Banach Space and Metric Geometry

度量空间，特别是那些所谓的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是重要的。对作为通勤者的运算符进行分类的工作最初是受到量子力学中的不确定性原理的推动(从数学的角度来看，这是因为运算符的乘法不同于数字的乘法，是不可交换的)。非线性现象在自然界中经常发生，但很难处理。这使得了解什么时候非线性实际上隐藏了潜在的线性结构是重要的，这是Banach空间的非线性研究的中心。该研究项目的部分内容是与德克萨斯农工大学的分析和概率论研讨会协调进行的，国际和平研究所是该研讨会的副主任。讲习班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。讲习班的活动包括研讨会、集中周、入门讲座和年度会议。主要研究人员和研讨会的其他参与者正在努力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扩展。要考虑的Banach空间和度量几何中的问题分为几个子类别：Banach空间上的算子的交换子，Banach空间的逼近性质，p-可积函数的有限和无限维空间的结构，Banach空间的非线性分类，离散度量几何，定量线性代数，以及无限维环境下的集值问题。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相联系，包括算子论、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传感。

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.