Invariants for Multivariate Operator Theory

摘要：许多物理现象都是用与物理变量及其变化率相关的积分方程和微分方程进行数学建模的。至少在一个近似上，这样的方程可以被看作是线性的，并且可以被看作作用于函数空间。这些空间通常有一个长度的概念，就像欧几里得空间那样，这就产生了所谓的希尔伯特空间。对这些空间上的算子或线性变换的研究使我们对物理学中的量子力学和电气工程中的系统理论的理解有了新的见解。算符理论与数学的其他部分，包括几何和拓扑学也有很强的相互作用。近年来，研究人员将大部分兴趣转向同时研究多个操作员，通常假设他们通勤。正如人们可能猜到的那样，建模的数学现象可能相当复杂，它们依赖于代数和几何的概念和概念。此外，所得到的问题和结果丰富了这些领域，并为研究这一多变量算子理论提供了有力的工具。自伴随多元算子理论的研究已经有几十年的历史了，而目前的研究主要是关于非自伴随的情况，它与代数和复几何有很强的联系。代数中模的基本概念适用于希尔伯特空间。我们考虑了一些例子，其中全纯向量束自然产生，曲率和曲率的高阶概念允许我们在算子理论中调解结果。在目前的建议中，主要的重点是商数模块和建模他们的结构和核函数使用一个新的射流束精确地为此目的而构建。当商模由超曲面决定时，至少对于函数在法向上消失到低阶的理想情况，是可以很好地控制的。将这一理解扩展到由具有更高co维数的品种所决定的其他理想是下一个重大步骤，并且将是未来两年工作的中心。

Abstract:Many physical phenomena are modeled mathematically using integral and differential equations which relate the physical variables and their rates of change. At least to a first approximation, such equations can be taken to be linear and can be profitably viewed as acting on spaces of functions. These space often possess a notion of length like that in Euclidean space which results in what is called a Hilbert space. The study of operators or linear transformations on such spaces has led to new insights in our understanding of quantum mechanics in physics and systems theory in electrical engineering. There are also strong interactions of operator theory with other parts of mathematics including geometry and topology. In recent years, researchers have turned much of their interest to studying more than one operator at a time, often assuming that they commute. As one might guess, the mathematical phenomena modeled can be quite intricate and they rest on notions and concepts from algebra and geometry. Moreover, the questions and results obtained have enriched these fields as well as providing powerful tools to study this multi-variable operator theory.While the study of self-adjoint multivariate operator theory is many decades old, the current proposal concerns the non-self adjoint case which has strong ties to algebraic and complex geometry. The basic notion of module from algebra is adapted to the Hilbert space setting. One considers examples in which holomorphic vector bundles arise naturally and curvature and higher order notions of curvature allow one to mediate results in operator theory. In the current proposal, the principal focus is on quotient modules and modeling their structure and kernel function using a new jet bundle constructed precisely for this purpose. The case when the quotient module is determined by a hyper surface is pretty well under control, at least for the ideal of functions vanishing to low order in the normal direction. Extending this understanding to other ideals determined by varieties with higher co dimension is the next large step and will be at the center of efforts over the next two years.