Noncommutative Rational Functions in Free Analysis

自由分析中的非交换有理函数

基本信息

批准号：
1954709
负责人：
Thomas Schlumprecht
金额：
$ 11.39万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954709&HistoricalAwards=false
关键词：
Noncommutative Rational Functions Free Analysis

项目摘要

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.

在行动的顺序通常很重要的是，我们的世界本质上是非共同的。例如，取暖和破裂鸡蛋会导致煮熟的鸡蛋或油炸鸡蛋，具体取决于手术的顺序。这就是为什么在数学中编码非共同性的物品在科学方面无所不在的原因。在许多领域，例如控制理论，量子信息理论和随机矩阵理论，有关物品及其集团的新兴问题被措辞，以便独立于矩阵大小。例如，控制系统被设计为黑匣子，其稳定性优先取决于输入数据的大小（矩阵），而仅取决于系统的设计和结构（矩阵的函数）。这种问题的常见框架是通过自由分析（无尺寸的“自由”）提供的，该框架研究了矩阵变量的功能。当仅使用变量和算术操作构建此类函数时，它被称为非交换性合理函数。该项目着重于非交通合理功能的分析，代数和几何方面及其对物品的评估。目的是应用新颖的协同技术来回答有关非交通理性功能的基本开放问题，将其决议应用于半决赛优化和控制理论，并通过有效的算法参与这些理论结果。该项目的目的是双重的。一方面，它考虑了有关自由分析和实际代数几何形状引起的非交流性合理函数的问题。它们的共同线程如下：给定矩阵评估的几何特征，对非交通合理函数，其结构可以推断出什么？这项研究着重于非共同合理功能的积极和奇异性集，它们之间的对称性以及它们之间存在理性图，并观察到控制理论中的非convex（硬）问题，并优化为凸（易于）。此外，该项目的这一部分介绍了非交通合理函数的自然扩展，例如非交通性杂构功能以及对作用于无限尺寸空间的操作员的合理功能。另一方面，非交通性有理函数形成了自由偏斜场，因此与几个基本的纯粹的代数主题有关，例如自由偏度领域的自动形态群，自由lüroth问题和自由algebra中的无特征性的freiheitssatz。该项目的第二部分提出了运用自由分析中的思想和技术来克服这些挑战的建议。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响评估标准来评估的珍贵支持。