Applications of polynomial families and free probability

多项式族和自由概率的应用

• 批准号: 0900935
• 负责人: Michael Anshelevich
• 金额: $ 10.5万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900935&HistoricalAwards=false
• 关键词: Applications; polynomial; families; free; probability

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.

PI的研究集中在泛函分析、概率论(交换和非交换)和组合学的界面上。在以前的工作中，PI发展了与自由概率理论相关的多项式族的机制。他现在建议使用这一机制来获得自由概率及其相关领域的一些应用。具体项目包括研究由多项式上的特殊迹态产生的von Neumann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。他还将研究两态自由概率理论中由高斯态对产生的算子代数，多项式族为其提供了额外的结构。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。他还将探讨另一类多项式族是否有概率解释的问题。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，非对易物体的概率论就一直是一个重要的研究领域。20世纪80年代，沃库列斯库开始研究自由概率理论，这种类型的理论在算子代数和随机矩阵理论中也有许多(有时是壮观的)应用，它本身在物理和信号处理中发挥着越来越重要的作用。另一方面，多项式在数学中是普遍存在的，尽管不能交换的变量中的多项式不太常见。该方案将非对易多项式的基本技巧与自由概率方法相结合，应用于算子代数和随机矩阵的研究。这个项目的一些部分非常适合本科生的研究，这会鼓励学生对数学的兴趣。国际和平研究所将继续举办研讨会，为青年研究人员提供传播工作和与同事见面的机会。最后，国际数学联合会将就提案中涉及的主题组织一次会议，目的是将来自不同数学领域的研究人员聚集在一起，实现互利互动。