The Geometry of Banach Spaces and Its Applications

首席研究员将研究巴拿赫空间的几何及其在许多其他数学领域的应用，特别是逻辑和集合论。巴拿赫空间理论的技术为算子理论、傅里叶分析和框架理论的研究提供了强有力的工具。更具体地说，研究者将研究经典巴拿赫空间（如p幂可积函数空间和p幂可和序列空间）之间算子的因式分解。讨论了具有Pelczynski分解的Banach空间上算子的交换子的本征分类。建立了度量空间间的Lipschitz p求和、Lipschitz p积分和Lipschitz p核算子的理论。具有Lipschitz提升性质的度量空间也将被研究。我们还将研究Schauder帧包含Schauder基的子序列的充分必要条件。巴拿赫空间为线性和非线性泛函分析、概率和优化提供了一个框架，在偏微分方程和量子力学的数学模型中具有重要的基础意义。因此，这一纯数学分支的进步影响深远，延伸到数学的许多领域，以及理论物理和与计算机科学和工程相关的多种应用。例如，度量空间之间的Lipschitz映射的研究对理论计算机科学产生了重大影响，量子力学中的不确定性原理最终是一个关于算子的对易子的定理。首席研究员还将支持一名研究生参与该项目，并优先从历史上代表性不足的背景中选择一名合格的学生。

The principal investigator will investigate the geometry of Banach spaces and its applications to many other areas of mathematics, especially logic and set theory. The techniques of Banach space theory provide strong tools for the study of operator theory, Fourier analysis and frame theory. More specifically, the investigator will study the factorizations of operators between classical Banach spaces such as the space of p-power integrable functions and p-power summable sequences. Intrinsic classifications of commutators of operators on Banach spaces with Pelczynski decomposition will be explored. The theory of Lipschitz p-summing, Lispchitz p-integral and Lipschits p-nuclear operators between metric spaces will be developed. Metric spaces with the Lipschitz lifting property will also be investigated. Necessary and sufficient conditions for Schauder frames to contain a subsequence that is a Schauder basis will be studied as well.Banach spaces provide a framework for linear and non-linear functional analysis, probability, and optimization and are of fundamental importance in partial differential equations and mathematical models in quantum mechanics. Hence, the impact of advances in this branch of pure mathematics is far-reaching, extending across many areas of mathematics as well as to theoretical physics and multiple applications related to computer science and engineering. For instance, the study of Lipschitz mappings between metrics spaces has shown great impact on theoretical computer science and the uncertainty principle in quantum mechanics is ultimately a theorem about commutators of operators. The principal investigator will also support one graduate student on this project, and priority will be placed on selecting a qualified student from historically underrepresented background.