Problems in Complex Analysis

复杂分析中的问题

• 批准号: 0705027
• 负责人: John Fornaess
• 金额: $ 33.6万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705027&HistoricalAwards=false
• 关键词: Problems; Complex; Analysis

The principal investigator plans to work on various problems in complex analysis. There are two main parts to the project. The first focuses on several complex variables, the second on complex dynamics. However, the line between these two areas is not clear-cut. Within the several complex variables portion, the central concept is the Cauchy-Riemann equation. The problems are concerned with either deriving estimates for this equation, or improving understanding of key concepts basic to it, or applying estimates for this equation to function theory. The second part of the project deals with complex dynamics. Its objective is to obtain an understanding of foliations by Riemann surfaces. These arise naturally from flows of holomorphic vector fields and lead to basic questions about currents. (The latter could just as well be thought of as belonging to several complex variables.) The principal investigator has a long-standing collaboration with Sibony in which the two develop the theory of complex dynamics. Most recently they have carried out some investigations of Riemann surface laminations. These occur in the iteration of a holomorphic map, in that its associated Julia set might have a lamination along which the map is more regular than it is in other directions. (Of course, laminations are also an interesting topic in their own right.) Until now it has been necessary for the study to impose extra conditions, such as hyperbolicity, on the singular points. The plan is to continue the study while allowing for more general singularities and to search for ergodic properties of the laminations. In two dimensions, positive closed currents can be approximated by currents of integration of Riemann surfaces. It would be useful for the study of dynamics in higher dimensions to have a similar geometric interpretation of currents. The PI and Sibony propose to work on this topic with Coman. The case of immediate interest is that of currents in three dimensions of bidimension (1,1).Mathematical analysis is an important tool for studying the world from a quantitative perspective. But already when one tries to find roots of polynomial equations, it becomes clear that complex numbers and complex analysis are needed. Hence it is important to develop complex analysis, which can then be used to build up other areas of mathematics (e.g., algebraic geometry, number theory). This project explores various aspects of complex analysis. The plan is for the principal investigator to work with two senior mathematicians, Diederich and Sibony. Together the three have a good overview of the fields of several complex variables and complex dynamics, which benefits not only their joint projects but also their work with younger mathematicians. They will work with two midcareer mathematicians (Coman and Lanzani) and with a group of postdocs (Heier, Herbig, Lee, Sahutoglu, Siano, and Wold). The project will also involve a number of graduate students, including five women, from both the University of Michigan and other institutions.

首席研究员计划研究复分析中的各种问题。该项目有两个主要部分。第一个侧重于几个复杂的变量，第二个复杂的动力学。然而，这两个领域之间的界限并不明确。在多复变量部分中，中心概念是柯西-黎曼方程。这些问题涉及到这个方程的估计，或提高对它的基本关键概念的理解，或将这个方程的估计应用于函数论。该项目的第二部分涉及复杂的动态。其目的是通过黎曼曲面获得对叶理的理解。这些自然产生于全纯向量场的流动，并导致关于电流的基本问题。(The后者也可以被认为是属于几个复杂的变量。首席研究员与Sibony有着长期的合作关系，两人共同开发了复杂动力学理论。最近，他们进行了一些调查的黎曼表面层。这些发生在全纯映射的迭代中，因为它的相关Julia集可能有一个层沿着，映射比它在其他方向上更规则。(Of当然，叠层本身也是一个有趣的话题。）到目前为止，该研究有必要对奇异点施加额外的条件，例如双曲性。计划是继续研究，同时允许更一般的奇点，并寻找遍历性能的叠层。在二维空间中，正闭合流可以近似为黎曼曲面积分流。如果对电流有类似的几何解释，对高维动力学的研究将是有用的。PI和Sibony建议与Coman一起研究这个主题。直接感兴趣的情况是二维（1，1）中三维电流的情况。数学分析是从定量角度研究世界的重要工具。但是，当人们试图找到多项式方程的根时，很明显需要复数和复分析。因此，重要的是要发展复杂的分析，然后可以用来建立数学的其他领域（例如，代数几何、数论）。这个项目探讨了复杂分析的各个方面。该计划是为首席研究员与两名高级数学家，Diederich和Sibony。三人一起对几个复变量和复杂动力学领域有了很好的概述，这不仅有利于他们的联合项目，也有利于他们与年轻数学家的合作。他们将与两个职业中期的数学家（科曼和Lanzani）和一组博士后（海尔，赫比格，李，Sahutoglu，Siano和沃尔德）。该项目还将涉及密歇根大学和其他机构的一些研究生，包括5名妇女。