Holomorphic and CR mappings in Several Complex Variables

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）和开发技术（如微局部分析），从几个复杂的变量。它们与量子场论和控制理论等物理学和工程学的各个领域有着密切的联系，并在磁流体力学和电气网络等主题中有着广泛的应用。在该项目中研究的厄米对称空间在数学物理中发挥着重要作用。例如，经典IV型域与未来管是双全纯等价的。四维复欧几里德空间中的未来管是现代宇宙学的基本对象，因为它是全纯相对论场的自然定义域。PI的研究将加深对此类物体几何结构的理解。主要研究者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，该项目的重点是研究几何，分析和代数方面的全纯和CR映射采用技术从偏微分方程，代数，微分几何，除了复杂的分析。 特别是，PI想研究复流形或CR流形之间映射的刚性，存在性和正则性问题，以及算术代数几何和复几何中出现的相关问题。PI还将研究复杂空间中域的Kahler几何与其边界的CR几何之间的内在联系。研究所需的技术来自于几个复变量（CR不变理论、Bergman核的渐近性态和Chern-Moser理论等）、Kahler几何和几何分析。该研究项目的目标是进一步了解几何函数理论在几个复杂的变量，以及它与代数几何，复几何，动力系统和数论方面的深刻联系。PI还希望该项目能够开发出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。该奖项反映了NSF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 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.