Noncommutative Rational Functions in Free Analysis

我们的世界本质上是不可交换的，因为行动的顺序通常很重要；例如，加热和打蛋可以得到煮鸡蛋或煎鸡蛋，这取决于操作的顺序。这就是为什么在数学中编码非交换性的矩阵在科学中无处不在的原因。在控制论、量子信息论和随机矩阵理论等许多领域中，关于矩阵及其集合的新问题都被表述为与矩阵大小无关。例如，一个控制系统被设计成一个黑盒，它的稳定性最好不取决于输入数据（矩阵）的大小，而只取决于系统的设计和结构（矩阵的函数）。这类问题的通用框架是由自由分析提供的（“自由”指的是尺寸自由），它研究矩阵变量中的函数。当这样的函数只使用变量和算术运算构建时，它被称为非交换有理函数。这个项目的重点是非交换有理函数的解析、代数和几何方面及其在矩阵上的评价。目标是应用新的协同技术来回答关于非交换有理函数的基本开放问题，将其解决方案应用于半定优化和控制理论，并将这些理论结果与有效的算法相结合。这个项目的目的是双重的。一方面，它考虑了自由分析和实际代数几何中出现的非交换有理函数的问题。它们的共同点是：给定一个非交换有理函数的矩阵求值的几何特征，可以推导出它的结构是什么？本文主要研究非交换有理函数的正集和奇异集，它们的对称性和它们之间的有理映射的存在性，以期将控制理论和优化中的非凸（难）问题转化为凸（易）问题。此外，这一部分的项目涉及非交换有理函数的自然扩展，如非交换亚纯函数和作用于无限维空间的算子上的有理函数。另一方面，非交换有理函数形成自由偏场，因此与自由偏场的自同构群、自由l<s:1>罗斯问题和自由代数中的无特征Freiheitssatz等几个基本的纯代数问题有关。该项目的第二部分建议应用自由分析的思想和技术来克服这些挑战。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.