New Developments at the Interface of Banach Algebras and Complex Analysis

该项目将推进复杂分析和功能分析界面的研究，为物理、工程和计算化学领域提供数学工具。几个长期存在的问题将研究使用新概念最近介绍的首席研究员。这一研究项目的成果将通过研讨会和会议上的演讲以及广泛发行的期刊上的出版物加以传播。首席研究员将继续指导相关领域的初级数学家，向他们提出由本研究项目激发的、与他们的特长相匹配的问题。他还将继续鼓励学生攻读STEM领域的大学学位。首席研究员将使用泛函分析和单变量和多变量复分析的方法来研究四个不同的、相关的数学领域的问题：几个复变量的电晕问题，多变量算子理论中的Arveson猜想，交换Banach代数中的解析结构问题，以及关于同态群的问题。研究的一个主要焦点将是交换巴拿赫代数和分析性之间的密切联系，尽管许多数学家已经研究了60多年，但在许多方面仍然是神秘的。最近，首席研究员发现了解析集的新概念：除了提供了对交换Banach代数解析结构的新视角外，该概念还提供了对电晕问题的新视角，并提出了一个可以描述为“非光滑复分析”的领域的发展。研究的另一个重点将是Banach代数在群作用下的不变量及其与Arveson关于Hilbert模的一个重要猜想的联系，该猜想在算子理论和C*-代数中具有广泛的分支。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will advance research at the interface of complex analysis and functional analysis, areas of mathematics that provide the mathematical tools used in physics, engineering, and computational chemistry. Several longstanding problems will be studied using new concepts recently introduced by the principal investigator. The results of this research project will be disseminated via talks in seminars and conferences and via publications in widely available journals. The principal investigator will continue to mentor junior mathematicians in related areas by suggesting to them problems motivated by this research project that are well matched to their particular strengths. He will also continue to encourage students to pursue university degrees in STEM fields. The principal investigator will use methods from functional analysis and complex analysis in one and several variables to study problems in four distinct, related areas of mathematics: the corona problem in several complex variables, Arveson's conjecture in multivariable operator theory, problems concerning analytic structure in commutative Banach algebras, and problems concerning homeomorphism groups. A major focus of the research will be the close connection between commutative Banach algebras and analyticity which, despite over 60 years of investigation by many mathematicians, remains in many ways mysterious. The principal investigator recently discovered a new concept of analytic set: in addition to providing a new perspective on analytic structure in commutative Banach algebras, this concept also provides a new perspective on the corona problem and suggests the development of a field that could be described as "nonsmooth complex analysis". Another major focus of the research will be Banach algebras invariant under group actions and their connections to an important conjecture of Arveson concerning Hilbert modules with wide ranging ramifications in operator theory and C*-algebras.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.