Banach Spaces: Theory and Applications

Banach 空间：理论与应用

基本信息

批准号：
1764343
负责人：
Thomas Schlumprecht
金额：
$ 26万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-05-01 至 2022-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764343&HistoricalAwards=false
关键词：
Banach Spaces Theory Applications

项目摘要

A focus of this project is the theory of Banach spaces and their geometry which provide an important conceptual framework to study problems in engineering, physics, signal processing, and the analysis of large data sets. The second main object of this investigation are "graphs". In computer science they are used to represent networks of communication, data organization, computational devices. An embedding of graphs, or more generally metric spaces, which may represent a large data set, into a Banach space, is necessary to be able to store this data set efficiently. Secondly it is necessary to provide the Banach space with the right coordinate system, it is therefore also necessary to be able to embed a Banach space, into a Banach space admitting the appropriate coordinate system. The problems that will be studied in this project revolve around the issue of embedding less structured mathematical objects like graphs or, more generally, metric spaces, into better structured objects like Banach spaces. On one hand the goal is to obtain information about the structure of the graph, from the property that it embeds in certain Banach spaces, and on the other hand deduce geometric properties of a Banach space, from the property that certain graphs embed or do not embed in it.The principal investigator will investigate possible extensions of Ribe's Program on metric characterizations of local properties of Banach spaces to characterizing asymptotic properties. An important question for example is the question whether or not the property of a Banach space to be reflexive can be metrically characterized. Other important properties to be considered are uniform asymptotic convexity and uniform asymptotic smoothness. The principal investigator will also study the problem of isomorphically embedding Banach spaces, having a certain property, into Banach spaces with a coordinates system like a Schauder basis or an unconditional basis. Finally the principal investigator will continue the study of closed sub-ideals of the spaces of operators on specific 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.

该项目的重点是Banach空间及其几何形状的理论，它们为研究工程，物理，信号处理以及大数据集的分析提供了重要的概念框架。该研究的第二个主要对象是“图”。在计算机科学中，它们用于表示通信，数据组织，计算设备网络。对于能够有效存储此数据集的必要条件，必须将图形的图形嵌入或更一般的度量空间嵌入到Banach空间中。其次，有必要为Banach空间提供正确的坐标系，因此，还必须能够将Banach空间嵌入到Banach空间中，以承认适当的坐标系。该项目中将研究的问题围绕嵌入较低的结构化数学对象（如图形）或更一般而言的度量空间等问题，将其嵌入到Banach Space等更好的结构化对象中。一方面，目标是从嵌入某些Banach空间中的属性以及从某些图形嵌入或不嵌入其中的属性中推导出Banach空间的几何特性的属性的信息。例如，一个重要的问题是一个问题，是否可以将BANACH空间的财产进行度量表征。要考虑的其他重要特性是均匀的渐近凸性和均匀的渐近平滑度。首席研究者还将研究具有一定特性的同构嵌入Banach空间的问题，该问题具有坐标系统（如Schauder基础或无条件基础）的Banach空间。最终，首席研究人员将继续研究经营者在特定Banach空间上的封闭次级思想。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。