Banach Spaces: Theory and Applications

该项目的重点是Banach空间及其几何理论，为研究工程，物理，信号处理和大型数据集分析问题提供了重要的概念框架。本研究的第二个主要对象是“图”。在计算机科学中，它们被用来表示通信网络，数据组织，计算设备。图的嵌入，或更一般的度量空间，这可能代表一个大的数据集，到一个Banach空间，是必要的，能够有效地存储这个数据集。其次，必须为Banach空间提供正确的坐标系，因此也必须能够将Banach空间嵌入到允许适当坐标系的Banach空间中。在这个项目中将研究的问题围绕着嵌入结构较少的数学对象，如图或更一般的度量空间，到更好的结构对象，如Banach空间的问题。一方面，我们的目标是从它嵌入某些Banach空间的属性中获得关于图的结构的信息，另一方面，推导出Banach空间的几何属性，从某些图嵌入或不嵌入它的属性。主要研究者将调查可能的扩展里贝的计划对度量特征的局部性质的Banach空间，以表征渐近特性.一个重要的问题，例如是问题是否属性的Banach空间是自反的可以度量的特点。其他要考虑的重要性质是一致渐近凸性和一致渐近光滑性。主要研究者还将研究问题的同构嵌入Banach空间，具有一定的属性，到Banach空间的坐标系，如Schauder基础或无条件的基础。最后，首席研究员将继续研究封闭的子理想的空间的运营商在特定的Banach spaces.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.