Partial Differential Equations in Several Complex Variables

自从几十年前Kohn和Hormander的开创性工作以来，偏微分方程一直是研究几个复变量的主要工具之一。更重要的是，其中一个领域的进步经常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-黎曼方程和切线柯西-黎曼方程。本项目所考虑的问题包括：复射影空间上的Cauchy-Riemann方程和切线Cauchy-Riemann方程，这些方程的带有曲率项的几何方面，以及它们与复流形中函数理论的关系。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杂几何产生影响。这些方程在复杂叶理理论中的新应用也在拓扑学、几何学和动力学方面产生了重要的结果。例如，主要研究者最近关于复射影空间中Levi平坦超曲面不存在的工作是动力系统中经典Poincare-Bendixson定理的全纯版本。这个项目要做的研究的一个重要方面在于，这些问题是几个不同领域的中心，即复变量、复杂几何和偏微分方程。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上的变化，控制它们的方程通常需要用复变量的语言来表示。许多这样的方程在几何背景下涉及多个复变量，还没有得到彻底的分析，许多相关的物理现象，如交流电场中存在圆锥形液滴，还没有得到令人满意的解释。首席研究员希望解决几何学和拓扑学中的这些遗留问题和新问题。她与来自世界不同地区不同地区的数学家互动。她曾在许多国家向学生和研究人员讲授这些学科的课程。这种跨学科和国际化的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，在非光滑区域上的所谓Dirichlet和Neumann问题的最新进展已经在其他学科中得到了应用，例如在物理和工程中。

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.