Geometry of Banach Spaces and Metric Spaces

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工作的概念框架。离散度量几何，包括PI建立的降维结果，在算法设计和压缩感知中是重要的。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重要，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与概率论研讨会协调进行的。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：经典Banach空间上有界线性算子的Banach代数的结构，Banach空间的逼近性质，Banach空间的非线性分类，以及离散度量几何。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，包括算子理论、群论、几何分析和线性代数。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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 dimension reduction results established by the PI, is important in the design of algorithms and in compressed sensing. 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. The efforts of the principal investigator 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: the structure of the Banach algebra of bounded linear operators on classical Banach spaces, approximation properties of Banach spaces, the non linear classification of Banach spaces, and discrete metric geometry. 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.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.