Problems in Function Theory with Applications

PI建议研究几何函数理论中的几个问题，这些问题具有共同的基本主题“共形不变量”，如双曲和小林度量、调和测度、格林函数和路径族的模。其中一个问题是研究非单连通域上p调和测度的维数性质。第二个问题是关于Chang-Marshall定理在空间上的推广，即解析函数的迹和它们在高维中的拟正则对应物，当被限制在边界上时的指数可积性。第三个问题研究了解析函数在一维和多维上的迭代，重点研究了复平面上单位圆盘和高维单位球的复杂动力学与双曲几何的相互作用。本研究还将利用上述共形不变量的性质，在大型网络的研究中获得具体应用。随着互联网的出现和分析大型数据库（所谓的海量数据集）的需要，这一领域变得更加突出。大多数人都熟悉的一个例子是搜索引擎。互联网搜索的工作方式是通过随机过程，不断地对网络进行采样，并定期返回平均值和其他统计数据。最简单的这种过程被称为随机爬行器或漫步器，控制其行为的数学推导来自几何函数理论中保形不变量的研究。PI正在进行的研究有望为将这种随机过程的行为与数据集的几何结构进行比较的任务带来新的工具。由于这些结果的广泛适用性，PI还将研究流行病爆发的问题。在这种情况下，PI已经从堪萨斯州立大学参与和社区发展中心获得了初步资助，与电气和计算机工程系的Scoglio教授和家庭研究系的Schumm教授合作开展了一个联合项目。我们的团队在堪萨斯州的沙努特市收集数据，并已经建立了一个“接触”网络，现在正在使用上面提到的共形不变量进行分析。最终目标是为沙努特市提供一套具体的指导方针，帮助该市官员缓解和管理流行病的爆发，尤其是源于农场的人畜共患疾病。

The PI proposes to study several problems in geometric function theory that have as common underlying theme "conformal invariants" such as the hyperbolic and Kobayashi metric, harmonic measure, Green's functions, and modulus of path families. One such problem consists in studying dimensionality properties of p-Harmonic measure on domains beyond simply-connected ones. A second question deals with generalizations of the Chang-Marshall theorem in space, namely with exponential integrability properties for the trace of analytic functions, and their quasiregular counterparts in higher dimensions, when restricted to the boundary. A third problems studies iteration of analytic functions in one and several dimensions with a focus on the interplay between complex dynamics and the hyperbolic geometry of the unit disk in the complex plane and of the unit ball in higher dimensions.This research will also draw on the properties of conformal invariants mentioned above to obtain concrete applications in the study of large networks. This is an area that has become more salient with the advent of the internet and the need to analyze large databases (so-called massive data-sets). One example that most people are familiar with is search-engines. The way internet searches work is through random processes that continually sample the web and periodically return averages and other statistics. The simplest such process is called a random crawler or walker and the mathematics that governs its behavior is derived from the study of conformal invariants in geometric function theory. The PI is conducting research that is expected to bring new tools to the task of comparing the behavior of such random processes to the geometry of the data-set. Because of the large applicability of such results the PI will also study the problem of epidemic outbreaks. In this context the PI has already obtained initial funding from the Center for Engagement and Community Development at Kansas State University for a joint project with Professor Scoglio in the Department of Electrical and Computing Engineering and Professor Schumm in the Department of Family Studies. Our team collected data in the city of Chanute, Kansas, and has already built a "contact" network, which is now being analyzed using the conformal invariants mentioned above. The ultimate goal is to provide the city of Chanute with a concrete set of directions that could help its city officials mitigate and manage an epidemic outbreak, especially one of zoonotic nature, originating on a farm.