Exceptional orthogonal polynomials: theory and applications

我目前的研究重点是例外正交多项式理论。经典正交多项式是一种基本的数学工具，其应用范围从原子物理和电路设计到科学计算中使用的基本算法。例外正交多项式是经典概念的一个新的和迷人的推广。它们也作为二阶微分方程的解出现，但允许由此产生的多项式族不包含所有次数的可能性。这种对理论假设的温和放松导致了更丰富的例子，并随之增加了超对称量子力学等应用的灵活性。早在2008-2009年，我就与合作者一起引入并开发了这些新技术。从那时起，该领域出现了快速增长，在应用和理论方面都有许多令人兴奋的发展。我的研究目标是专注于特殊正交多项式的理论基础，以期探索与数学理论现有分支的新联系。

My current research focus is the theory of exceptional orthogonal polynomials. Classical orthogonal polynomials are a fundamental mathematical tool with applications ranging from atomic physics and the design of electrical circuits to fundamental algorithms utilized in scientific computation. Exceptional orthogonal polynomials are a new and fascinating generalization of the classical concept. They also arise as solutions of second-order differential equations, but allow for the possibility that the resulting family of polynomials does not contain every degree. This mild relaxation of the theoretical assumption leads to a much richer set of examples with an attendant increase of flexibility in applications such as super-symmetric quantum mechanics. Working with collaborators, I introduced and developed these novel techniques back in 2008-2009. Since then the field has seen rapid growth with many exciting developments both in applications and theory. My research goals is to focus on the theoretical underpinnings of exceptional orthogonal polynomials with a view towards exploring novel connections with existing branches of mathematical theory.