Mathematical Sciences: Interactions Between Functional Analysis, Harmonic Analysis and Probability

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9424396 Montgomery-Smith This project involves work on two different fronts. The first involves harmonic analysis on abelian groups whose dual is ordered. The project involves taking traditional results from harmonic analysis, specifically those concerning analyticity and the conjugate function, and generalizing them to a new setting. This project will be based on a synthesis of a classification theorem of Hahn on ordered groups together with modern techniques such as martingale theory, Browninan motion and the geometry of Banach spaces. The second project involves spectral properties of semigroups. This work has applications to the asymptotic behavior of differential equations in the case where the coefficients are infinite dimensional. Functional analysis seeks to understand the rigorous side of the kind of analysis that was needed to study functions that arise as solutions of differential equations. Harmonic analysis is concerned with studying the Fourier transform of functions. Probability, theory originally invented to understand events of chance, is now a technical subject related to measure theory. This project involves a blending of theses areas of mathematics. The results will be of use to a broad spectrum of mathematicians with the potential for application in systems theory.