Investigations on the Corona Problem and a Study of Multi-Parameter Harmonic Analysis

Vanderbilt University, Nashville TN

Investigators

Abstract

This proposal will conduct research on problems inspired by questions from multi-parameter harmonic analysis and the Corona Problem. There are several questions connected to the Corona problem that will be addressed. First, questions about the Stable Rank of certain Banach algebras will be addressed. This will have implications in Control Theory as well as Complex Analysis. Second, questions related to the Operator Corona Theorem will be considered. Results here will be both important in the one and several complex variable setting. Additionally, there are hopes to extend some of the techniques to the several complex variable setting, which could shed light on the Corona problem there. In terms of multi-parameter harmonic analysis, the questions that will be addressed are connected to important issues arising from higher commutators. In a broad sense, the questions considered are an attempt to extend results from classical harmonic analysis to the multi-parameter setting. This area of research has only recently opened. The proposed research project fits into the general framework of understanding problems through the application of harmonic analysis. In recent years, the full power of modern tools from harmonic analysis has been applied to problems in function theory and operator theory, with excellent results. The Corona Problem is an abstract mathematical question about the algebraic structure of a function space. Its answer and the techniques used in the solution have implications in other areas of analysis and applications in the real world. It has proved to be a fundamental question in operator theory and has played an important role in questions from Control Theory. Multi-parameter harmonic analysis is a very natural generalization of standard harmonic analysis. However, this generalization provides a complex structure to many of the spaces and ideas that one finds in harmonic analysis. Many of the tools of harmonic analysis are unavailable in the multi-parameter setting. Only recently, questions in these areas have become more accessible due to new technical tools developed. The full usefulness of these tools has yet to be explored and fully utilized. These are both very important areas of analysis, with many important questions remaining. Answers to the questions proposed would be extremely beneficial. A deeper study of these areas is certainly warranted. The above research will have broad impact on other areas of mathematics as well as to problems in the real world. Harmonic analysis is an area inspired by applications. Commutator estimates arise naturally in physics through the use of Div-Curl Lemmas in the study of electromagnetism via Maxwell's Equations. Also, tools used in this proposal, in particular smooth wavelet systems, also have applications outside mathematics. Recently, they have been utilized in National Security to compress biometric data such as fingerprints and have also found applications in medical imaging.