Mathematical Sciences: Banach Spaces and Convexity Theory

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This Research Opportunities for Women award supports Professor Werner's work in Banach space theory. One part of her project concerns properties of Banach spaces that can be characterized by the existence of appropriate renormings, for instance the property of containing a subspace isomorphic to the space of absolutely summable sequences. She will also investigate the geometry of convex bodies in Euclidean space. 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 Werner's project is concerned with how the ability to rescale the distance measurement in certain ways impinges on the geometrical structure of the space.