The Geometry of Banach Spaces and Its Applications

主要研究者将研究Banach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。Banach空间理论为算子理论、Fourier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Banach空间之间的算子的因式分解，例如p-幂可积函数和p-幂可和序列的空间。将探讨具有Pelczynski分解的Banach空间上的算子的分解子的内在分类。本文将发展度量空间之间的Lipschitz p-求和、Lipschitz p-积分和Lipschitz p-核算子的理论。度量空间的Lipschitz提升性质也将被调查。还将研究Schauder框架包含Schauder基子序列的充分必要条件。Banach空间为线性和非线性泛函分析、概率和优化提供了框架，在偏微分方程和量子力学数学模型中具有根本重要性。因此，在纯数学的这个分支中的进步的影响是深远的，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个应用。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巨大的影响，量子力学中的测不准原理最终是关于算子分解子的定理。主要研究者还将支持一名研究生参与该项目，并优先考虑从历史上代表性不足的背景中选择合格的学生。

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.