CAREER: An Integrated Proposal Based on The Corona Problem

该项目的研究目标是对电晕问题进行更深入的研究，使用在相关分析领域开发的工具和技术，目的是解决开放和重要的问题。电晕问题可以表述为一个关于解析函数的特定代数中矩阵的左可逆性的问题。此外，它还包含了算子理论、复杂微分几何、泛函分析和交换代数等领域的公式。电晕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，它出现在现实世界的应用中，通过使用控制理论来解决工程问题。本研究计划利用这些广泛分析领域的知识和技术，提供一系列工具来解决本项目中提出的具有挑战性的问题。拟议的研究基于首席研究员最近做出的重大贡献，并侧重于与电晕问题相关的关键问题。特别是，首席研究员将通过与问题相关的规范向量束的曲率来解决将电晕问题与复杂微分几何相关的问题。对于解析函数的更一般的乘数代数，将进一步研究电晕问题。最后，通过解析函数环的稳定秩的计算，探讨其与电晕问题和控制理论的联系。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论坛，让他们在学术和专业上相互交流和学习。研讨会分为三个阶段，包括网络讲座、工作小组和最后的会议。主要目标是增加全国研究生、博士后研究人员和高级教师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，并通过与更资深的研究人员进行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组成部分相结合：前沿研究将为互联网研讨会提供许多选择的主题，研讨会参与者可能会在未来的研究项目中合作。在这个项目中调查的研究问题的解决方案将在复杂分析、函数理论、调和分析和算子理论中有无数的应用。它们不仅将为进一步的数学研究开辟道路，而且还将对现实世界的想法，特别是在控制理论领域有重要的应用。教育部分旨在通过创建一个任何研究人员都可以参与的开放和包容的研究论坛，扩大孤立的研究人员和代表性不足的群体的参与。参与者将能够与他们所在领域的一些顶级专家一起工作，并最好得到他们的指导，而不受地理位置的限制。

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.