Numerical solution of Riemann--Hilbert problems: random matrix theory and orthogonal polynomials

In recent developments, a general approach for solving Riemann--Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this numerical algorithm with an approach to compute Fredholm determinants, we are able to calculate level densities and gap statistics for general finite-dimensional unitary ensembles. We also include a description of how to compute the Hastings--McLeod solution of the homogeneous Painlevé II equation.