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

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.