CAREER: An Integrated Proposal Based on The Corona Problem

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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.