CAREER: An Integrated Proposal Based on The Corona Problem

职业：基于新冠问题的综合提案

• 批准号: 1603246
• 负责人: Brett Wick
• 金额: $ 3.66万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-10-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1603246&HistoricalAwards=false
• 关键词: CAREER; Integrated; Proposal; Based; Corona

The research objective of this project is to conduct a deeper study of the Corona Problem, using tools and techniques developed in interrelated areas of analysis, with a goal of settling open and important questions. The Corona Problem can be phrased as a question about left invertibility of matrices in particular algebras of analytic functions. Additionally, it has formulations in the areas of operator theory, complex differential geometry, functional analysis, and commutative algebra. The Corona Problem has served as an impetus for research in four main areas of analysis: complex analysis, function theory, harmonic analysis, and operator theory. Additionally, it arises in real-world applications through the use of control theory to engineering questions. This research program utilizes knowledge and techniques from these broad areas of analysis to provide an array of tools with which to approach the challenging questions raised in this project. The proposed research is based on recent, significant contributions made by the principal investigatorr and focuses on key questions connected to the Corona Problem. In particular, the principal investigator will address questions that relate the Corona Problem to complex differential geometry via the curvature of canonical vector bundles associated with the problem. The Corona Problem will additionally be studied for more general multiplier algebras of analytic functions. Finally, the connection with the Corona Problem and control theory will be explored via the computation of the stable rank of rings of analytic functions. The project's educational component creates a novel "Internet Analysis Seminar" that provides a forum for researchers in these areas to interact and learn from one another, both academically and professionally. The seminar includes three phases involving Internet lectures, working groups, and a final conference. A primary goal is to increase the collaborative learning and mentoring between graduate students, postdoctoral researchers, and senior faculty across the country. The seminar takes the standard dissemination of research results further, providing an open, inclusive setting for junior mathematicians to learn new research concepts and apply them through group projects with more senior researchers. Moreover, the project integrates the principal investigator's current and future research with an ambitious educational component: the cutting-edge research will provide many of the topics selected for the Internet seminar, and seminar participants will likely collaborate on future research projects. Solutions to the research questions investigated in this project will have countless applications in complex analysis, function theory, harmonic analysis, and operator theory. Not only will they open the way to additional mathematical inquiry, but they will also have significant application to real-world ideas, in particular in the area of control theory. The educational component seeks to broaden the participation of isolated researchers and underrepresented groups by creating an open and inclusive research forum in which any researcher can participate. Participants will be able to work with, and optimally be mentored by, some of the top experts in their fields regardless of geographic location.

该项目的研究目标是使用相互关联的分析领域开发的工具和技术对电晕问题进行更深入的研究，以解决开放和重要问题。可以将电晕问题作为有关矩阵特别是分析函数代数的左逆性问题的问题。 此外，它在操作者理论，复杂的差异几何形状，功能分析和交换代数的领域中还具有制剂。 电晕问题已成为四个主要分析领域研究的动力：复杂的分析，功能理论，谐波分析和操作者理论。此外，它通过将控制理论用于工程问题而在现实世界应用中产生。该研究计划利用这些广泛的分析领域的知识和技术提供了一系列工具，以解决该项目中提出的具有挑战性的问题。 拟议的研究是基于主要的调查工作做出的最近的重大贡献，并重点介绍了与电晕问题有关的关键问题。 特别是，主要研究者将通过与该问题相关的规范矢量束的曲率解决将电晕问题与复杂差异几何形状联系起来的问题。 还将研究电晕问题，以了解分析功能的更通用的乘数代数。 最后，将通过计算分析功能的稳定等级来探索与电晕问题和控制理论的联系。 该项目的教育组成部分创建了一个新颖的“互联网分析研讨会”，该研讨会为这些领域的研究人员提供了一个论坛，可以在学术和专业上相互互动和学习。 该研讨会包括涉及互联网讲座，工作组和最终会议的三个阶段。 一个主要目标是增加研究生，博士后研究人员和全国高级教师之间的协作学习和指导。研讨会进一步促进了研究结果的标准传播，为初级数学家提供了一个开放的，包容性的环境，以学习新的研究概念，并通过与更多高级研究人员的小组项目应用它们。此外，该项目将主要研究者的当前和未来研究与雄心勃勃的教育组成部分结合在一起：尖端研究将为互联网研讨会选择的许多主题提供，研讨会参与者可能会在未来的研究项目上合作。该项目研究的研究问题解决方案将在复杂分析，功能理论，谐波分析和操作员理论中具有无数的应用。 他们不仅会为其他数学询问开辟道路，而且还将在控制理论领域特别适用于现实世界的思想。教育部分旨在通过创建一个开放的包容性研究论坛来扩大孤立的研究人员和代表性不足的群体的参与，任何研究人员都可以参与其中。 参与者将能够与某些领域的一些顶级专家合作，并最佳地指导，无论地理位置如何。