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

多重连通域就是数学家所说的有洞的区域。在物理学中，这些洞可以对应许多不同的东西，比如流体中的气泡或大气中旋转运动的区域（例如，风暴系统或飓风），或者争夺共同食物供应的不同细菌群。因此，在日常现象的研究中，多重连通域的数学概念出现在许多不同的地方。要理解这些现象，就必须研究和理解它们的数学模型。这需要数学函数的知识和专门为这些现象发生的多重连接领域量身定制的技术。不幸的是，过去在多重连接域中发展了函数数学的数学家们，并没有很好地将他们的结果的意义传达给那些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，如果一个人能够成功地翻译这些数学结果，并证明它们适用于各种日常现象，那么研究这些数学结果的科学家就可以使用强大的新技术，使他们的工作更容易，并带来新的见解。本研究建议继续在这一运动中发展和应用经典函数理论和复杂分析的数学结果来解决现实问题。

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.