Nonlinear Geometry of Banach Spaces and Metric Spaces

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

The project proposes to study which Banach spaces have the class of their linear subspaces and/or linear quotients closed under uniform or Lipschitz maps. It also proposes to study the coarse geometry of these Banach spaces and their subsets. Coarse maps were introduced by Gromov to study the geometry of groups in the large scale. The particular subsets that the project will focus on will be discrete metric spaces with bounded geometry, including expanders. As the project goes along these lines, the geometry of general metric spaces will be studied as well. The general aim is to extend the techniques from Banach space theory into the study of metric spaces. One of the goals will be to understand metric uniform convexity more fully. Nonlinear Banach Space Theory is a branch of Banach Space Theory that studies the geometry of Banach spaces and their subsets under nonlinear maps. Banach Space Theory is a mature branch of Functional Analysis with well-developed techniques. The nonlinear theory rejuvenates it by bringing it into the service of other areas of Mathematics such as Algebraic Geometry, Geometric Group Theory and Theoretical Computer Science. It is one of the tools used in the study of the Novikov conjecture, a conjecture that spans several areas of Mathematics already, by studying the metric geometry of one object intimately tied to the manifold: its fundamental group. It also contributes to Computer Science by helping find a simpler geometry for the many metrics that appear in practice, like in Data Mining for example. In the meantime, it deepens the understanding of the primary objects of Banach Space Theory itself. The research proposed in the present project will touch all of these aspects of Nonlinear Banach Space Theory.