Advances in rational operations in free analysis

Drexel University, Philadelphia PA

Investigators

Abstract

The order of actions or operations typically matters; for example, one should first wash the clothes and then dry them, not the other way around. In other words, operations typically do not commute; this is why matrices, which encode noncommutativity in mathematics, are omnipresent in science. While matrix and operator theory has been profoundly developed in the past, the fast-evolving technological advances raise new challenges that have to be addressed. Concretely, expanding quantum technologies, complex control systems, and new resources in optimization and computability pose questions about ensembles of matrices and their features that are independent of the matrix size. The common framework for studying such problems is provided by free analysis ("free" as in size-free), which investigates functions in matrix and operator variables. This project focuses on such functions that are built only using variables and arithmetic operations, and are therefore called noncommutative polynomials and rational functions. While these are more tangible and computationally accessible than general noncommutative functions, most of their fundamental features are yet to be explored. The scope of the project is to investigate noncommutative rational functions and their variations, develop a theory that allows resolving open problems about them, and finally apply these resolutions to tackle emerging challenges in optimization, control systems, and quantum information. This project provides research training opportunities for graduate students. The scope of this project is twofold. Firstly, the project aims to answer several function-theoretic open problems on rational operations in noncommuting variables. Among these are singularities and vanishing of rational expressions in bounded operator variables, geometric and structural detection of composition in noncommutative rational functions using control-theoretic tools, noncommutative tensor-rational functions and their role in computational complexity, and existence of low-rank values of noncommutative polynomials with a view towards noncommutative algebraic geometry and approximate zero sets. These fundamental problems call for new synergistic methods that combine complex analysis, representation theory, algebraic geometry and operator theory. Secondly, the project aims to advance the framework of positivity and optimization in several operator variables without dimension restrictions, where the objective functions and constraints are noncommutative polynomials and their variations. The approach to this goal leads through functional analysis, real algebraic geometry and operator algebras. Moreover, the project seeks to apply these new optimization algorithms in quantum information theory, to study nonlinear Bell inequalities in complex quantum networks and the self-testing phenomenon in device-independent certification and cryptographic security. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.