Arveson-Douglas Conjecture and Its Applications

阿维森-道格拉斯猜想及其应用

• 批准号: 2101370
• 负责人: Yi Wang
• 金额: $ 6.55万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-16 至 2022-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101370&HistoricalAwards=false
• 关键词: Arveson; Douglas; Conjecture; Its; Applications

This project focuses on an area of research called the Arveson-Douglas Conjecture, which belongs to the general field of operator theory. It embodies many challenges and new, exciting mathematics. The conjecture was originally made around 2000, and a large body of literature has been accumulated on the subject while many unsolved problems remain. It has connections with and applications to many other parts of mathematics; for example, it is connected to index theory, which attracts the attention of many researchers from various fields. Another connection is to the holomorphic extension problem in several complex variables, another important branch of mathematical analysis. The extension problem itself goes back to the 1970s, and recent progress on the Arveson-Douglas Conjecture sheds new light on this old problem. An important part of any research in mathematics is the search for new tools and methods, and the principal investigator will take concrete steps toward that goal. On the technical side, the Arveson-Douglas Conjecture makes strong connections with geometry and several complex variables. The concrete steps in this research include: the study of a recently defined property - the asymptotic stable division property - for submodules; a generalized version of the holomorphic extension problem; identifying index elements and index formulas on modules; extending known results to strongly pseudoconvex domains. The research is aimed at building machinery that allows one to treat global properties of complex analytic sets, especially algebraic sets, through local analysis. New techniques involving tools from operator theory, harmonic analysis, several complex variables and topology will be developed to treat these problems. The principal investigator will also study Toeplitz operators and Toeplitz algebra on strongly pseudoconvex domains.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.

这个项目集中在一个叫做Arveson-Douglas猜想的研究领域，它属于算子理论的一般领域。它包含了许多挑战和新的、令人兴奋的数学。这个猜想最初是在2000年左右提出的，关于这个主题已经积累了大量的文献，但仍有许多未解决的问题。它与数学的许多其他部分有联系和应用；例如，它与指标理论相联系，引起了各个领域许多研究者的关注。另一个联系是几个复变量的全纯扩展问题，这是数学分析的另一个重要分支。可拓问题本身可以追溯到20世纪70年代，而Arveson-Douglas猜想的最新进展为这个老问题提供了新的视角。任何数学研究的一个重要部分是寻找新的工具和方法，首席研究员将采取具体步骤实现这一目标。在技术方面，Arveson-Douglas猜想与几何和几个复杂变量有很强的联系。本文研究的具体步骤包括：研究了子模的渐近稳定除法性质；全纯扩展问题的一个广义版本识别模块上的索引元素和索引公式；将已知结果扩展到强伪凸域。该研究旨在建立一种机制，允许人们通过局部分析来处理复杂分析集，特别是代数集的全局性质。涉及算子理论、谐波分析、几个复杂变量和拓扑学等工具的新技术将被开发来处理这些问题。首席研究员还将研究强伪凸域上的Toeplitz算子和Toeplitz代数。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。

项目成果

The Helton-Howe trace formula for submodules

• DOI: 10.1016/j.jfa.2021.108997
• 发表时间: 2021-07
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Quanlei Fang;Yi Wang;Jingbo Xia

Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion

• DOI: 10.1016/j.aim.2021.107890
• 发表时间: 2021-09
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Yi Wang;Jingbo Xia

Essential commutants on strongly pseudo-convex domains

强伪凸域上的基本交换子

• DOI: 10.1016/j.jfa.2020.108775
• 发表时间: 2021
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Wang, Yi;Xia, Jingbo
• 通讯作者: Xia, Jingbo