Loewner theory and holomorphic mappings in several variables

单叶函数理论(即复平面上单位圆盘上的一对一复解析函数)是单复变中最美丽的话题之一。关于这类函数的极值问题，有许多值得注意的定理。也许最著名的是关于单叶函数的泰勒级数系数的比伯巴赫猜想，该猜想由Louis de Brange在1985年解决，但有时仍被称为比伯巴赫猜想。用来解决比伯巴赫猜想的最重要的技术之一是一种被称为洛夫纳方法的变分技术，在这种方法中，单叶函数被嵌入到由微分方程式控制的这种函数的扩展流中。*我研究的主要目的是看一元单叶函数理论的哪些方面可以推广到几个复变量，哪些不能。在可以推广的情况下，通常需要新的证明方法。在几个变量上的Loewner方法肯定是这样的，这是我的主要兴趣之一。这种类型的结果有助于更深入地理解一元单叶函数理论，并有助于研究多个复变量的映射问题的各种方法。*结果将有助于维护一个知识库，以便在社会需要时解决技术问题。他们将对从事复杂分析的数学家特别感兴趣。

The theory of univalent functions (i.e. one-to-one complex-analytic functions on the unit disc in the complex plane) is one of the most beautiful topics in one complex variable. There are many remarkable theorems dealing with extremal problems for this class of functions. Perhaps the most famous is the Bieberbach Conjecture for the Taylor series coefficients of univalent functions, solved in 1985 by Louis de Branges, but still sometimes known as the Bieberbach Conjecture. One of the most important techniques used to solve the Bieberbach Conjecture is a variational technique known as the Loewner method, in which a univalent function is embedded in an expanding flow of such functions governed by a differential equation. ******The main objective of my research is to see which aspects of univalent function theory in one variable can be generalized to several complex variables, and which cannot. In cases where a generalization is possible, it is frequently the case that new methods of proof are needed. This is certainly the case with the Loewner method in several variables, one of my main interests. Results of this type contribute to a deeper understanding of univalent function theory in one variable, and to the variety of methods available to study mapping questions in several complex variables. ******The results will contribute to maintaining a knowledge base which is available to solve technical problems when needed by society. They will have particular interest for mathematicians working in complex analysis.