Mathematical Sciences: Complex Analysis and Potential Theory

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In this project the principal investigator will study a number of problems in complex analysis and potential theory. In particular, he will investigate the global behavior of solutions of holomorphic Cauchy problems, the spectra of various potential theoretic integral operators, the cyclicity of the shift operator on Bergman spaces, and the reflection of harmonic functions in higher dimensions. The techniques used to attack such a diversity of problems are drawn from the theory of partial differential equations, complex analysis and potential theory. Historically the areas of mathematics known as the theory of partial differential equations and the theory of complex variables have been closely associated with each other. Indeed much of nineteenth century mathematics revolved around this relationship between partial differential equations and complex variables, with each area having a profound influence on the other. In this project the principal investigator will use this symbiotic relationship between the two areas to study a number of interesting problems involving harmonic functions in several space dimensions. ***//