect to a real ideal. These results will lead to striking links between the geometry of a real hypersurface in complex Euclidean space and basic questions in algebra. Mapping theorems in one complex dimension have long played a central role in mathematics, physics, and engineering. The proposed work can be regarded as developing the fundamental ideas in higher dimensions, where the situation becomes much more subtle and new phenomena arise. The resulting work leads to a promising and unusual combination of analysis, geometry, and algebra. In 1900 the mathematician David Hilbert set out a list of 23 problems. These problems continue to drive much of current day mathematical research. The famous 17th problem was solved in the 1920s. D'Angelo's work has led to Hermitian analogues of this problem, where new difficulties appear and connections to many areas of mathematics arise. D'Angelo will continue to mentor young mathematicians (including graduate students) in these topics and to organize and attend conferences. He will also continue teaching a new complex analysis course he has developed for Honors freshmen.

约翰德安杰洛将继续他的研究厄米的形式和他们的应用程序，以几个复杂的变量和CR几何。这项工作是在数学，CR几何复杂性理论的发展领域的基础，同时也使连接到数学的其他分支。起点关注球和超二次曲面之间的CR映射;它继续研究CR映射的复杂性的各种概念，埃尔米特对称函数的签名对，以及有限酉群的表示理论。应用包括希尔伯特第17问题的埃尔米特类似物，围绕代数集上埃尔米特多项式的新矩阵正性条件。这项工作是密切相关的一个非线性形式的柯西-施瓦茨不等式和概念的厄米特零的一套关于一个真实的理想。这些结果将导致一个真实的超曲面在复杂的欧几里德空间的几何和代数的基本问题之间的惊人的联系。一维复映射定理在数学、物理和工程学中一直扮演着重要的角色。所提出的工作可以被看作是在更高维度中发展基本思想，在那里情况变得更加微妙，新的现象出现。由此产生的工作导致了一个有前途的和不寻常的组合分析，几何和代数。1900年，数学家大卫·希尔伯特列出了一个包含23个问题的清单。这些问题继续推动着当今的数学研究。著名的第17个问题在20世纪20年代得到了解决。D 'Angelo的工作导致了这个问题的Hermitian类似物，出现了新的困难，并与许多数学领域产生了联系。德安杰洛将继续指导年轻的数学家（包括研究生）在这些主题，并组织和参加会议。他还将继续教授他为荣誉新生开发的新的复杂分析课程。