Exceptional orthogonal polynomials: theory and applications

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.