Applications of random matrix theory to probabilistic aspects of operator algebras

算符代数理论是研究量子力学的数学框架。它有许多应用于数学的各个部分，并与其他科学领域（物理学，量子信息……）进行了新的互动。在过去的几十年里，研究算子代数的结构问题一直是一个巨大的研究和进步的对象。我的研究对象是集中在这些代数的“渐近概率”方面。算子代数理论中许多重要的问题多年来一直没有得到解决，如cones嵌入问题。我的主要动机之一是试着找出哪些算子代数可以用矩阵在矩的意义上近似，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，以及它如何在连续时间过程的水平上扩展。该方法主要依赖于随机矩阵理论的概率技术和表示理论方法。特别是，矩阵积分理论是一个工具，我打算做一个基本的使用。此外，我计划利用相关矩阵积分的图解展开（费曼图）。继Biane和Speicher的方法以及最近在Horn问题上的突破之后，另一种具有潜在关键用途的方法是渐近组合学。我期望这一提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项目中产生，就像过去已经出现的那样（与理论物理学家和天文学家）。

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).