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New Developments at the Interface of Banach Algebras and Complex Analysis

Banach代数与复分析接口的新进展

基本信息

批准号：
1856010
负责人：
Alexander Izzo
金额：
$ 23.43万
依托单位：
Bowling Green State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1856010&HistoricalAwards=false
关键词：
New Developments Interface Banach Algebras

项目摘要

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.

这个项目将推进复杂分析和泛函分析的接口研究，这两个数学领域提供了在物理、工程和计算化学中使用的数学工具。将使用首席研究员最近提出的新概念来研究几个长期存在的问题。这一研究项目的成果将通过研讨会和会议上的演讲以及在广泛可获得的期刊上发表的出版物来传播。首席研究人员将继续指导相关领域的初级数学家，向他们建议这项研究项目所激发的问题，这些问题与他们的特殊优势很好地匹配。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泛函分析和一元多元复分析的方法来研究四个不同的相关数学领域的问题：多复变量的日冕问题，多元算子理论中的Arveson猜想，关于交换Banach代数的解析结构的问题，以及关于同胚群的问题。研究的一个主要焦点将是交换Banach代数与解析性之间的密切联系，尽管许多数学家进行了60多年的研究，但在许多方面仍然是神秘的。这位首席研究者最近发现了一个新的解析集概念：除了为交换Banach代数的解析结构提供了一个新的视角外，这个概念还提供了一个关于冠冕问题的新视角，并提出了一个可以描述为“非光滑复分析”的领域的发展。研究的另一个主要焦点将是群作用下的Banach代数不变，以及它们与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NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。