Noncommutative Multivariable Operator Theory

物理科学和工程中的许多问题都可以用非对易函数来建模。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如何设计执行所需任务的系统或如何使其性能最大化的重要信息。这个项目的动机是相对最近世界范围内对多变量算子理论和函数论的非对易方面的兴趣，以及它们与经典函数论、代数和调和分析的相互作用。本项目的目的是将基本思想从分析、代数和几何扩展到非对易的背景下，并在科学和工程中找到应用。对非对易函数及其生成的代数的研究是本项目的目标，在自由概率、内插、优化与控制以及系统理论方面具有潜在的应用。首席研究人员预计，该项目的结果将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。该项目是首席研究者正在进行的计划的继续，该计划旨在发展Sz.-Nagy-FOIAs关于非对易区域和多个非对易变量簇的压缩理论的自由类比，并发展这些区域上的自由全纯函数理论。该项目致力于加强对具有普适模型和具有丰富解析函数理论的非对易多域和簇的结构的理解，并将它们的分类向前推进到自由双全纯等价。随之而来的是对这些多域上的自由全纯函数的研究，重点是几何方面及其与双曲几何的联系。这个项目最突出的特点是非对易多域和簇的结构、由相应的普适模型算子生成的算子代数和这些多域上的非对易解析函数理论之间的相互作用。此外，这项研究立足于经典的多元复变函数论和复代数几何。该项目主要研究以下问题：非交换多域和簇到自由双全纯等价的分类以及相关泛代数到完全等距同构的分类；普适模型、不变子空间和交换提升；非交换多域上的酉不变量(例如曲率、欧拉特征和熵)；非交换多域上的双曲几何；多域上的自由全纯函数；非交换球和复合算子的自由全纯自映射。

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.