Quantum Groups, Special Functions, and Integrable Probability

表示论起源于对代数对象的数学研究，这些对象来自对物理和概率系统的研究。量子群是一大类这样的对象，它们封装了统计力学中某些高度对称模型的结构。虽然量子群起源于统计物理学，但它们的基本结构也出现在概率论、弦论和组合学中。本课题将应用量子群理论，研究量子可积系统和随机矩阵的概率模型等物理模型中的微分方程的解，具体研究量子仿射代数和q-KZB系统、Macdonald理论及其仿射推广、可积随机矩阵模型等。 在以前的工作中，研究人员将量子仿射代数的交织迹与某些θ超几何积分联系起来，并提出了所谓的仿射麦克唐纳图。 该项目的第一部分将使用表示理论的方法来证明这些和相关的命题。 第二部分从有限型量子群的角度研究了Whittaker和Macdonald-Koornwinder函数。 该项目的第三部分将量子可积系统的方法应用于统计学中某些样本协方差矩阵特征值的渐近研究。

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.