Advances in rational operations in free analysis

自由分析中理性运算的进展

• 批准号: 2348720
• 负责人: Jurij Volcic
• 金额: $ 15.5万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348720&HistoricalAwards=false
• 关键词: Advances; rational; operations; free; analysis

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.

行动或操作的顺序通常很重要；例如，人们应该先洗衣服，然后晾干，而不是反过来。换句话说，操作通常不会交换；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。虽然矩阵和算子理论在过去已经得到了深刻的发展，但快速发展的技术进步提出了必须解决的新挑战。具体地说，扩展的量子技术、复杂的控制系统以及优化和可计算性方面的新资源提出了关于矩阵集合及其与矩阵大小无关的特征的问题。研究这类问题的通用框架是由自由分析提供的（“自由”在尺寸上是自由的），它研究矩阵和算子变量中的函数。这个项目的重点是这样的函数，只使用变量和算术运算，因此被称为非交换多项式和有理函数。虽然它们比一般的非交换函数更具体，更易于计算，但它们的大多数基本特征还有待探索。该项目的范围是研究非交换有理函数及其变化，发展一种理论，允许解决关于它们的开放问题，并最终应用这些解决方案来解决优化，控制系统和量子信息中的新挑战。本项目为研究生提供研究训练机会。这个项目的范围是双重的。首先，该项目旨在回答非交换变量的有理操作的几个函数理论开放问题。其中包括有界算子变量中有理表达式的奇异性和消失性，使用控制理论工具对非交换有理函数组成的几何和结构检测，非交换张量-有理函数及其在计算复杂性中的作用，以及从非交换代数几何和近似零集的角度看待非交换多项式的低秩值的存在性。这些基本问题需要结合复分析、表示理论、代数几何和算子理论的新的协同方法。其次，该项目旨在提出无维度限制的若干算子变量的正性和优化框架，其中目标函数和约束为非交换多项式及其变化。实现这一目标的途径是通过泛函分析、实代数几何和算子代数。此外，该项目寻求将这些新的优化算法应用于量子信息理论，研究复杂量子网络中的非线性贝尔不等式以及设备无关认证和加密安全中的自我测试现象。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。