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

本计画将针对多参数谐波分析与电晕问题所启发的问题进行研究。 有几个问题连接到电晕的问题，将得到解决。 首先，某些Banach代数的稳定秩的问题将得到解决。 这将在控制理论以及复杂性分析中产生影响。 其次，将考虑与算子电晕定理有关的问题。 这里的结果在一个和多个复变量设置中都很重要。 此外，有希望将一些技术扩展到几个复杂的变量设置，这可能会揭示电晕问题。 在多参数谐波分析方面，将解决的问题是连接到更高的谐波产生的重要问题。 从广义上讲，考虑的问题是一种尝试，从经典的谐波分析的多参数设置的结果。 这一领域的研究只是最近才开始的，所提出的研究项目符合通过应用调和分析来理解问题的一般框架。 近年来，调和分析的现代工具的全部力量已被应用于函数论和算子论中的问题，并取得了很好的结果。 电晕问题是一个关于函数空间代数结构的抽象数学问题。 它的答案和解决方案中使用的技术对真实的世界中的其他分析和应用领域具有影响。 它已被证明是算子理论中的一个基本问题，并在控制论问题中发挥了重要作用。 多参数谐波分析是标准谐波分析的自然推广。 然而，这种概括为调和分析中发现的许多空间和思想提供了一个复杂的结构。 许多谐波分析工具在多参数设置中不可用。 直到最近，由于开发了新的技术工具，这些领域的问题才变得更容易获得。 这些工具的全部效用尚待探索和充分利用。 这些都是非常重要的分析领域，还有许多重要的问题有待解决。 回答所提出的问题将是极为有益的。 对这些领域进行更深入的研究当然是必要的。 上述研究不仅对真实的世界的问题，而且对数学的其他领域也将产生广泛的影响。 谐波分析是一个受应用启发的领域。 交换子估计在物理学中通过使用Div-Curl引理在电磁学研究中通过麦克斯韦方程自然产生。 此外，在这个建议中使用的工具，特别是光滑小波系统，也有数学以外的应用。 最近，它们已被用于国家安全，以压缩生物特征数据，如指纹，并在医学成像中也有应用。

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.