Banach Space and Metric Geometry

在Banach空间和度量几何的问题要考虑分为几个子类：算子的Banach空间，逼近性质的Banach空间，结构的有限和无限维空间的p-可积函数，非线性分类的Banach空间，离散度量几何，定量线性代数，和集群值问题在无限维设置。 这些主题是Banach空间和度量空间几何的核心，并与数学中的许多其他领域，包括算子论，群论，几何分析和线性代数联系在一起。度量空间，特别是Banach空间，形成了数学家，科学家和工程师在研究涉及估计或近似的问题时的概念框架。 离散度量几何，包括降维结果已建立的PI，是重要的算法设计和压缩感知。对可交换算符的分类工作最初是由量子力学中的不确定性原理（从数学的角度来看，这是因为某些算符不可交换）推动的。非线性现象在自然界中经常发生，但很难处理。 这使得理解非线性何时实际上隐藏了潜在的线性结构变得很重要，这是Banach空间非线性研究的核心。该研究项目的部分内容与德克萨斯A M大学的分析和概率论研讨会协调，该研讨会由PI指导。该研讨会鼓励研究人员和学徒在不同的数学领域之间的互动，汇集了研究生和初级和高级博士后参与者在几个分析领域。 工作坊的活动包括研讨会，集中周，介绍性讲座和年度会议。 主要研究人员和讲习班其他参加者的努力正在帮助打破不同数学领域之间的障碍，并促进纯数学向其他科学，特别是计算机科学的推广。

The problems in Banach space and metric geometry to be considered fall into several subcategories: commutators of operators on Banach spaces, approximation properties of Banach spaces, the structure of finite and infinite dimensional spaces of p-integrable functions, the non linear classification of Banach spaces, discrete metric geometry, quantitative linear algebra, and cluster value problems in an infinite dimensional setting. These topics are at the heart of the geometries of Banach spaces and of metric spaces and make contact with many other areas within mathematics, including operator theory, group theory, geometric analysis, and linear algebra.Metric spaces, especially Banach spaces, form the conceptual framework in which mathematicians, scientists, and engineers work when investigating problems that involve estimation or approximation. Discrete metric geometry, including dimensional reduction results which have been established by the PI, is important in the design of algorithms and in compressed sensing. Work on the classification of operators which are commuters was originally motivated by the uncertainty principle in quantum mechanics (which, from a mathematical point of view, arises because certain operators do not commute). Non linear phenomena often occurs in nature but is difficult to deal with. This makes it important to understand when non linearity actually conceals underlying linear structure, and this is central to the non linear study of Banach spaces. Parts of this research project are coordinated with the Workshop in Analysis and Probability Theory at Texas A&M University, which is directed by the PI. The Workshop encourages interactions among researchers and apprentices in different mathematical fields by bringing together graduate students and junior and senior postdoctoral participants in several areas of analysis. Activities of the Workshop include seminars, Concentration Weeks, introductory lectures, and an annual conference. The efforts of the principal investigators and other participants in the Workshop are helping to break down barriers between different areas of mathematics and also promote the outreach of pure mathematics to other sciences, especially to computer science.