Noncommutative Multivariable Operator Theory

物理科学和工程中的许多问题都可以用非交换函数来建模。这些函数用于编码有关物理系统的信息，因此研究它们的各个方面可以揭示有关如何设计执行所需任务的系统或如何最大化其性能的重要信息。这个项目的动机是最近世界范围内对多变量算子理论和函数理论的非交换方面的兴趣，以及它们与经典函数、代数和调和分析理论的相互作用。本项目旨在将分析、代数和几何的基本思想扩展到非交换环境中，并在科学和工程中寻找应用。非交换函数及其生成的代数的研究是该项目的目标，在自由概率、插值、优化和控制以及系统理论方面具有潜在的应用。首席研究员希望这个项目的结果能够在数学的几个领域之间建立新的联系，并在数学物理中得到应用。该项目的另一个重要目标是吸引研究生参加PI的研究项目，并帮助德克萨斯大学圣安东尼奥分校建立数学博士项目。拟议的项目是首席研究员正在进行的开发Sz的免费模拟程序的延续。-Nagy-Foias关于非交换域和若干非交换变量变异的缩并理论，并发展了这些域上的自由全纯函数理论。本项目致力于提高对具有普遍模型和丰富解析函数理论的非交换多域和变体的结构的认识，并在其分类到自由生物全纯等价方面取得进展。这伴随着对这些多域上的自由全纯函数的研究，重点是几何方面以及与双曲几何的联系。本课题最突出的特点是研究了非交换多域和变异的结构、由相应的通用模型算子生成的算子代数以及这些多域上的非交换解析函数理论之间的相互作用。此外，本研究是锚定在经典复变函数理论在几个变量和复杂代数几何。该项目主要研究以下问题：非交换多域和变异到自由生物全纯等价的分类和相关的全代数到完全等距同构的分类；全称模型、不变子空间和交换子提升非交换多域上的酉不变量（如曲率、欧拉特征和熵）；非交换多边形上的双曲几何多域上的自由全纯函数非交换球的自由全纯自映射与复合算子。

Many problems in the physical sciences and engineering can be modeled by noncommutative functions. Such functions are used to encode information about physical systems, so studying various aspects of them could reveal important information about how to design systems that perform desired tasks or how to maximize their performance. The motivation for this project is the relatively recent worldwide interest in the noncommutative aspects of multivariable operator theory and function theory, and their interplay with the classical theory of functions, algebras, and harmonic analysis. The present project aims at extending fundamental ideas from analysis, algebra, and geometry to the noncommutative context and finding applications in science and engineering. The study of noncommutative functions and the algebras that they generate, which is the goal of the project, has potential applications to free probability, interpolation , optimization and control, and systems theory. The principal investigator expects the results of the project to make new connections between several areas of mathematics and to have applications in mathematical physics. Another important objective of the project is to attract graduate students to the PI's research program and help build a Ph.D. program in mathematics at the University of Texas-San Antonio. The proposed project is a continuation of the ongoing program of the principal investigator to develop a free analogue of the Sz.-Nagy-Foias theory of contractions for noncommutative domains and varieties in several noncommuting variables and to develop the theory of free holomorphic functions on these domains. The project is devoted to enhancing the understanding of the structure of the noncommutative polydomains and varieties that admit universal models and have rich analytic function theory, and to make advances towards their classification up to free biholomorphic equivalence. This is accompanied by the study of free holomorphic functions on these polydomains with the emphasis on geometric aspects and the connection with the hyperbolic geometry. The most prominent feature of this project is the interaction between the structure of the noncommutative polydomains and varieties, the operator algebras generated by the corresponding universal model operators, and the noncommutative analytic function theory on these polydomains. Moreover, this study is anchored in classical complex function theory in several variables and in complex algebraic geometry. The project focuses on the following problems: classification of noncommutative polydomains and varieties up to free biholomorphic equivalence and the classification of the associated universal algebras up to completely isometric isomorphisms; universal models, invariant subspaces, and commutant lifting; unitary invariants on noncommutative polydomains (e.g., the curvature, the Euler characteristic, and the entropy); hyperbolic geometry on noncommutative polyballs; free holomorphic functions on polydomains; free holomorphic self-maps of noncommutative balls and composition operators.