Combinatorial Methods in Free Probability

自由概率中的组合方法

• 批准号: 0400860
• 负责人: Michael Anshelevich
• 金额: $ 7.5万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2006-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400860&HistoricalAwards=false
• 关键词: Combinatorial; Methods; Free; Probability

Intellectual merit: Anshelevich's research lies at the interface between functional analysis, probability, and combinatorics. He will use combinatorial methods to obtain new results in free probability theory. For example, he will develop a theory of free Appell polynomials, and use it to obtain new results about limiting behavior of averages of stationary correlated random sequences. He will also use methods from functional analysis to obtain new results in probability theory. For example, he will investigate the chaos decomposition for general Levy processes using the machinery of Fock spaces. Finally, he will use methods from non-commutative probability to obtain new results in combinatorics and special functions theory. For example, he will investigate the linearization coefficients for the q-analogs of the Meixner orthogonal polynomials. Matrices do not commute. The importance of this fact in our interpretation of nature was realized with the invention of quantum mechanics. That theory has also used probabilistic ideas and language since its inception. Non-commutative probability is the probability theory of non-commuting objects, such as matrices and operators. In the 1980s Voiculescu introduced free probability theory, which is a particular type of such a non-commutative theory. Its origins lie in the, currently quite abstract, theory of von Neumann algebras of free groups, but also in the theory of random matrices, directly applicable in an number of areas of physics and signal processing. Anshelevich's expertise lies primarily in functional analysis, thus one of the consequences of this research will be to gain a deeper understanding of a number of fields with potential connections to free probability. Also, some of the combinatorial problems considered are well-suited for undergraduate research, and will be used to encourage interest in mathematics among students.

智力价值：Anshelevich的研究在于功能分析，概率和组合学之间的接口。他将使用组合的方法，以获得新的成果，在自由概率论。例如，他将开发一个理论的自由阿佩尔多项式，并使用它来获得新的结果的限制行为的平均平稳相关的随机序列。他还将使用方法从功能分析，以获得新的成果，概率论。例如，他将研究混沌分解一般利维过程使用的机械福克空间。最后，他将使用非交换概率的方法来获得组合学和特殊函数理论的新结果。 例如，他将研究Meixner正交多项式的q-类似物的线性化系数。 矩阵不对易。随着量子力学的发明，这一事实在我们对自然的解释中的重要性得以实现。该理论自诞生以来也使用了概率思想和语言。非交换概率是关于非交换对象的概率论，如矩阵和算子。在20世纪80年代Voiculescu介绍了自由概率论，这是一个特殊类型的这种非交换理论。它的起源在于，目前相当抽象，冯诺依曼代数理论的自由群，但也在理论的随机矩阵，直接适用于一些领域的物理和信号处理。 Anshelevich的专业知识主要在于功能分析，因此这项研究的结果之一将是更深入地了解一些与自由概率有潜在联系的领域。此外，一些被认为是非常适合本科研究的组合问题，并将用于鼓励学生对数学的兴趣。