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学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。它们也可以作为二阶微分方程的解出现，但允许这样一种可能性，即所得到的多项式族不包含每一次。这种对理论假设的温和放松导致了一组更丰富的例子，同时在超对称量子力学等应用中也增加了灵活性。**在2008-2009年，我与合作者一起引入并开发了这些新技术。*从那时起，该领域迅速发展，在应用和理论方面都取得了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.