Alfredo Deaño : Recent results and some problems on matrix-valued orthogonal polynomials

Abstract:

 In this talk we present some recent results and work in progress in the theory of matrix orthogonal polynomials (MVOPs). We are interested in algebraic and differential identities for MVOPs and their connection with integrable systems, as well as in the asymptotic analysis of MVOPs as their degree tends to infinity. In the first part, we present the ideas used by Casper and Yakimov in connection with the matrix Bochner problem, and in the second part, we use the Riemann-Hilbert formulation and the Deift-Zhou method of steepest descent. In the matrix case, this asymptotic analysis presents some features that are different from the scalar case and can be of independent interest.