Nonlinear and noncommutative perspectives on Banach space theory

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For many centuries the classical mathematical tools of basic calculus and algebra have been widely used in the study of many real-world phenomena. However there are two important assumptions that play a crucial role in their usefulness that do not always hold. First, calculus can only be used when, like the surface of the earth, the quantities of interest "appear to be flat" from up close. In addition, many properties in algebra use the fact that when multiplying numbers, changing the order of the factors does not affect their product. Unfortunately there many real-world situations that cannot be modeled adequately under these assumptions. For example, when an online merchant recommends a particular product based on previous purchases, the information that is relevant consists of quantities whose variations can be more accurately described as "jumps": whether or not we buy a particular product causes a sudden change in quantity sold. In quantum physics, or physics at nanoscopic scales, the order in which factors are multiplied can affect the product. The PI proposes in this project to develop modern versions of existing mathematical tools in the absence of those seemingly simple assumptions. The particular tools that will be considered are relevant for computer science and quantum physics.The PI will carry on a program that studies the counterparts of various aspects of Banach space theory (mainly related to study of ideals of operators and approximation properties) in the context of metric spaces and operator spaces. This will be done by combining methods and techniques from the local theory of Banach spaces with modern nonlinear and noncommutative approaches. The nonlinear aspects include using Lipschitz-free spaces to investigate whether or not certain approximation properties for Banach spaces are invariant under Lipschitz isomorphisms; and also answering quantitative questions about embedding finite metric spaces into Hilbert spaces. On the noncommutative side, the project aims at proving composition theorems for certain classes of mappings acting between operator spaces, constructing almost Euclidean subspaces of finite-dimensional spaces of trace-class operators, and defining a Radon-Nikodym property for operator spaces.