The Geometric Function Theory and its Applications

本项目提出了复变分析中几何函数理论及其应用的进一步研究。几何函数理论的目标是在定量方法失败时获得被研究对象的定性特征，它由几个主题表示。(1)研究复流形上全纯环空间的商，这些商决定了复流形的几个内在性质。引入不同的等价关系，可以构造从类似于经典基本群的全纯基本群到复流形的展开的对象，其中所有的全纯函数和多重亚调和函数都可以扩展。在这个方向上的最初工作已经显示出分析、几何和拓扑学的有趣混合。(2)紧集上的多势理论。在这种情况下，像Monge-Ampere方程这样的主要工具不起作用，而许多经典问题都自然地归结为紧致情况。特别地，它是极大多次调和函数的多次调和纤维的存在性问题。由于已知通常不存在通过复曲线的纤颤，PI将通过紧集来寻找纤颤，使得最大多重亚调和函数对这些集的限制是多重调和的。(3)超凸域上的Hardy和Bergman空间理论。这是另一个可以用几何方法处理的话题。这里的主要目标是Poisson核与域的几何以及复合算子的应用之间的联系。作为几何函数论的应用，PI将研究诸如一致代数、整函数的代数相依性和逼近理论等相关学科。如果19世纪和20世纪数学的重大进步是由于我们更好的量化技能，那么现代数学变得太复杂了，无法用公式来描述。例如，像分形图这样重要的物体不能用任何方程来描述。这个项目打算用几何学来研究复杂的数学问题。几何作为数学的一部分，目的是描述不同物体之间的定性联系，而不是定量联系。然而，几何联系可以提供有关研究对象的定量信息。该项目的主要目标是开发工具，当涉及复数时，该工具可以允许我们找到函数的属性和其属性域的属性之间的几何联系。反过来，这些联系的知识将使我们更好地理解数学的其他领域，如：复空间的几何、超越数和多项式逼近函数的理论。作为一个更广泛的影响，这个项目的资金将支持锡拉丘兹大学分析小组的基础设施，该小组在历史上一直非常强大。这项研究提出的一些问题将导致研究生的博士论文主题，而另一些问题将提供一个让本科生参与数学研究的机会。这笔资金将用于允许博士生参加学习和传播自己成果的会议。将作为提案的一部分编写的一本教科书将允许中等规模大学的学生接触到这门重要的学科。

This project proposes the further study of the geometric function theory in complex analysis and its applications. The geometric function theory, whose goal is to obtain qualitative characteristics of investigated objects when quantitative approach fails, is represented by several topics. (1) Studies of quotients of holomorphic loop spaces on complex manifolds, which are responsible for several intrinsic properties of the manifolds. Introducing different equivalence relations it is possible to construct objects ranging from holomorphic fundamental groups similar to classical fundamental groups to expansions of complex manifolds, where all holomorphic and plurisubharmonic functions can be extended. The initial work in this direction has shown an interesting mixture of analysis, geometry and topology. (2) Pluripotential theory on compact sets. In this setting such major tool as the Monge-Ampere equation does not work, while many classical questions have a natural reduction to the compact case. In particular, it is a problem of the existence of pluriharmonic fibrations of maximal plurisubharmonic functions. Since it is known that, in general, fibrations by complex curves do not exist, the PI will look for fibrations by compact sets such that the restrictions of a maximal plurisubharmonic function to these sets are pluriharmonic. (3) The theory of Hardy and Bergman spaces on hyperconvex domains. This is another topic that can be approached by geometric methods. Here the primary goal is the connection of the Poisson kernels with the geometry of the domains and applications to composition operators. As applications of the geometric function theory the PI is going to study such related subject as: uniform algebras, algebraic dependence of entire functions and approximation theory.If the major advances in mathematics achieved in 19th and 20th centuries were due to our better quantitative skills, the modern mathematics became too complicated to be described by formulas. For example, such important objects as fractals cannot be described by any equations. This project intends to study complex mathematical problems using geometry. Geometry, as a part of mathematics, aims to describe qualitative, rather than quantitative, links between different objects. However, the geometric links can provide quantitative information about the subjects of study. The main goal of the project is to develop tools which can allow us to find geometric links between properties of functions and properties of their domains when complex numbers are involved. In its turn the knowledge of these links will lead us to better understanding of other areas of mathematics such as: the geometry of complex spaces, the transcendental numbers and the theory of approximation of functions by polynomials. As a broader impact, funding for this project will support the infrastructure of the Syracuse University's analysis group, which has historically been very strong. Some of the questions raised by this research will lead to PhD dissertation topics for graduate students, and others will provide an opportunity to involve undergraduate students in mathematical research. The funds will be used to allow PhD students to attend conferences for learning and disseminating their own achievements. A textbook which will be written as a part of the proposal will allow students in mid-size universities to access this important subject.