Problems in Function Theory with Applications

PI建议研究几何函数论中的几个问题，这些问题的基本主题是“共形不变量”，如双曲和小林度量、调和度量、格林函数和路族的模。其中一个问题是研究单连通区域以外区域上p-调和测度的维度性质。第二个问题涉及张马歇尔定理在空间中的推广，即解析函数的迹的指数可积性，以及当限制在边界上时，它们在高维上的拟正则对应。第三个问题研究一维和多维解析函数的迭代，重点研究复动力学与复平面中单位圆盘的双曲几何和高维单位球的双曲几何之间的相互作用。这项研究还将利用上述共形不变量的性质来获得在大型网络研究中的具体应用。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游器，控制其行为的数学是从几何函数论中对共形不变量的研究得出的。PI正在进行研究，预计将为将这种随机过程的行为与数据集的几何形状进行比较的任务带来新的工具。由于这些结果的适用性很大，PI还将研究疫情暴发问题。在这方面，国际和平协会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金，用于与电气和计算工程系的Scoglio教授和家庭研究系的Schumm教授共同开展的项目。我们的团队在堪萨斯州的查努特市收集了数据，并已经建立了一个“联系”网络，现在正在使用上面提到的共形不变量进行分析。最终目标是为查努特市提供一套具体的方向，帮助该市官员缓解和控制源自农场的疫情，特别是人畜共患病的疫情。

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.