Metric spaces, especially those so-called 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 that has been established by the PI and a collaborator, is important in the design of algorithms and in compressed sensing. Work on the classification of operators that are commuters was originally motivated by the uncertainty principle in quantum mechanics (which, from a mathematical point of view, arises because multiplication of operators, unlike multiplication of numbers, is not commutative). Nonlinear phenomenon 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 nonlinear study of Banach spaces. Parts of this research project are coordinated with the Workshop in Analysis and Probability Theory at Texas A&M University, of which the PI is Associate Director. The Workshop encourages interactions among researchers and apprentices in different mathematical fields by bringing together graduate students and junior and senior postdoctoral participants in several areas of analysis. Activities of the Workshop include seminars, Concentration Weeks, introductory lectures, and an annual conference. The efforts of the principal investigators 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: commutators of operators on Banach spaces, approximation properties of Banach spaces, the structure of finite and infinite dimensional spaces of p-integrable functions, the nonlinear classification of Banach spaces, discrete metric geometry, quantitative linear algebra, and cluster value problems in an infinite dimensional setting. 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, as well as with other areas of science, including the design of algorithms and compressed sensing.