Free probability, polynomial families, and applications

自由概率是一个相对年轻的理论（始于20世纪80年代），有许多强有力的结果和应用于算子代数、随机矩阵和其他领域的研究。另一方面，多项式的研究可以追溯到代数的最开始，特别是正交多项式在数学、物理和工程中广泛使用。也许令人惊讶的是，通过福克空间表示、多项式的鞅性质、随机积分和特殊函数，这两个研究领域之间存在着许多联系。这两个领域也发展了许多推广和扩展：一方面是算子值自由概率论和其他非交换概率论；另一方面，算子值正交多项式、矩阵参数多项式、多重非交换变量多项式等。本项目将通过一系列具体问题来探讨上述对象，强调项目标题中两个领域之间的联系。它还将涉及到其他领域的应用，特别是随机矩阵。矩阵的一个基本性质是它们不能交换。自量子力学诞生以来，非交换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开始了自由概率论的研究，这种类型的理论在算子代数和随机矩阵理论（它本身在物理学和信号处理中起着越来越重要的作用）上也有许多，有时是惊人的应用。另一方面，多项式在数学中是无处不在的，尽管不交换变量的多项式不太熟悉。这个项目将继续研究这两个学科之间的相互作用，将每个学科的技术应用到另一个学科的研究中。作为该项目的一部分，PI将继续组织研讨会和会议，汇集来自不同领域的研究人员和学生。他也将继续他的工作与本科生的研究课题，形成这个项目的一部分。

Free probability is a relatively young theory (started in the 1980s), with numerous powerful results and applications to the study of operator algebras, random matrices, and other fields. On the other hand, the study of polynomials goes back to the very beginnings of algebra, and orthogonal polynomials in particular are used throughout mathematics, physics, and engineering. Perhaps surprisingly, there are numerous relations between these two fields of study, through Fock space representations, martingale properties of polynomials, stochastic integrals, and special functions. Both fields have also developed numerous generalizations and extensions: operator-valued free probability theory and other non-commutative probability theories on one hand; and on the other, operator-valued orthogonal polynomials, polynomials of matrix argument, polynomials in multiple non-commuting variables, etc. This project will explore, through a series of specific problems, the objects mentioned above, emphasizing the connection between the two fields in the project title. It will also involve applications to other fields, notably random matrices.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 (which itself plays 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 project will continue the study of the mutual interaction between these two subjects, applying techniques from each one to the study of the other. As part of the project, the PI will continue organizing seminars and conferences bringing together researchers and students from different fields. He will also continue his work with undergraduate students on research topics forming part of this project.