Invariants for Multivariate Operator Theory

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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.