Noncommutative Multivariable Operator Theory

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.

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.