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.