New Developments at the Interface of Banach Algebras and Complex Analysis

该项目将推进复杂分析和泛函分析接口的研究，这些数学领域提供了物理，工程和计算化学中使用的数学工具。 几个长期存在的问题将使用新的概念，最近引入的主要研究者进行研究。 这一研究项目的结果将通过研讨会和会议的讲座以及通过广泛发行的期刊上的出版物传播。 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问题，这些问题与他们的特殊优势非常匹配。 他还将继续鼓励学生攻读STEM领域的大学学位。 主要研究人员将使用方法从功能分析和复杂的分析在一个和几个变量研究问题在四个不同的，相关领域的数学：电晕问题在几个复杂的变量，Arveson的猜想在多变量算子理论，问题有关的解析结构在交换Banach代数，和问题有关同胚群。 一个主要重点的研究将是密切联系之间的交换Banach代数和解析，尽管超过60年的调查，许多数学家，仍然在许多方面的神秘。 首席研究员最近发现了一个新的概念解析集：除了提供一个新的视角解析结构在交换Banach代数，这个概念也提供了一个新的视角电晕问题，并建议发展一个领域，可以被描述为“非光滑复分析”。 该研究的另一个主要重点将是Banach代数不变下的群体行动和他们的连接到一个重要的猜想Arveson关于希尔伯特模块与广泛的分支在运营商理论和C*-algebras.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.