Applications of random matrix theory to probabilistic aspects of operator algebras

随机矩阵理论在算子代数概率方面的应用

• 批准号: 341303-2007
• 负责人: Collins, Benoit
• 金额: $ 0.87万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363705
• 关键词: Applications; random; matrix; theory; probabilistic

Operator Algebra Theory is the mathematical framework for the study of quantum mechanics.It has many applications to various parts of mathematics and renewed interaction with other scientific areas (physics, quantum information...). The question of studying the structure of operator algebras has been the object of tremendous research and progresses during the last decades. The object of my research is to focus on 'asymptotic probabilistic' aspects of these algebras. Very important questions have been left open for many years in Operator Algebra Theory, such as Connes' embedding problem. One of my leitmotiv would be to try to figure out which operator algebras can be approximated by matrices in the sense of moments, and how good can these approximations be. In the same vein, I plan to investigate which non-commutative random variables can be approximated by random matrices, and how this extends at the level of continuous time processes. The method proposed relies mainly on probabilistic techniques inspired from Random Matrix Theory and on representation theoretic methods. In particular, Matrix Integral theory is a tool I intend to make a fundamental use of. Also, I plan to make use of diagrammatic expansion (Feynman diagrams) of relevant matrix integrals. An other method of potential crucial use is asymptotic combinatorics following methods of Biane and Speicher, and recent breakthroughs in the Horn problem. I expect that the operator algebraic motivations of this proposal will provide new applications and tools to random matrix theory. I also believe that interdisciplinary collaborations with physicists could arise from my research program, as it has already arosen in the past (with theoretical physicists and astronomers).

算子代数理论是研究量子力学的数学框架。它在数学的各个部分都有许多应用，并与其他科学领域（物理学，量子信息......）重新互动。算子代数的结构问题是近几十年来研究和发展的热点。我的研究对象是集中在'渐近概率'方面的这些代数。算子代数理论中的一些重要问题，如Connes的嵌入问题，一直是一个悬而未决的问题。我的主旨之一是试图弄清楚哪些算子代数可以用矩意义上的矩阵近似，以及这些近似有多好。同样，我计划研究哪些非交换随机变量可以用随机矩阵近似，以及这如何在连续时间过程的水平上扩展。所提出的方法主要依赖于随机矩阵理论和表示论方法启发的概率技术。特别是，矩阵积分理论是一个工具，我打算作出基本使用。此外，我计划利用相关矩阵积分的图解展开（费曼图）。另一种潜在的关键用途是渐近组合学的方法Biane和Speicher，以及最近的突破霍恩问题。我期望这个建议的算子代数动机将为随机矩阵理论提供新的应用和工具。我也相信，与物理学家的跨学科合作可以从我的研究计划中产生，因为它已经在过去（与理论物理学家和天文学家）。