Geometry of Banach spaces and metric spaces

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The principal investigator and his graduate students will investigate problems in the geometry of Banach spaces and the geometry of metric spaces. The problems to be considered fall into several subcategories; namely, commutators of operators on a Banach space, non linear factorization of linear operators, non linear classification of Banach spaces, discrete metric geometry, approximation properties, and miscellaneous problems. Sample problems are: 1. If a Banach space has a Pelczynski decomposition and the space of bounded linear operators on the space has a unique maximal ideal, must every bounded linear operator that is not a commutator be the sum of a non zero scalar plus an operator in the maximal ideal? 2. Must a Lipschitz complemented subspace of a separable Banach space be linearly complemented? 3. Estimate the smallest dimension k such that every n point subset of the space of integrable functions embeds, with distortion at most D, into the k dimensional space normed by the sum of the absolute values of the coordinates. (Here k will depend on both n, the number of points in the subset, and D, the allowable distortion).The topics treated in this project make contact with many areas of analysis and areas outside of analysis, including operator theory, group theory, geometric analysis, and theoretical computer science. One of the graduate students is involved in developing the conceptual framework for understandings mappings of large scale, high dimensional data sets into the normed vector spaces that are widely used in science and engineering. This work and other aspects of the discrete metric geometry part of the project can be used in the design of algorithms. Moreover, positive results on dimension reduction will have applications in compressed sensing. The questions on the factorization of non linear operators are connected to recurring problems in geometric analysis, while the linear finite dimensional investigation belong as much to convex geometry as to analysis. Aspects of the non linear classification problems may be as important for geometric group theory as for geometric functional analysis.