Free harmonic analysis and applications

这一研究建议是在非对易概率中提出的，它是纯数学的一个分支，通常被认为是概率论的非对易平行。该理论植根于量子力学的数学基础，并与另一个数学领域密切相关，称为泛函分析，在泛函分析中，人们研究拓扑向量空间和这些空间之间的函数。粗略地说，这里的非交换性意味着通常的函数(或随机变量)，比如X和Y，被向量空间上的矩阵或算子取代，因此熟悉的恒等式XY=YX不再成立。(非交换的)随机变量的这一特征使得在它们之间引入各种独立的概念并随后研究它们的概率行为成为可能。最著名的例子是自由独立的概念和相应的自由概率，这是本提案所基于的。 申请人建议研究自由概率的调和分析方面，以及它在双自由概率理论、自由局部极限定理和无限遍历理论等相关领域的推广和应用。 沃库列斯库在2013年引入了双自由的概念，这是一种适用于非交换随机变量对的非交换独立性的概念。这一领域目前正处于快速发展之中，迄今为止，大多数对无双卷积和基本无双卷积的治疗是组合的。与经典或自由概率理论相比，目前的文献缺乏对双自由概率的调和分析方法。这项拟议研究的主要目标是发展这样一种理论，重点是双自由无限可分分布。 在最近与黄浩伟合作的一项工作中，申请人获得了一些结果，表明经典的无穷可分定律极限理论，由于Levy和Khintchine，在变量对交换的框架下，有一个完美的双自由模拟。申请人相信，他的方法和通过极限定理的方法可以用来处理双自由调和分析的一般问题，而不需要交换性假设。进一步发展上述结果，对于更好地理解双无关性并探索其与随机矩阵的联系是非常重要的。该项目的成功将使加拿大站在这一研究的前沿，并对自由概率的调和分析方面带来影响。 该提案还包含研究生或博士后研究员可以访问的几个未回答的问题。申请者将使用NSERC发现补助金为HQP的培训做出贡献，特别是在博士生和博士后研究员层面上。

This research proposal is in non-commutative probability, a branch of pure mathematics that is often regarded as a non-commutative parallelism of probability theory. The theory has its roots in the mathematical foundation of quantum mechanics and is closely related to another area of mathematics, called functional analysis, in which one studies topological vector spaces and functions between such spaces. Roughly, the non-commutativity here means that usual functions (or random variables), say, X and Y, are replaced by matrices or operators on vector spaces so that the familiar identity XY=YX no longer holds. This feature of (non-commutative) random variables makes it possible to introduce various notions of independence among them and study their probabilistic behaviour thereafter. The most famous example of such is the notion of free independence and the corresponding free probability, on which this proposal is based on. The applicant proposes to study the harmonic analysis aspect of free probability, as well as its extension and applications to related fields such as bi-free probability theory, free local limit theorems, and infinite ergodic theory. Voiculescu introduced in 2013 the notion of bi-freeness, which is a suitable notion of non-commutative independence for pairs of non-commutative random variables. This area is currently under a rapid development, and most treatments of bi-freeness and the underlying bi-free convolution are combinatorial to date. In contrast to the classical or free probability theories, the literature lacks a harmonic analysis approach to bi-free probability at this moment. The primary goal of this proposed research is to develop such a theory, with an emphasis on bi-freely infinitely divisible distributions. In a recent joint work with Hao-Wei Huang, the applicant obtained some results to show that the classical limit theory of infinitely divisible laws, due to Levy and Khintchine, has a perfect bi-free analogue in the framework of commuting pairs of variables. The applicant believes that his methodology and the approach through limit theorems can be used to treat the general question of bi-free harmonic analysis, without the commutativity assumption. It is quite important that the aforementioned results be further developed in order to understand the bi-freeness better and explore its connections with random matrix. A success in this program would put Canada at the frontier of this research line, and brings impact to harmonic analysis aspect of free probability. This proposal also contains several unanswered questions that are accessible to graduate students or postdoc fellows. The applicant will use the NSERC Discovery Grant to contribute to the training of HQP, especially at the level of PhD students and Postdoctoral Fellows.