Banach Space and Metric Geometry

度量空间，特别是那些所谓的巴拿赫空间，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工作的概念框架。离散度量几何，包括由PI和合作者建立的降维，在算法设计和压缩感知中很重要。对通勤算子的分类工作最初是由量子力学中的不确定性原理激发的（从数学的角度来看，这是因为算子的乘法不像数字的乘法，是不可交换的）。非线性现象在自然界中经常发生，但又难以处理。这使得理解非线性何时隐藏了潜在的线性结构变得很重要，这也是巴拿赫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农工大学的分析与概率论研讨会协调，PI是该研讨会的副主任。研讨会鼓励不同数学领域的研究人员和学徒之间的互动，将研究生和初级和高级博士后参与者聚集在几个分析领域。工作坊的活动包括研讨会、专注周、介绍讲座和年度会议。研讨会的主要研究者和其他参与者的努力有助于打破数学不同领域之间的障碍，并促进纯数学向其他科学，特别是计算机科学的扩展。要考虑的巴拿赫空间和度量几何中的问题分为几个子类：巴拿赫空间上算子的对易子、巴拿赫空间的近似性质、p可积函数的有限维和无限维空间的结构、巴拿赫空间的非线性分类、离散度量几何、定量线性代数和无限维设置中的聚类值问题。这些主题是巴拿赫空间和度量空间几何的核心，并与数学中的许多其他领域联系，包括算子理论、群论、几何分析和线性代数，以及其他科学领域，包括算法设计和压缩感知。

Metric spaces, especially those so-called 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 dimension reduction that has been established by the PI and a collaborator, is important in the design of algorithms and in compressed sensing. Work on the classification of operators that are commuters was originally motivated by the uncertainty principle in quantum mechanics (which, from a mathematical point of view, arises because multiplication of operators, unlike multiplication of numbers, is not commutative). Nonlinear phenomenon 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 nonlinear study of Banach spaces. Parts of this research project are coordinated with the Workshop in Analysis and Probability Theory at Texas A&M University, of which the PI is Associate Director. 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.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 nonlinear 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, as well as with other areas of science, including the design of algorithms and compressed sensing.