Banach space theory treats infinite-dimensional phenomena within the framework of rigorous mathematics. It has historical connections to more concrete applied subjects because, for instance, solutions to differential equations are often conveniently thought of as inhabiting an appropriate Banach space. In physics, one frequently encounters systems with infinitely many degrees of freedom which must be modeled in Hilbert space, a special type of Banach space. Two interesting things to do with a Banach space are to slice it (technically, project it onto a complemented subspace) and to approximate it by finite dimensional subspaces, in which one can so to speak draw pictures and do geometry. The principal investigator for this project will work on a famous open problem that asks what the slices of a certain standard Banach space look like, and will investigate the geometry of finite-dimensional subspaces of Banach spaces.
