Noncommutative Functions, Algebra and Operator Analysis

不可交换性是指操作的顺序很重要；先穿袜子后穿鞋和先穿鞋后穿袜子是完全不同的。非交换性在量子力学的基础上发挥了重要作用，导致了泛函分析这一数学分支的发展。从物理学的起源开始，泛函分析已经获得了自己的生命，并在量子信息论、量子计算以及控制和系统理论等领域找到了令人兴奋的应用。最近，工程文献中出现的问题（关于半定规划和线性矩阵不等式）导致了泛函分析中一个新的分支领域的发展，即非交换函数理论。这个令人兴奋的新分支领域位于非交换代数、泛函分析和算子理论的交叉点，并且具有广泛的应用。非交换函数理论不是处理完全抽象的数学对象，而是使用被称为矩阵的具体结构，这是科学、工程和工业中使用的基本非交换对象之一。本项目旨在加深对非交换函数理论的理解，从而扩大其与相关数学领域的联系。它将有助于早期研究的专业发展，并提供非交换函数理论领域中已解决和未解决问题的最新列表。该项目的主要目标之一是增加非交换函数理论、非交换代数和算子理论之间的相互作用。近年来，通过非交换函数理论、复分析和算子理论等技术，非交换代数的一些定理和研究取得了一些进展。感兴趣的对象将是非交换有理函数，以及它们在矩阵和算子上的求值如何揭示函数及其生成的偏场的代数信息；非交换有理映射的注入域及其雅可比矩阵的可逆性集与非交换有理映射是否诱导自由偏场的自同构之间有很强的联系。另一方面，非交换代数的结果（如非交换有理函数的实现）将用于加深对数学分析主题的理解。利用非交换代数和非交换函数理论的新思想将为最优多项式近似和相关领域的研究提供新的见解。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。

Noncommutativity is the idea that the order of operations matters; socks before shoes is very different than shoes before socks. Noncommutativity played a fundamental role in the foundation of quantum mechanics, leading to the development of the branch of mathematics known as functional analysis. From its origins in physics, functional analysis has gained a life of its own and has found exciting applications in areas such as quantum information theory and quantum computing as well as control and systems theory. More recently, questions (about semi-definite programming and linear matrix inequalities) arising in the engineering literature have led to the development of a new subfield of functional analysis known as noncommutative function theory. This exciting new subfield sits at the intersection of noncommutative algebra, functional analysis, and operator theory, and it enjoys a wide variety of applications. Rather than deal with completely abstract mathematical objects, noncommutative function theory employs the use of concrete structures known as matrices, which are one of the fundamental noncommutative objects used in science, engineering and industry. This project aims to deepen the understanding of noncommutative function theory and consequently augment its connections to related mathematical fields. It will assist in the professional development of early researches and provide an up-to-date list of both solved and unsolved problems in the field of noncommutative function theory.             One of the main goals of this project is to increase the interplay between noncommutative function theory, noncommutative algebra, and operator theory. Recently, theorems and advances have been made in noncommutative algebra through techniques in noncommutative function theory, complex analysis, and operator theory. Objects of interest will be noncommutative rational functions and how their evaluations on matrices and operators reveals algebraic information about the functions and the skew field they generate; there is a strong connection between the injectivity domains of noncommutative rational mappings, the invertibility sets of their Jacobian matrices, and whether the noncommutative rational mapping induces an automorphism of the free skew field. On the other hand, results from noncommutative algebra (such as the realization of a noncommutative rational function) will be used to deepen the understanding of topics in mathematical analysis. Employment of novel ideas from noncommutative algebra and noncommutative function theory will provide new insights into the study of optimal polynomial approximants and related areas.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.