Banach Spaces: Theory and Applications

一般度量空间，特别是图，用于表示通信、数据组织和计算设备的网络。为了分析、使用和存储它们，有必要了解它们的“结构”，用数学语言来说，这意味着了解它们的“几何”。例如，这意味着用坐标系统将它们嵌入到更好的结构空间中。巴拿赫空间，是实现这一目标的概念框架。这个项目的一个主要焦点是寻找在计算机科学中使用的度量空间嵌入到某些巴拿赫空间中。这种度量空间的例子是Lamplighter图，它为旅行推销员问题建模，以及运输成本度量空间，它为产品的最优分配建模。另一方面，从某个度量空间对给定巴拿赫空间的可嵌入性或不可嵌入性，可以推导出该巴拿赫空间的重要几何性质。该奖项将支持首席研究员对这些主题的研究，并通过培养研究生为美国劳动力发展做出贡献。项目的第一部分关注某些度量空间，如Lamplighter空间和Transportation Cost空间，是否可以在可积函数的Banach空间中粗糙、均匀或双lipschitzly嵌入，并估计这些嵌入的畸变。在第二部分中，研究者试图通过某些度量图的可嵌入性或不可嵌入性来找到Banach空间的几个同构性质的度量特征。研究将集中于有限渐近性质，目的是建立近年来成功的关于局部性质度量表征的Ribe程序的渐近版本。研究者还将继续他对反身性的度量解释的研究，希望找到表征反身性的度量测试空间。研究巴拿赫空间及其几何的一个核心部分是研究其上的（线性和有界）算子。最近发现，对于一大类可分离的巴拿赫空间，算子空间的闭子的基数是连续统子集集合的基数。这是，通过简单的集合理论的考虑，最大的可能。PI将研究更多类巴拿赫空间算子空间的闭理想。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

General metric spaces, especially graphs, are used to represent networks of communication, data organization, and computational devices. In order to analyze, use and store them it is necessary to understand their “Structure”, in mathematical language this means to understand their “Geometry”. For example this means, to embed them into better structured spaces with a coordinate system. Banach spaces, are the conceptual framework to accomplish that. A main focus of this project is finding embeddings of metric spaces, used in Computer Science, into certain Banach spaces. Examples for such metric spaces are Lamplighter Graphs, which model the traveling salesman problem, and the Transportation Cost Metric Space, which models the optimal distribution of products. On the other hand, from the embeddability, or non-embeddability of certain metric spaces into a given Banach space, one can deduce important geometric properties of that Banach space. This award will support the principal investigator's research on these topics and also contribute to US workforce development through the training of graduate students. The first part of the project is concerned with the question whether or not certain metric spaces, like the Lamplighter space and Transportation Cost space, can be coarsely, uniform or bi-Lipschitzly embedded in the Banach space of integrable functions, and estimate the distortion of these embeddings. In the second part, the investigator seeks to find metric characterizations of several isomorphic properties of Banach spaces via the embeddabilty, or non embeddability, of certain metric graphs. The research will concentrate on finite asymptotic properties with the goal of establishing an asymptotic version of the in recent years successful Ribe Program on metric characterizations of local properties. The investigator will also continue his investigation on the metric interpretations of reflexivity, with the hope of finding metric test spaces which characterize reflexivity. A central part of studying Banach spaces and their geometry is the study of (linear and bounded) operators on them. Very recently it was discovered that for a large class of separable Banach spaces, the cardinality of closed subideals of the space of operators is the cardinality of the collection of subsets of the continuum. This is, by simple set theoretic considerations, the maximal possible. The PI will investigate the closed ideals of spaces of operators of more classes of Banach spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.