Problems in Complex Analysis

首席研究员计划研究复杂分析中的各种问题。这个项目有两个主要部分。前者关注几个复杂变量，后者关注复杂动力学。然而，这两个领域之间的界限并不明确。在几个复变量部分中，中心概念是柯西-黎曼方程。这些问题要么是关于推导这个方程的估计，要么是关于提高对它的关键概念的理解，要么是关于将这个方程的估计应用到函数理论中。项目的第二部分涉及复杂的动力学。它的目标是通过黎曼曲面获得对叶理的理解。它们自然地从全纯矢量场的流动中产生，并导致关于电流的基本问题。（后者也可以被认为属于几个复杂变量。）首席研究员与Sibony长期合作，两人共同发展了复杂动力学理论。最近，他们进行了一些黎曼表面层合的研究。这种情况发生在全纯映射的迭代中，因为它的相关Julia集合可能有一个分层，沿着这个分层，映射比在其他方向上更规则。（当然，叠片本身也是一个有趣的话题。）到目前为止，对奇异点的研究有必要施加额外的条件，如双曲性。计划是继续研究，同时允许更多的一般奇点，并寻找层合的遍历性质。在二维中，正闭合电流可以用黎曼曲面的积分电流来近似表示。如果对电流有类似的几何解释，将有助于高维动力学的研究。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）的三维电流。数学分析是从定量角度研究世界的重要工具。但是当一个人试图求多项式方程的根时，很明显需要复数和复分析。因此，发展复分析是很重要的，它可以用来建立其他数学领域（例如，代数几何，数论）。这个项目探讨了复杂分析的各个方面。计划是让首席研究员与两位资深数学家Diederich和Sibony合作。他们三人对几个复杂变量和复杂动力学领域有一个很好的概述，这不仅有利于他们的共同项目，也有利于他们与年轻数学家的合作。他们将与两位处于职业生涯中期的数学家（Coman和Lanzani）以及一组博士后（Heier, Herbig, Lee, Sahutoglu， Siano和Wold）合作。该项目还将涉及来自密歇根大学和其他机构的一些研究生，其中包括五名女性。

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.