Partial Differential Equations and Several Complex Variables

复数变量中两个最重要的方程是柯西-黎曼方程和诱导切向柯西-黎曼方程。在过去的几十年里，对这些方程的理解一直是复杂分析研究的焦点。本课题研究的问题包括复流形上的柯西-黎曼方程和切向柯西-黎曼复，特别是复射影空间（紧化且具有正曲率）和负弯曲流形上的柯西-黎曼复。理解这些方程在曲率条件下的几何方面及其与复杂流形中函数理论的关系是复杂分析和几何中最具挑战性和最重要的问题。在几何或非光滑环境中对几个复杂变量的研究为拓扑学、叶理理论、复杂动力学、代数和复杂几何等问题提供了有趣的新问题和新的见解。复杂几何理论才刚刚开始发展，肖将继续在这个方向上努力。她还将继续研究将几何测度理论和谐波分析应用于非光滑域的几个复杂变量。自从一个多世纪前庞加莱和哈托格斯的开创性工作以来，复数变量领域在现代数学中发挥了重要作用。在过去的几十年里，偏微分方程的使用已经成为研究一些复杂变量以及复杂几何的主要工具。拟议活动的更广泛影响是，这些问题是分析、几何和拓扑学与应用数学和物理应用的交叉点。除了在提案中描述的数学领域之外，最近在非光滑域上的狄利克雷和诺伊曼问题的进展已经在其他学科如物理和工程中得到了应用。霍奇定理是经典狄利克雷原理的扩展，狄利克雷原理是由传热问题引起的能量最小化问题的典型解。角楔域定理的最新应用已经应用于电动力学和其他工程和物理领域。PI将在她指导学生的工作中使用所有这些想法，并撰写一篇文章，使偏微分方程的一些主题更容易被更广泛的数学家所理解，特别是那些从事几何和复杂分析的数学家。

Two of the most important equations in several complex variables are the Cauchy-Riemann equations and the induced tangential Cauchy-Riemann equations. The understanding of these equations have been the focal point of research in complex analysis in the past few decades. The problems addressed in this project include the Cauchy-Riemann equations and the tangential Cauchy-Riemann complex on complex manifolds, especially on complex projective spaces (which is compact and with positive curvature) and negatively curved manifolds. Understanding the geometric aspects of these equations under the curvature conditions and their relations with function theory in complex manifolds are some of the most challenging and important problems in complex analysis and geometry. The study of several complex variables in a geometric or non-smooth setting has provided interesting new questions with fresh insight to problems in topology, foliation theory, complex dynamics, algebraic and complex geometry. Complex geometric theory has only just begun to develop and Shaw will continue her efforts in this direction. She will also continue her research on applying the geometric measure theory and harmonic analysis to several complex variables for non-smooth domains. Since the pioneering work of Poincare and Hartogs more than a century ago, the field of several complex variables has played a major role in modern mathematics. The use of partial differential equations has been the main tool for studying several complex variables, as well as complex geometry in the past few decades. The broader impacts from the proposed activity are that these problems are at the intersection of analysis, geometry and topology with applications in applied mathematics and physics. Other than the mathematical areas described in the proposal, recent progress in the Dirichlet and Neumann problem on nonsmooth domains has found applications in other disciplines like physics and engineering. The Hodge theorem is an extension of the classical Dirichlet Principle, the canonical solution to the energy minimizing problem arising from the heat transfer problem. Recent applications of the theorem on domains with corners and wedges have been used in electrokinetics and other fields in engineering and physics. The PI will use all of these ideas in her work mentoring students and the writing of a text that makes some of these partial differential equations topics more accessible to a wider range of mathematicians, especially those working in geometry and complex analysis.