This project deals with a number of problems concerning Banach space theory and applications to other areas of mathematics. Part of the project deals with the theory of extensions of Banach spaces and related problems about the existence of extensions of linear operators between Banach spaces. For example, one central problem concerns identifying those subspaces of the classical Lebesgue space of integrable functions that have the property that every operator into a Hilbert space can be extended to the whole space. This amounts to finding subspaces that verify Grothendieck's theorem and is equivalent to the problem of finding spaces with the property that every extension by a Hilbert space is trivial. Similar problems for spaces of continuous functions in place of Hilbert space are also of interest. A second part of the project deals with the nonlinear theory of Banach spaces; a typical open problem here is whether two Lipschitz isomorphic separable Banach spaces are always linearly isomorphic. A third part deals with the study of sectorial operators and possible applications of that study to partial differential equations.
Extensions of Banach spaces have a geometrical interpretation in finite dimensions. If, given a centrally symmetric convex set, one only has information about a slice in some directions and the shadow cast in the perpendicular directions, one cannot reconstruct the origin set. However, the hope is to reconstruct the body as accurately as possible, even though this information is incomplete. This is a concrete way of looking at many questions in analysis that arise in quite different forms. The nonlinear theory of Banach spaces and the related Lipschitz structure of metric spaces has potential applications in theoretical computer science in the handling of large data sets. Sectorial operators and semigroups are a basic tool in the study of partial differential equations of evolution type.
