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式和微分方程式进行数学建模的。至少在第一次近似下，这样的方程可以被认为是线性的，并且可以有益地被视为作用于函数空间。这些空间通常具有欧几里得空间中的长度概念，这导致了所谓的希尔伯特空间。对这种空间上的算符或线性变换的研究使我们对物理学中的量子力学和电子工程中的系统论有了新的理解。算符理论与数学的其他部分也有很强的相互作用，包括几何学和拓扑学。近年来，研究人员将大部分兴趣转向同时研究不止一个运营商，通常假设他们通勤。正如人们可能猜测的那样，所模拟的数学现象可能相当复杂，它们依赖于代数和几何中的概念和概念。虽然自伴多元算子理论的研究已经有了几十年的历史，但目前的研究主要针对与代数和复杂几何有着密切联系的非自伴情形。代数中模的基本概念适用于Hilbert空间环境。我们考虑这样的例子：全纯向量丛是自然产生的，而曲率和高阶曲率概念允许我们在算子理论中调解结果。在当前的提案中，主要的焦点是商模，并使用为此目的而构造的新的喷丛来建模它们的结构和核函数。当商模由超曲面确定时，至少对于函数在法线方向上消失到低阶的理想来说，这种情况是很好地控制的。将这一理解扩展到由具有更高余维的品种确定的其他理想是下一大步，将是未来两年努力的中心。

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.