Holomorphic and CR mappings in Several Complex Variables

这项拟议的研究与数学、理论物理和应用科学的许多其他领域相互作用。PI将研究几何对象(如CR流形)，并开发由几个复杂变量产生的技术(如微局部分析)。它们与量子场论、控制论等物理和工程领域有着密切的联系，在磁流体力学、电网络等领域有着广泛的应用。该项目中研究的厄米对称空间在数学物理中起着基础性的作用。例如，类型IV的经典区域与未来的管是双全纯等价的。四维复欧几里德空间中的未来管是现代宇宙学中的一个基本对象，因为它是全纯相对论领域的自然定义域。PI的研究将加深对这类物体几何结构的理解。主要研究人员建议研究复分析和柯西黎曼几何中的映射问题。更准确地说，除了复数分析外，该项目还重点研究了全纯映射和CR映射的几何、解析和代数方面的知识，方法包括偏微分方程组、代数和微分几何。特别是，PI想要研究复流形或CR流形之间的映射的刚性、存在性和正则性问题，以及在算术代数几何和复几何中出现的相关问题。PI还将研究复空间中区域的Kahler几何与其边界的CR几何之间的内在联系。这项研究所需要的技术来自几个复变量(CR不变量理论、Bergman核的渐近性质和Chern-Moser理论等)、Kahler几何和几何分析。该研究项目的目的是加深对多复变几何函数论的理解，以及它与代数几何、复几何、动力系统和数论等方面的深刻联系。PI还期望该项目开发出能够影响这些领域的新方法和新想法，并为研究生和博士后提供有趣的研究课题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The proposed research has interplay with many other fields of mathematics, theoretical physics, and applied science. The PI will study geometric objects (such as CR manifolds) and develop techniques (such as microlocal analysis) that arise from several complex variables. They are deeply connected with various areas in physics and engineering such as quantum field theory and control theory, and have extensive applications to topics including magnetic hydrodynamics and electrical networks. Hermitian symmetric spaces studied in the project play a fundamental role in mathematical physics. For instance, the classical domain of type IV is biholomorphically equivalent to the future tube. The future tube in four-dimensional complex Euclidean space is a basic object in modern cosmology as it is the natural defining domain for holomorphic relativistic fields. The PI's research will deepen the understanding of the geometric structure of such objects. The principal investigator proposes to study mapping problems in complex analysis and Cauchy Riemann geometry. More precisely, the project focuses on studying the geometric, analytic and algebraic aspects of holomorphic and CR mappings by employing techniques from partial differential equations, algebra, and differential geometry, in addition to complex analysis. In particular, the PI would like to investigate rigidity, existence and regularity problems for mappings between complex or CR manifolds,as well as related questions that arise in arithmetic algebraic geometry and complex geometry. The PI will also study the intrinsic connections between the Kahler geometry of a domain in a complex space and the CR geometry of its boundary. The techniques that are needed for this study come from several complex variables (CR invariant theory, asymptotic behavior of Bergman kernel and Chern-Moser theory, etc), Kahler geometry and geometric analysis. The objective of the research project is to further the present understanding of geometric function theory in several complex variables, as well as its profound connections with aspects of algebraic geometry, complex geometry, dynamical systems, and number theory. The PI also expects the project to develop substantially new methods and ideas that will influence these areas, and provide interesting research topics for graduate students and postdocs as well.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.