Noncommutative Function Theory and Multivariable Operator Theory

非交换函数论和多变量算子理论

• 批准号: 418585-2012
• 负责人: Kennedy, Matthew
• 金额: $ 1.24万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635294
• 关键词: Noncommutative; Function; Theory; Multivariable; Operator

The field of operator theory originated with physicists in the early part of the 20th century as a mathematical framework for modeling phenomenon in quantum mechanics, and it has since become a fundamental component of modern theoretical physics. Operator theory has also become an indispensable part of the mathematical landscape, with important connections to other areas in mathematics. It is these connections that have increasingly become the focus of modern operator theory.My research concerns multivariable operator theory, which is the study of more than one operator at a time. To date, most of the research in multivariable operator theory has focused on the theory of families of commuting operators, where there are deep connections to classical areas of mathematics like function theory, commutative algebra and algebraic geometry. However, in recent years, there has been a great deal of interest in developing noncommutative analogues of these classical areas. It is natural to expect that there should be similarly deep connections between these new developments and the theory of noncommuting families of operators.Classical commutative mathematics guides our intuition and motivates our development of noncommutative mathematics. However, the perspective we gain from working in the noncommutative setting also gives us a better understanding of the commutative setting. My proposal concerns this interplay between the commutative and the noncommutative in the area of multivariable operator theory.

算子理论领域起源于 20 世纪初期的物理学家，作为量子力学现象建模的数学框架，从此成为现代理论物理学的基本组成部分。算子理论也已成为数学领域不可或缺的一部分，与数学的其他领域有着重要的联系。正是这些联系日益成为现代算子理论的焦点。我的研究涉及多变量算子理论，即同时研究多个算子。迄今为止，多变量算子理论的大部分研究都集中在交换算子族理论上，该理论与函数论、交换代数和代数几何等数学经典领域有着深厚的联系。然而，近年来，人们对开发这些经典领域的非交换类似物产生了很大的兴趣。很自然地期望这些新发展与非交换算子族理论之间应该存在类似的深刻联系。经典交换数学指导我们的直觉并激励我们非交换数学的发展。然而，我们从非交换环境中获得的视角也让我们对交换环境有了更好的理解。我的建议涉及多变量算子理论领域中交换律和非交换律之间的相互作用。