Free harmonic analysis and applications

免费谐波分析和应用

• 批准号: RGPIN-2016-03796
• 负责人: Wang, JiunChau
• 金额: $ 1.09万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635287
• 关键词: Free; harmonic; analysis; applications

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.

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。