Noncommutative Function Theory and Multivariable Operator Theory

算符理论起源于20世纪初的物理学家，作为量子力学中模拟现象的数学框架，此后它已成为现代理论物理的基本组成部分。算符理论也已经成为数学版图中不可或缺的一部分，与数学中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，它是一次研究多个算子。到目前为止，多变量算子理论的研究大多集中在交换算子族的理论上，这些算子与函数论、交换代数和代数几何等经典数学领域有着深刻的联系。然而，近年来，人们对开发这些经典领域的非对易类似物产生了极大的兴趣。这些新的发展与非对易算子族理论之间应该有同样深刻的联系，这是很自然的。经典的交换数学指导我们的直觉，并激励我们发展非对易数学。然而，我们从非对易环境中获得的视角也让我们对对易环境有了更好的理解。我的建议涉及到多变量算子理论中对易和非对易之间的相互作用。

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.