Partial Differential Equations in Several Complex Variables

多个复变量的偏微分方程

• 批准号: 0801200
• 负责人: Mei-Chi Shaw
• 金额: $ 17.25万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801200&HistoricalAwards=false
• 关键词: Partial; Differential; Equations; Several; Complex

Since the pioneering work of Kohn and Hormander several decades ago, partial differential equations have been among the main tools for studying several complex variables. Morever, advances in one of these fields has frequently triggered advances in the other. Two of the most important examples of such equations are the Cauchy-Riemann equations and the tangential Cauchy-Riemann equations. The problems under consideration in this project include: the Cauchy-Riemann equations and the tangential Cauchy-Riemann equations on complex projective spaces, the geometric aspects of these equations with curvature terms, and their relationship to function theory in complex manifolds. The behavior of these equations under the curvature condition has not been explored systematically earlier, and an understanding of them in this context holds promise for impact on algebraic and complex geometry. New applications of these equations to complex foliation theory also yield important results in topology, geometry, and dynamics. For example, the principal investigator's recent work on the nonexistence of Levi-flat hypersurfaces in complex projective spaces is a holomorphic version of the classical Poincare-Bendixson theorem in dynamical systems. An important aspect of the research that will be done in this project rests in the fact that these problems are central to several different areas, namely, complex variables, complex geometry, and partial differential equations. These classical fields in mathematical analysis have contributed to our understanding of many important phenomena in physics. Since physical problems quite often involve variations in both time and space, the equations that govern them often need to be expressed in the language of complex variables. Many such equations involving several complex variables in a geometric setting have yet to be thoroughly analyzed, and many of the associated physical phenomena, such at the existence of a conic drop in an alternating-current electric field, have yet to be satisfactorily explained. The principal investigator hopes to tackle such remaining and new problems in geometry and topology. She interacts with mathematicians from many different areas in different parts of the world. She has taught courses on the subjects to students and researchers alike in many countries. Such interdisciplinary and international approaches bring new perspective and greater depth to each of the fields involved. In particular, recent progress in the so-called Dirichlet and Neumann problem on nonsmooth domains has found application in other disciplines, for instance, in physics and engineering.

自Kohn和Hormander几十年前的开创性工作以来，偏微分方程一直是研究多复变量的主要工具之一。然而，其中一个领域的进步往往会引发另一个领域的进步。这类方程的两个最重要的例子是柯西-黎曼方程和切向柯西-黎曼方程。在这个项目中考虑的问题包括：柯西-黎曼方程和切向柯西-黎曼方程的复射影空间，这些方程的几何方面与曲率项，以及它们的关系，函数理论在复杂的流形。这些方程在曲率条件下的行为以前没有被系统地探索过，在这种情况下对它们的理解有望对代数和复几何产生影响。新的应用这些方程复杂的叶理理论也产生了重要的结果，拓扑结构，几何和动力学。例如，首席研究员最近关于复射影空间中列维平坦超曲面的不存在性的工作是动力系统中经典庞加莱-本迪克森定理的全纯版本。一个重要的方面的研究，将在这个项目在于事实上，这些问题是中央的几个不同的领域，即，复杂的变量，复杂的几何，偏微分方程。数学分析中的这些经典领域有助于我们理解物理学中的许多重要现象。由于物理问题经常涉及时间和空间的变化，因此控制它们的方程经常需要用复变量的语言来表达。许多这样的方程涉及几个复杂的变量在几何设置尚未得到彻底的分析，许多相关的物理现象，如在存在的圆锥形下降在交流电场，尚未得到满意的解释。首席研究员希望解决几何和拓扑学中的这些遗留和新问题。她与来自世界不同地区的许多不同领域的数学家进行互动。她曾在许多国家为学生和研究人员教授有关这些主题的课程。这种跨学科和国际方法为所涉及的每个领域带来了新的视角和更大的深度。 特别是，最近的进展，在所谓的狄利克雷和诺依曼问题的非光滑域已发现在其他学科的应用，例如，在物理学和工程学。