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、电路设计到科学计算中的基本算法，特殊正交多项式是经典概念的一种新的、令人着迷的推广。他们也出现作为解决方案的二阶微分方程，但允许的可能性，由此产生的家庭多项式不包含每一个程度。这种对理论假设的温和放松导致了更丰富的例子集，同时也增加了超对称量子力学等应用的灵活性。我在2008年引进并开发了这些新技术-从那时起，该领域已经看到了快速增长，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理论特殊正交多项式的基础与viewtoward探索新的连接与现有的数学理论分支。

My current research focus is the theory of exceptional orthogonalpolynomials. Classical orthogonal polynomials are a fundamentalmathematical tool with applications ranging from atomic physics andthe design of electrical circuits to fundamental algorithms utilizedin scientific computation.Exceptional orthogonal polynomials are a new and fascinatinggeneralization of the classical concept. They also arise as solutionsof second-order differential equations, but allow for the possibilitythat the resulting family of polynomials does not contain everydegree. This mild relaxation of the theoretical assumption leads to amuch richer set of examples with an attendant increase of flexibilityin 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 viewtowards exploring novel connections with existing branches of mathematical theory.