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: 0955432
• 负责人: Brett Wick
• 金额: $ 44.94万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0955432&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利用在相关分析领域开发的工具和技术对日冕问题进行更深入的研究，目的是解决公开和重要的问题。CORONA问题可以表述为一个关于矩阵的左可逆性的问题，特别是解析函数的代数。此外，它还在算子理论、复微分几何、泛函分析和交换代数等领域有公式。日冕问题推动了四个主要分析领域的研究：复分析、函数理论、调和分析和算子理论。此外，它还出现在实际应用中，通过使用控制理论来解决工程问题。这项研究计划利用这些广泛的分析领域的知识和技术来提供一系列工具，用来解决本项目中提出的具有挑战性的问题。拟议的研究是基于首席调查员最近作出的重大贡献，并侧重于与日冕问题有关的关键问题。特别是，主要研究人员将通过与问题相关的正则向量丛的曲率来解决将日冕问题与复杂微分几何联系起来的问题。另外，我们还将研究更一般的解析函数乘子代数的电晕问题。最后，通过计算解析函数环的稳定秩来探讨它与日冕问题和控制论的联系。该项目的教育部分创建了一个新颖的“互联网分析研讨会”，为这些领域的研究人员提供了一个在学术和专业方面相互交流和相互学习的论坛。研讨会包括三个阶段，包括互联网讲座、工作组和期末会议。主要目标是加强全国研究生、博士后研究人员和高级教职员工之间的协作学习和指导。研讨会进一步规范了研究成果的传播，为初级数学家提供了一个开放、包容的环境，让他们学习新的研究概念，并通过与更多资深研究人员的小组项目来应用这些概念。此外，该项目将首席研究员当前和未来的研究与雄心勃勃的教育部分结合在一起：尖端研究将提供许多为互联网研讨会选择的主题，研讨会参与者可能会就未来的研究项目进行合作。本项目所研究问题的解决方案将在复变分析、函数论、调和分析和算子理论中有无数的应用。它们不仅将为更多的数学探索开辟道路，而且还将在现实世界的想法中有重要的应用，特别是在控制理论领域。教育部分寻求扩大孤立的研究人员和代表性不足的群体的参与，方法是建立一个开放和包容的研究论坛，任何研究人员都可以参与。参与者将能够与各自领域的一些顶尖专家合作，并得到他们的最佳指导，而不受地理位置的影响。