Beijing Institute of Mathematical Sciences and Applications

Orthogonal polynomials provide a fundamental framework connecting random matrix theory, spectral analysis, and integrable systems. This course introduces the analytic structures of orthogonal polynomials and explores their applications in modern mathematical physics.

We begin with the basic theory of orthogonal polynomials, including recurrence relations, kernel functions, and zero distributions. The connection between orthogonal polynomials and random matrix ensembles will be discussed through logarithmic potential theory and equilibrium measures, which describe the limiting distribution of eigenvalues.

The course then introduces the Riemann–Hilbert formulation of orthogonal polynomials and its role in deriving asymptotic results through the nonlinear steepest descent method. Further connections with special functions, Painlevé equations, and integrable hierarchies will also be explored.

Applications to random matrices, universality phenomena, critical asymptotic behavior, and integrable probability will be discussed.