Banach Spaces: Theory and Applications

一般度量空间，尤其是图，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，有必要理解它们的“结构”，在数学语言中，这意味着理解它们的“几何”。例如，这意味着将它们嵌入到具有坐标系的结构更好的空间中。Banach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计算机科学中使用的度量空间到某些Banach空间的嵌入。这类度量空间的例子有Lantighter图和运输成本度量空间，前者为旅行商问题建模，后者为产品的最优分配建模。另一方面，从某些度量空间在给定的Banach空间中的可嵌入性或不可嵌入性，可以推导出该Banach空间的重要几何性质。该奖项将支持首席研究员在这些主题上的研究，并通过研究生培训为美国劳动力发展做出贡献。项目的第一部分是关于某些度量空间，如Lantighter空间和运输成本空间，是否可以粗略地、一致地或双Lipschitzly地嵌入到可积函数的Banach空间中的问题，并估计这些嵌入的失真。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寻找Banach空间的几个同构性质的度量刻画。这项研究将集中在有限渐近性质上，目的是建立近年来成功的关于局部性质的度量刻画的Ribe计划的渐近版本。研究者还将继续他对自反性的度量解释的研究，希望找到表征自反性的度量测试空间。研究Banach空间及其几何的一个核心部分是研究其上的(线性的和有界的)算子。最近发现，对于一大类可分Banach空间，算子空间的闭子理想的基数就是连续统子集的集合的基数。通过简单的集合论考虑，这是可能的最大值。PI将调查更多类Banach空间的算子的闭理想。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势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.