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 The conjecture was originally made around 2000, and a large body of literature has been accumulated on the subject while many unsolved problems remain.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.

该项目着重于一个名为Arveson-Douglas猜想的研究领域，该领域属于一般的操作员理论领域。它体现了许多挑战和新的令人兴奋的数学。该猜想最初是在2000年左右做出的，并且在该主题上积累了大量文献，但仍然存在许多未解决的问题。它与数学的许多其他部分具有连接和应用；例如，它与索引理论有关，该理论吸引了许多来自各个领域的研究人员的注意。 在几个复杂变量（数学分析的另一个重要分支）中，与全体形态扩展问题的另一个联系是。 扩展问题本身可以追溯到1970年代，阿尔维森·道格拉斯（Arveson-Douglas）的最新进展为这个旧问题提供了新的启示。 任何数学研究的重要组成部分是寻找新的工具和方法，主要研究人员将朝着该目标采取具体步骤。在技​​术方面，Arveson-Douglas猜想与几何形状和几个复杂变量建立了牢固的联系。这项研究的具体步骤包括：针对子模型的最近定义的财产研究 - 渐近稳定划分财产；全体形态扩展问题的广义版；识别模块上的索引元素和索引公式；将已知结果扩展到强烈的伪共元域。该研究旨在通过局部分析来处理复杂分析集（尤其是代数集）的复杂分析集（尤其是代数集）的全球特性的构建机械。将开发涉及运营商理论，谐波分析，几个复杂变量和拓扑的工具的新技术来处理这些问题。首席研究人员还将研究Toeplitz运营商和Toeplitz代数强烈的伪有源领域。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子的评估来支持的，并具有更广泛的影响。

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

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

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