Applications of polynomial families and free probability

PI的研究集中在泛函分析、概率论（交换和非交换）和组合学的接口。在以前的工作中，PI已经开发了与自由概率理论相关的多项式族的机器。他现在建议使用这一机制，以获得一些应用在自由概率和相关领域。具体项目包括研究来自多项式上特殊迹态的冯诺依曼代数，研究这些态的自由Fisher信息，以及它们与随机矩阵理论中的多矩阵模型的联系。他还将调查运营商代数所产生的高斯对状态的两个国家的自由概率理论，其中多项式家庭提供额外的结构。相反，PI计划使用概率技术来研究多项式族的组合性质，例如它们的线性化系数。他还将探讨是否进一步类多项式家庭的问题有概率解释。矩阵不可交换是矩阵的一个基本性质。自量子力学诞生以来，非对易物体的概率论一直是一个重要的研究领域。在20世纪80年代，Voiculescu开始调查的自由概率理论，这种类型的理论也有许多（有时壮观）应用算子代数和理论的随机矩阵，本身发挥越来越重要的作用，物理和信号处理。另一方面，多项式在数学中是普遍存在的，尽管不交换变量的多项式不太熟悉。这个建议适用于非交换多项式的基本技术，结合自由概率的方法，研究算子代数和随机矩阵。该项目的部分内容非常适合本科生研究，鼓励学生对数学的兴趣。PI将继续组织研讨会，为年轻研究人员提供传播其工作和会见同事的机会。最后，PI将组织一次关于提案中所涵盖主题的会议，目的是将来自不同数学领域的研究人员聚集在一起，进行互利的互动。

The PI's research is concentrated at the interface of functional analysis, probability theory (commutative and non-commutative), and combinatorics. In previous work, the PI has developed the machinery of polynomial families associated to Free Probability theory. He now proposes to use this machinery to obtain a number of applications in free probability and related fields. Specific projects include the investigation of von Neumann algebras coming from special tracial states on polynomials, the study of the free Fisher information of these states, as well as their connection with multi-matrix models in the theory of random matrices. He will also investigate operator algebras arising from Gaussian pairs of states in the two-state free probability theory, for which polynomial families provide extra structure. Conversely, the PI plans to use probabilistic techniques to investigate combinatorial properties of polynomial families, such as their linearization coefficients. He will also explore the issue of whether further classes of polynomial families have probabilistic interpretations. It is a fundamental property of matrices that they may not commute. Since the beginning of quantum mechanics, probability theory of non-commuting objects has been an important field of research. In the 1980s, Voiculescu started the investigation of free probability theory, a theory of this type which also has numerous (sometimes spectacular) applications to operator algebras and the theory of random matrices, itself playing an increasingly important role in physics and signal processing. On the other hand, polynomials are ubiquitous in mathematics, although polynomials in variables which do not commute are less familiar. This proposal applies fundamental techniques of non-commutative polynomials, combined with methods from free probability, to the study of operator algebra and random matrices. Parts of this project are well-suited for undergraduate research, which encourages interest in mathematics among students. The PI will continue to organize a seminar, which provides opportunities for young researchers to disseminate their work and meet colleagues. Finally, the PI will organize a conference on topics covered in the proposal, with the goal of bringing together researchers from different fields of mathematics, resulting in mutually beneficial interactions.