An Algebraic and Asymptotic Study on Matrix Valued Orthogonal Polynomials, and Their Applications

Description

Orthogonal polynomials are one of the classical subjects in both pure and applied mathematics. They play a significant role in various areas of mathematics and physics, such as approximation theory, numerical analysis, combinatorics, statistics, integrable systems, random matrix theory, quantum mechanics, quantum field theory, and so on. With the persistent development in related scientific fields, generalizations of classical orthogonal polynomials, such as Jacobi polynomials, Hermite polynomials, and Laguerre polynomials, have been explored continuously in the literature. One notable class of generalizations is the so-called matrix valued orthogonal polynomials, which introduces higher dimension as well as non-commutativity into the framework. Recently, matrix valued orthogonal polynomials find important applications in non-commutative integrable systems, as well as random tiling models. Although breakthrough has been made, there are still many significant problems which remain unsolved. In this project, we will explore various properties of matrix valued orthogonal polynomials. We aim to investigate their algebraic properties and their connections with non-commutative integrable systems, such as the non-Abelian Toda lattice and non-commutative Painlevé equations. Another main target of the study is to understand their asymptotic behavior as the polynomial degree tends to infinity. Furthermore, we aim to explore their applications in the study of random tiling models.