Function theory in multiply-connected domains & applications to physical systems

多连通域是数学家们所说的带洞区域。在物理学中，这些洞可以对应许多不同的东西，例如流体中的气泡或大气中的漩涡运动区域（例如风暴系统或飓风）或竞争共同食物供应的不同细菌群。因此，多连通域的数学概念出现在日常现象研究的许多不同地方。为了理解这些现象，有必要研究和理解它们的数学模型。这需要专门针对这些现象发生的多连通域的数学函数和技术的知识。不幸的是，过去发展了多连通域中函数数学的数学家们并没有很好地将他们的结果转化为对描述和研究日常现象（如流体中的气泡或风暴系统的运动）感兴趣的科学家。然而，PI最近的工作表明，如果人们能够成功地翻译这些数学结果，并证明它们适用于这些各种日常现象，那么研究它们的科学家就可以使用强大的新技术，使他们的工作变得更加容易，并导致新的见解。本研究建议继续在这场运动中发展和应用经典函数论和复分析的数学成果，以解决现实生活中的问题。

Multiply-connected domains are what mathematicians call regions with holes . In physics, the holes can correspond to lots of different things such as air bubbles in fluids or regions of swirling motion (for example, storm systems or hurricanes) in the atmosphere or different clusters of bacteria competing for a common food supply. Thus, the mathematical concept of a multiply-connected domain occurs in many different places in the study of everyday phenomena. To understand such phenomena, it is necessary to study and understand mathematical models of them. This requires a knowledge of mathematical functions and techniques specially tailored to the multiply-connected domains in which these phenomena are taking place. Unfortunately, mathematicians in the past who have developed the mathematics of functions in multiply-connected domains have not done a very good job of translating the significance of their results to scientists interested in describing and studying everyday phenomena such as bubbles in fluids or the motion of storm systems. Yet, recent work by the PI has shown that if one can successfully translate these mathematical results and demonstrate their applicability to these various everyday phenomena, powerful new techniques become available to those scientists who study them, making their jobs much easier and leading to new Insights. This research proposes to continue in this crusade to develop and apply the mathematical results of classicalfunction theory and complex analysis to real-life problems.