Orthogonal Polynomials and Painlevé Equations

The Riemann-Hilbert formulation of orthogonal polynomials provides a crucial bridge between disparate areas of mathematics, allowing tools developed in the context of integrable systems to be available for the study of orthogonal polynomials and random matrix theory. A Riemann-Hilbert formulations is also at the heart of the Inverse Scattering transform. In this case, the isolated singularities of the data of the RH problem are well know to play a special role, for example they identify the structurally stable components of the solution. Using the unified transform of Fokas, one can give such formulations also for boundary value problems. The class of linear constant-coefficient PDEs is a special example of integrable PDEs, hence boundary value problems for such PDEs admit a Riemann-Hilbert formulation. I will discuss the important role that the singularities of such RH problems play, in particular their importance in elucidating the spectral structure of linear differential operators in one and two variables. Orthogonal polynomials and integral transforms Ana Loureiro (University of Kent) Abstract. In this talk I will explain how operating with certain integral transforms over polynomial sequences is a useful tool to obtain and deduce properties of one sequence based on the other. A special attention will be given to certain d-orthogonal polynomial sequences, which basically are polynomial sequences satisfying a recurrence relation of order d + 1. When d = 1, we recover the orthogonal case. Examples of some known polynomial sequences with a plank of applications will be used to illustrate the usefulness of the technique. Among the targeted sequences, some semiclassical polynomials will arise, whose weights correspond to certain deformation of the classical weights.