Banach Spaces: Theory and Applications

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General metric spaces, especially graphs, are used to represent networks of communication, data organization, and computational devices. In order to analyze, use and store them it is necessary to understand their “Structure”, in mathematical language this means to understand their “Geometry”. For example this means, to embed them into better structured spaces with a coordinate system. Banach spaces, are the conceptual framework to accomplish that. A main focus of this project is finding embeddings of metric spaces, used in Computer Science, into certain Banach spaces. Examples for such metric spaces are Lamplighter Graphs, which model the traveling salesman problem, and the Transportation Cost Metric Space, which models the optimal distribution of products. On the other hand, from the embeddability, or non-embeddability of certain metric spaces into a given Banach space, one can deduce important geometric properties of that Banach space. This award will support the principal investigator's research on these topics and also contribute to US workforce development through the training of graduate students. The first part of the project is concerned with the question whether or not certain metric spaces, like the Lamplighter space and Transportation Cost space, can be coarsely, uniform or bi-Lipschitzly embedded in the Banach space of integrable functions, and estimate the distortion of these embeddings. In the second part, the investigator seeks to find metric characterizations of several isomorphic properties of Banach spaces via the embeddabilty, or non embeddability, of certain metric graphs. The research will concentrate on finite asymptotic properties with the goal of establishing an asymptotic version of the in recent years successful Ribe Program on metric characterizations of local properties. The investigator will also continue his investigation on the metric interpretations of reflexivity, with the hope of finding metric test spaces which characterize reflexivity. A central part of studying Banach spaces and their geometry is the study of (linear and bounded) operators on them. Very recently it was discovered that for a large class of separable Banach spaces, the cardinality of closed subideals of the space of operators is the cardinality of the collection of subsets of the continuum. This is, by simple set theoretic considerations, the maximal possible. The PI will investigate the closed ideals of spaces of operators of more classes of 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.