Banach Spaces: Theory and Applications

Logistics planning, including optimal distribution of products, leads to questions about maps with weighted distances, and routes that minimize these distances. Transportation cost spaces, also known as Lipschitz-free spaces, Wasserstein spaces, Arens-Eals spaces, and Earthmover spaces, have been used to model such problems. They can be viewed as a framework to study nonlinear metric spaces by embedding them isometrically and linearly densely into Banach spaces and provide powerful tools to study the nonlinear geometry of Banach spaces using well-known linear techniques for nonlinear problems. These spaces play a fundamental role in many areas of applied mathematics, engineering, physics, computer science, finance, and social sciences. Finding an optimal embedding is known to be a computationally hard problem and it has become a central problem in computer science to find low distortion embeddings. Using methods from the structure theory of Banach spaces and computational graph theory, the investigator’s goal is to achieve more precise estimates of these embeddings. He will obtain a deeper understanding of the structure of these spaces, which will result in several applications to the areas mentioned above. The principal investigator plans to organize conferences as well as mentor Ph.D. students as a part of this project. A crucial connection exists between the L1-distortion of Transportation Cost Spaces and stochastic embeddings of the underlying metric space into trees. The investigator will further study this connection to obtain lower and upper estimations on the distortion. The second part of the project represents a contribution to Lindenstrauss’s program in determining Banach spaces that are primary, and that cannot be decomposed into essentially different subspaces. The investigator will continue to determine primary function spaces. This project concentrates on studying the primarity and related factorization properties of function spaces in two parameters, combining methods from Functional and Harmonic Analysis and Probability Theory. 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.