Quantum Groups, Special Functions, and Integrable Probability

量子群、特殊函数和可积概率

• 批准号: 2039183
• 负责人: Yi Sun
• 金额: $ 0.52万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2039183&HistoricalAwards=false
• 关键词: Quantum; Groups; Special; Functions; Integrable

Representation theory originated as the mathematical study of algebraic objects that arise from the study of of physical and probabilistic systems. Quantum groups are a large class of such objects that encapsulate the structure of certain highly symmetric models in statistical mechanics. Though the origins of quantum groups are in statistical physics, their underlying structures appear also in probability theory, string theory, and combinatorics. This project will apply the theory of quantum groups to study solutions of differential equations that appear in certain physical models known as quantum integrable systems and probabilistic models of random matrices.In more detail, the project focuses on quantum affine algebras and q-KZB systems, Macdonald theory and its affine generalizations, and integrable random matrix models. In previous work, the investigator related traces of intertwiners of quantum affine algebras to certain theta hypergeometric integrals and proposed so-called affine Macdonald conjectures. The first part of the project will use a representation-theoretic approach to prove these and related conjectures. The second part of the project studies Whittaker and Macdonald-Koornwinder functions from the perspective of finite-type quantum groups. The third part of the project applies methods motivated by quantum integrable systems to the asymptotic study of eigenvalues of certain sample covariance matrices arising in statistics.

代表理论起源于对物理和概率系统研究产生的代数对象的数学研究。量子组是一大批此类对象，它们封装了统计力学中某些高度对称模型的结构。尽管量子组的起源是统计物理学的，但它们的基本结构也出现在概率理论，弦理论和组合学中。该项目将应用量子组的理论来研究某些物理模型中出现的微分方程的解决方案，这些差分方程式被称为量子整合系统和随机矩阵的概率模型。在更详细的情况下，该项目着重于量子仿射代数和Q-KZB系统，麦克唐纳德理论及其亲属概括及其可用的随机矩阵模型。 在先前的工作中，研究者将量子仿射代数的互换器与某些theta超几何积分和拟议的所谓仿生麦克唐纳猜想相关联。 该项目的第一部分将使用表示理论方法来证明这些和相关的猜想。 从有限型量子组的角度来看，项目研究的第二部分Whittaker和MacDonald-Koornwinder的功能。 该项目的第三部分应用了由量子整合系统激励的方法，用于对统计中某些样品协方差矩阵的特征值的渐近研究。