Partial Differential Equations in Several Complex Variables

复数分析是数学中最重要的领域之一。一元复分析是一门成熟而美好的学科。它在数学和物理的各个分支中都有广泛的应用，包括代数(代数基本定理)、拓扑学(Riemann映射定理)、数论(素数定理)、流体力学、光学和静电学(共形映射)。在过去的一百年里，人们对多复变的全纯函数进行了较新的研究。多个复变量的函数理论与一个复变量的理论有根本的不同。它的发展影响了偏微分方程组、调和分析和复杂几何的现代分析。该项目的结果将对数学界的很大一部分人产生广泛的吸引力。它已经在涉及非光滑边界的现代技术中得到了应用，例如电喷雾和散射。这位首席研究员致力于开发一本关于偏微分方程式的复变量理论的教科书，以使这些有趣的主题不仅对数学，而且对物理科学和工程的广泛学生来说更容易理解。该项目为研究生提供培训机会，首席调查员积极参与各级学生的教学和培训。该项目的目标是在几个复杂的变量中探索几个不同和相互关联的领域。重点讨论了几个复变量中最重要的两个方程：Cauchy-Riemann方程和切线Cauchy-Riemann方程。关于这些方程的经典结果要求区域具有光滑边界，或者环境空间是欧几里得或斯坦因空间。该项目的很大一部分是关于具有非光滑边界的区域和复杂流形。主要研究者利用调和分析和几何测度论对Lipschitz域上的Cauchy-Riemann方程进行了研究。这些问题包括Dolbeault上同调群的Hausdorff性质，Levi-平坦超曲面和复叶化，非光滑区域和复流形上的Runge逼近和函数理论。这些问题是几个复杂变量、复杂几何和偏微分方程式的核心问题。该奖项反映了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。

Complex analysis is one of the most important areas of mathematics. Complex analysis in one variable is a mature and beautiful subject. It has broad applications to various branches of mathematics and physics, including algebra (Fundamental Theorem of Algebra), topology (Riemann Mapping Theorem), number theory (Prime Number Theorem), fluid mechanics, optics and electrostatics (Conformal Mappings). Holomorphic functions in several complex variables were studied more recently, in the past one hundred years. Function theory in several complex variables is fundamentally different from the theory of one complex variable. Its development has influenced modern analysis in partial differential equations, harmonic analysis and complex geometry. The results of the project will have wide appeal to a large section of the mathematics community. It has already found applications in modern technologies that involve non-smooth boundaries such as electrospray and scattering. The principal investigator is engaged in the development of a textbook on one complex variable theory for partial differential equations to make these interesting topics more accessible to a wide range of students not just in mathematics, but also in physical sciences and engineering. The project provides training opportunities for graduate students and the principal investigator is actively involved in teaching and training students at all levels. The goal of the project is to explore several diverse and interconnected areas in several complex variables. The focus is on two of the most important equations in several complex variables: the Cauchy-Riemann equations and the tangential Cauchy-Riemann equations. Classical results on these equations require that the domains have smooth boundary, or the ambient spaces are Euclidean or Stein. A large part of the project is on domains with non-smooth boundary and on complex manifolds. The principal investigator has initiated the study of the Cauchy-Riemann equations on Lipschitz domains using harmonic analysis and geometric measure theory. The problems include the Hausdorff property of Dolbeault cohomology groups, Levi-flat hypersurfaces and complex foliation, Runge approximation and function theory on non-smooth domains and complex manifolds. These problems are central to several complex variables, complex geometry and partial differential equations.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.