Exceptional orthogonal polynomials: theory and applications

我目前的研究重点是例外正交理论 多项式 经典正交多项式是一个基本的 数学工具，应用范围从原子物理学和 电路设计到所用的基本算法 在科学计算中。 特殊的正交多项式是一种新的和迷人的 这是对经典概念的概括。 它们也作为解决方案出现 二阶微分方程，但允许的可能性， 因此，多项式族不包含每个 ℃下 理论假设的这种轻微放松导致了 更丰富的示例集，同时增加了灵活性 在诸如超对称量子力学的应用中。 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， 探索与现有数学理论分支的新联系。

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.