Exceptional orthogonal polynomials: theory and applications

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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.