Banach Spaces: Theory and Applications

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A focus of this project is the theory of Banach spaces and their geometry which provide an important conceptual framework to study problems in engineering, physics, signal processing, and the analysis of large data sets. The second main object of this investigation are "graphs". In computer science they are used to represent networks of communication, data organization, computational devices. An embedding of graphs, or more generally metric spaces, which may represent a large data set, into a Banach space, is necessary to be able to store this data set efficiently. Secondly it is necessary to provide the Banach space with the right coordinate system, it is therefore also necessary to be able to embed a Banach space, into a Banach space admitting the appropriate coordinate system. The problems that will be studied in this project revolve around the issue of embedding less structured mathematical objects like graphs or, more generally, metric spaces, into better structured objects like Banach spaces. On one hand the goal is to obtain information about the structure of the graph, from the property that it embeds in certain Banach spaces, and on the other hand deduce geometric properties of a Banach space, from the property that certain graphs embed or do not embed in it.The principal investigator will investigate possible extensions of Ribe's Program on metric characterizations of local properties of Banach spaces to characterizing asymptotic properties. An important question for example is the question whether or not the property of a Banach space to be reflexive can be metrically characterized. Other important properties to be considered are uniform asymptotic convexity and uniform asymptotic smoothness. The principal investigator will also study the problem of isomorphically embedding Banach spaces, having a certain property, into Banach spaces with a coordinates system like a Schauder basis or an unconditional basis. Finally the principal investigator will continue the study of closed sub-ideals of the spaces of operators on specific Banach spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.