Hermitian Forms and CR Geometry

John D‘Angelo将继续他对厄米特形式及其在几个复变量和CR几何中的应用的研究。这项工作奠定了数学的一个发展领域--认知几何中的复杂性理论的基础，同时也使得与数学的其他分支建立了联系。本文的出发点是球面与超二次曲面之间的CR映射，接着研究了CR映射的各种复杂性概念，厄米特对称函数的符号对，以及有限酉群的表示理论。应用包括希尔伯特第17问题的厄米模拟，围绕代数集上厄米多项式的新的矩阵正性条件。这项工作与柯西-施瓦茨(Cauchy-Schwartz)不等式的一种非线性形式以及集合相对于真实理想的厄米零性的概念密切相关。这些结果将导致复欧氏空间中的实超曲面的几何与代数中的基本问题之间的显著联系。长期以来，复数维上的映射定理在数学、物理和工程中一直扮演着中心角色。拟议的工作可以被认为是在更高的维度上发展基本思想，那里的情况变得更加微妙，新的现象出现。由此产生的工作导致了分析、几何和代数的一种有希望的和不同寻常的组合。1900年，数学家大卫·希尔伯特列出了一份包含23个问题的清单。这些问题继续推动着当今的许多数学研究。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厄米特类比，出现了新的困难，并与数学的许多领域产生了联系。D‘Angelo将继续在这些主题上指导年轻数学家(包括研究生)，并组织和参加会议。他还将继续教授他为荣誉新生开发的一门新的复杂分析课程。

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.