Higher Order Freeness of Random Matrices

随机矩阵理论是数学研究的一个新领域，它受到数学金融、多元统计、量子引力、电子工程以及纯数学中问题的启发。其范围从风险建模和投资组合管理延伸到黎曼zeta函数的零点研究。 我最近在分析随机矩阵的涨落时有了一个重要发现，我把这个理论称为高阶自由度。自由独立性是一种新的独立性，旨在与矩阵值随机变量。高阶自由度对涨落矩的作用与Voiculescu的自由度对普通矩的作用相同。必须对这些事态发展采取后续行动;加拿大必须发展和保持这方面的专门知识。 高阶自由度是训练HQP的一个很好的领域，因为它有很多方面：组合，分析，拓扑，以及它与上面提到的应用程序的联系。这意味着具有非常不同背景的学生可以快速进入主题并做出贡献，同时学习数学思想的力量和优雅。和我一起工作的学生在学术界和工业界都取得了成功。

Random matrix theory is a new area of mathematical research motivated by problems in mathematical finance, multivariate statistics, quantum gravity, electrical engineering, as well as pure mathematics. Its scope extends from risk modelling and portfolio management to the study of the zeros of the Riemann zeta function. I recently made an important discovery in analyzing the fluctuations of random matrices; I call this theory higher order freeness. Free independence is a new kind of independence that is designed to work with matrix valued random variables. Higher order freeness does for fluctuation moments what Voiculescu's freeness did for ordinary moments. It is quite important that these developments be followed up; and that Canada develop and maintain expertise in this area. Higher order freeness is an excellent area for the training of HQP as it as many aspects: combinatorial, analytic, topological, as well as it connections to the applications mentioned above. This means students with very diverse backgrounds can quickly enter the subject and make a contribution, while at the same time learning the power and elegance of mathematical thought. Students working with me have gone on to successful careers in academia and industry.