Partial Differential Equations and Several Complex Variables

偏微分方程和多个复变量

• 批准号: 1101415
• 负责人: Mei-Chi Shaw
• 金额: $ 22万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101415&HistoricalAwards=false
• 关键词: Partial; Differential; Equations; Several; Complex

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.

多复变函数中最重要的两个方程是柯西-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分析研究的焦点。在这个项目中解决的问题包括Cauchy-Riemann方程和切Cauchy-Riemann复形，特别是在复流形上， 复射影空间（紧致且具有正曲率）和负曲率流形。 理解这些方程在曲率条件下的几何方面及其与复流形中函数论的关系 是复杂分析和几何中一些最具挑战性和最重要的问题。在几何或非光滑环境中对多个复变的研究提供了有趣的新问题 以新的视角来看待 拓扑学， 叶理理论，复动力学，代数 复杂的几何形状。复杂的几何 理论才刚刚开始发展， Shaw将继续朝着这个方向努力。她还将继续她的研究应用几何测度理论和谐波分析几个复杂的变量为非光滑域。自从庞加莱和哈托格斯在世纪前的开创性工作以来，多复变领域在现代数学中发挥了重要作用。在过去的几十年里，偏微分方程的使用一直是研究多个复变量以及复杂几何的主要工具。拟议活动的更广泛的影响是，这些问题是在分析，几何和拓扑学与应用数学和物理学的应用交叉。 除了提案中描述的数学领域外，非光滑域上的狄利克雷和诺依曼问题的最新进展在物理和工程等其他学科中也有应用。霍奇定理是经典狄利克雷原理的推广，狄利克雷原理是热传导问题中能量最小化问题的标准解。角域和楔域定理的最新应用已被用于电动力学和工程物理的其他领域。PI将在她的工作中使用所有这些想法来指导学生并编写一篇文本，使更广泛的数学家，特别是那些从事几何和复分析工作的数学家更容易理解其中一些偏微分方程主题。