Geometry of Invariants and Mappings in Several Complex Variables

对不变几何的研究是数学的基础。不变几何出现在许多不同的领域，如几个复杂的变量，偏微分方程，以及代数，复杂和微分几何。该研究项目研究了在研究几个复杂变量时出现的特定几何，并与数学物理中的当前主题有很深的联系，包括量子场论、广义相对论和弦理论，以及在系统工程和控制理论等方面的应用。本研究所需的思想和技术来自各种数学领域，包括复杂分析和几何、偏微分方程和微分几何；同时，本项目开发的工具也将为这些领域提供信息。本项目也为研究生和博士后提供了有趣的研究课题。这个数学研究项目的目标是研究复杂变量（更一般地说，CR结构）中的一般实子流形及其映射的几何、解析和代数方面。本研究将探讨高阶局部不变量的紧CR流形上的全局消失的几何结果，该流形是Fefferman的复Monge-Ampere方程的Cheng-Yau解平滑扩展到边界的障碍。在三维空间中，该不变量与Bergman核渐近展开中对数项边界上的迹相吻合；因此，这个问题也与拉马达诺夫猜想的一个强形式有关。PI将研究紧致CR 3-流形上CR脐点的存在性，特别是对Chern-Moser提出的在复2空间的有界严格伪凸域上CR脐点是否总是存在的问题进行修正；虽然一般来说答案是否定的，但如果定义域与球是微分同态的，这个问题仍然是开放的。PI还将讨论CR图的存在性、唯一性、规律性等相关问题。他将考虑Levi非简并超曲面到高维非简并超二次曲面的CR映射的背景。本研究将加深我们对构成Levi非简并超曲面理论中平面模型的超二次曲面的CR子流形结构的理解。这项工作还应该提供关于这些超曲面的局部CR几何如何影响其CR映射的各种性质的见解，例如非简并性和刚性的概念。PI还将通过研究定义CR映射到射流束上的奇异Pfaffian系统的系统的延拓，继续研究无穷型一般子流形之间的CR映射。他将专注于理解最近发现的现象，即对于无限型超曲面，生物全纯、形式和光滑CR等价分类是不同的。这些研究有望揭示无限型流形之间CR映射的性质，并有助于更好地理解在这种情况下产生的Pfaffian系统。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The study of invariant geometries is fundamental in mathematics. Invariant geometries arise in many different areas, such as several complex variables, partial differential equations, and algebraic, complex, and differential geometry. This research project investigates a particular geometry that arises in the study of several complex variables and has deep connections with current topics in mathematical physics, including quantum field theory, general relativity, and string theory, as well as applications in, e.g., systems engineering and control theory. The ideas and techniques needed for this study are drawn from a variety of mathematical areas, including complex analysis and geometry, partial differential equations, and differential geometry; at the same time, the tools developed in this project will inform these areas as well. The project also provides interesting research topics for graduate students and postdocs. The goal of this mathematics research project is to study geometric, analytic, and algebraic aspects of generic real submanifolds in complex varieties (more generally, of CR structures) and their mappings. The work will investigate the geometric consequences of global vanishing on a compact CR manifold of a higher order local invariant that arises as the obstruction to smooth extension to the boundary of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation. In three dimensions, this invariant coincides with the trace on the boundary of the log-term in the asymptotic expansion of the Bergman kernel; hence this problem is also connected with a strong form of the Ramadanov Conjecture. The PI will study the existence of CR umbilical points on compact CR 3-manifolds, especially a revised version of a question by Chern-Moser, which asks if such points always exist on bounded strictly pseudoconvex domains in complex 2-space; although the answer is 'no' in general, the question is still open if the domain is diffeomorphic to the ball. The PI will also consider questions regarding existence, uniqueness, and regularity of CR maps, as well as related questions. He will consider the context of CR maps of a Levi nondegenerate hypersurface into a nondegenerate hyperquadric of higher dimension. This study will enhance our understanding of the CR submanifold structure of the hyperquadrics, which constitute the flat models in the theory of Levi nondegenerate hypersurfaces. The work should also provide insight into how the local CR geometry of such hypersurfaces affects various properties, such as notions of nondegeneracy and rigidity, of their CR maps. The PI will also continue study of CR maps between generic submanifolds of infinite type by investigating the prolongation of the system defining CR maps to a singular Pfaffian system on the jet bundle. He will focus on understanding the recently discovered phenomenon that for infinite type hypersurfaces the biholomorphic, formal, and smooth CR equivalence classifications are different. These investigations are expected to shed new light on the nature of CR maps between infinite type manifolds and lead to a better understanding of the Pfaffian systems arising in this context.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.