The Geometric Function Theory and its Applications

本项目旨在进一步研究复分析中的几何函数理论及其应用。几何函数理论的目标是在定量方法失效时获得研究对象的定性特征，它由几个主题表示。(1)研究复流形上全纯圈空间的导子，它们决定了复流形的若干内在性质。 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，其中所有的全纯和pluissubharmonic函数都可以被扩展。在这个方向上的初步工作显示了一个有趣的混合分析，几何和拓扑。(2)紧集上的多势理论。在这种情况下，像蒙格-安培方程这样的主要工具就不起作用了，而许多经典问题自然地简化为紧凑的情况。特别地，它是极大多重次调和函数的多重调和纤维化的存在性问题。由于已知，一般来说，不存在由复曲线引起的纤维化，PI将寻找由紧集引起的纤维化，使得最大多次调和函数对这些集的限制是多调和的。(3)超凸域上的哈代和Bergman空间理论。这是另一个可以用几何方法处理的问题。这里的主要目标是连接的泊松内核与几何的域和应用程序的组合操作。作为几何函数论的应用，PI将研究一致代数，整函数的代数依赖和逼近理论等相关主题。如果说19世纪和20世纪数学的重大进步是由于我们更好的定量技能，那么现代数学变得太复杂而无法用公式描述。例如，像分形这样重要的物体不能用任何方程来描述。该项目旨在使用几何来研究复杂的数学问题。几何学作为数学的一部分，旨在描述不同对象之间的定性联系，而不是定量联系。然而，几何链接可以提供关于研究对象的定量信息。该项目的主要目标是开发工具，使我们能够找到函数属性和涉及复数时其域属性之间的几何联系。反过来，知识的这些联系将使我们更好地了解其他领域的数学，如：几何复杂的空间，超越数和理论的逼近函数的多项式。作为更广泛的影响，该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。本研究提出的一些问题将导致博士论文的研究生课题，和其他将提供一个机会，让本科生参与数学研究。这些资金将用于让博士生参加会议，学习和传播自己的成就。作为提案的一部分，将编写一本教科书，使中等规模大学的学生能够接触到这一重要学科。

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.