Noncommutative Rational Functions in Free Analysis

Texas A&M University, College Station TX

Investigators

Abstract

Our world is essentially noncommutative in the sense that the order of actions often matters; for example, heating and cracking an egg can result in either a boiled egg or a fried egg, depending on the order of the operations. This is the reason why matrices, which encode noncommutativity in mathematics, are omnipresent in science. In many areas, such as control theory, quantum information theory and random matrix theory, the emerging questions about matrices and their ensembles are phrased so as to be independent of the matrix size. For example, a control system is designed as a black box, and its stability preferably does not depend on size of the input data (matrices) but only on the design and the structure of the system (a function of matrices). The common framework for such problems is provided by free analysis ("free" as in size-free), which studies functions in matrix variables. When such a function is built using only variables and arithmetic operations, it is called a noncommutative rational function. This project focuses on analytic, algebraic and geometric aspects of noncommutative rational functions and their evaluations on matrices. The goal is to apply novel synergistic techniques to answer fundamental open questions about noncommutative rational functions, apply their resolutions to semidefinite optimization and control theory, and accompany these theoretical results with efficient algorithms. The aim of this project is twofold. On one hand, it considers questions about noncommutative rational functions that arise from free analysis and real algebraic geometry. Their common thread is the following: given a geometric feature of matrix evaluations of a noncommutative rational function, what can be deduced about its structure? This research focuses on positivity and singularity sets of noncommutative rational functions, their symmetries and existence of rational maps between them, with a view towards transforming non-convex (hard) problems in control theory and optimization into convex (easy) ones. Furthermore, this part of the project addresses natural extensions of noncommutative rational functions, such as noncommutative meromorphic functions and rational functions on operators acting on infinite dimensional spaces. On the other hand, noncommutative rational functions form a free skew field and are therefore related to several fundamental purely algebraic topics, such as the automorphism group of the free skew field, the free Lüroth problem, and the characteristic-free Freiheitssatz in a free algebra. The second part of this project proposes to apply ideas and techniques from free analysis to overcome these challenges. 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.