Professor Jarosz will investigate a number of open problems concerning maps between various function spaces and algebras that preserve or only slightly distort metric structure. The common thread is to discover algebraic, geometric, lattice, or other properties shared by two such spaces when there is an isometry or small bound isomorphism from one onto the other. These problems are closely related to small perturbations of Banach algebras, generalizations of the Banach-Stone theorem, geometry of Banach spaces, and quasiconformal mappings. The theory of Banach spaces is about analysis in infinitely many dimensions. For each finite dimension n, there is only one n-dimensional space, in most respects as well understood in general as in the familiar cases when n is 1, 2, or 3. These are used to keep track of situations in which there are only finitely many degrees of freedom. When the number of degrees of freedom becomes infinite, as would for instance be necessary to describe the configuration of a piece of string, there are many possible reasonable choices for the ambient space in which to perform analysis. A key element of structure that distinguishes these spaces one from another is the way in which distance is measured. Professor Jarosz's project is concerned with the relationship of distance measurement to algebraic and geometric properties in such spaces, in particular the persistence of these properties under small distortions of distance.
