Geometry of Banach Spaces and Metric Spaces

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Metric spaces, especially Banach spaces, form the conceptual framework in which mathematicians, scientists, and engineers work when investigating problems that involve estimation or approximation. Discrete metric geometry, including dimension reduction results established by the PI, is important in the design of algorithms and in compressed sensing. Non-linear phenomena often occurs in nature but is difficult to deal with. This makes it important to understand when non linearity actually conceals underlying linear structure, and this is central to the non linear study of Banach spaces. Parts of this research project are coordinated with the Workshop in Analysis and Probability Theory at Texas A&M University. The efforts of the principal investigator and other participants in the Workshop are helping to break down barriers between different areas of mathematics and also promote the outreach of pure mathematics to other sciences, especially to computer science.The problems in Banach space and metric geometry to be considered fall into several subcategories: the structure of the Banach algebra of bounded linear operators on classical Banach spaces, approximation properties of Banach spaces, the non linear classification of Banach spaces, and discrete metric geometry. These topics are at the heart of the geometries of Banach spaces and of metric spaces and make contact with many other areas within mathematics, including operator theory, group theory, geometric analysis, and linear algebra.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.