Loewner theory and holomorphic mappings in several variables

Loewner 理论和多变量的全纯映射

• 批准号: RGPIN-2015-04290
• 负责人: Graham, Ian
• 金额: $ 1.02万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612881
• 关键词: Loewner; theory; holomorphic; mappings; several

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.

单叶函数（即复平面上单位圆上的一对一复解析函数）理论是单复变函数中最美丽的课题之一。有许多显着的定理处理极值问题的这类功能。也许最著名的是比伯巴赫猜想的泰勒级数系数的单叶函数，解决了在1985年由路易斯德布兰日，但有时仍然被称为比伯巴赫猜想。用于解决比伯巴赫猜想的最重要的技术之一是被称为Loewner方法的变分技术，其中一个单叶函数嵌入在由微分方程控制的此类函数的扩展流中。 我的研究的主要目的是看看在一个变量的单叶函数理论的哪些方面可以推广到多个复杂的变量，而不能。在可能推广的情况下，通常需要新的证明方法。这当然是Loewner方法在多个变量的情况下，我的主要兴趣之一。这种类型的结果有助于更深入地了解单叶函数理论在一个变量，并提供各种方法来研究映射问题在几个复杂的变量。 研究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。他们将特别感兴趣的数学家在复杂的分析工作。