Nonlinear Geometry of Banach Spaces and Metric Spaces

该项目建议研究哪些Banach空间的线性子空间和/或线性商类在一致映射或Lipschitz映射下是闭的。它还建议研究这些Banach空间及其子集的粗几何。格罗莫夫引入粗图来研究大尺度上的群的几何。该项目将关注的特定子集将是具有有界几何图形的离散度量空间，包括扩张器。随着项目沿着这些路线进行，一般度量空间的几何也将被研究。其总体目标是将Banach空间理论的技巧扩展到度量空间的研究中。目标之一将是更全面地理解度量一致凸性。非线性Banach空间理论是Banach空间理论的一个分支，研究非线性映射下Banach空间及其子集的几何问题。Banach空间理论是泛函分析的一个成熟的分支，有着成熟的技术。非线性理论通过将其应用于其他数学领域，如代数几何、几何群论和理论计算机科学，使其重新焕发活力。它是研究诺维科夫猜想的工具之一，诺维科夫猜想已经跨越了数学的几个领域，通过研究与流形密切相关的一个对象的度量几何：它的基本群。它还通过帮助为实践中出现的许多指标找到更简单的几何图形来为计算机科学做出贡献，例如在数据挖掘中。同时，也加深了对Banach空间理论本身主要对象的理解。本项目提出的研究将涉及到非线性Banach空间理论的所有这些方面。

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.