Nonlinear Geometry of Banach Spaces and Metric Spaces

本课题拟研究哪些Banach空间具有闭于一致映射或Lipschitz映射下的线性子空间和/或线性商的类。并对这些巴拿赫空间及其子集的粗几何进行了研究。粗图是Gromov为了研究大比例尺群的几何而引入的。该项目将关注的特定子集将是具有有界几何的离散度量空间，包括展开器。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。总的目标是将巴拿赫空间理论的技术推广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。非线性巴拿赫空间理论是巴拿赫空间理论的一个分支，主要研究非线性映射下巴拿赫空间及其子集的几何问题。巴拿赫空间理论是泛函分析中一个成熟的分支，具有发达的技术。非线性理论通过将其引入代数几何、几何群论和理论计算机科学等其他数学领域而使其焕发活力。它是诺维科夫猜想（Novikov conjecture）研究中使用的工具之一，诺维科夫猜想已经跨越了几个数学领域，通过研究与流形密切相关的一个物体的度量几何：它的基本群。它还有助于为实践中出现的许多指标找到更简单的几何形状，例如在数据挖掘中。同时，也加深了对巴拿赫空间理论本身主要对象的认识。本项目的研究将涉及非线性巴拿赫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.