The Geometry of Banach Spaces and Its Applications

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The principal investigator will investigate the geometry of Banach spaces and its applications to many other areas of mathematics, especially logic and set theory. The techniques of Banach space theory provide strong tools for the study of operator theory, Fourier analysis and frame theory. More specifically, the investigator will study the factorizations of operators between classical Banach spaces such as the space of p-power integrable functions and p-power summable sequences. Intrinsic classifications of commutators of operators on Banach spaces with Pelczynski decomposition will be explored. The theory of Lipschitz p-summing, Lispchitz p-integral and Lipschits p-nuclear operators between metric spaces will be developed. Metric spaces with the Lipschitz lifting property will also be investigated. Necessary and sufficient conditions for Schauder frames to contain a subsequence that is a Schauder basis will be studied as well.Banach spaces provide a framework for linear and non-linear functional analysis, probability, and optimization and are of fundamental importance in partial differential equations and mathematical models in quantum mechanics. Hence, the impact of advances in this branch of pure mathematics is far-reaching, extending across many areas of mathematics as well as to theoretical physics and multiple applications related to computer science and engineering. For instance, the study of Lipschitz mappings between metrics spaces has shown great impact on theoretical computer science and the uncertainty principle in quantum mechanics is ultimately a theorem about commutators of operators. The principal investigator will also support one graduate student on this project, and priority will be placed on selecting a qualified student from historically underrepresented background.