Course overview: The goal of the course is to familiarize the students with the description of the asymptotic of orthogonal polynomials that employs the so-called “Riemann Hilbert Method”. The method was introduced in the late nineties and saw the main application in the first proof of “universality” results in Random Matrix Theory. In the literature, the specific technique is variably referred to as “nonlinear steepest descent method” or “Deift-Zhou method”.

A particular appeal of the method is that it applies to a wider range of problems which include asymptotic studies for nonlinear integrable wave equations like the Korteweg-deVries equation or the nonlinear Schrödinger equation, used in oceanography and nonlinear optics, respectively.

The goal of the course is, however, focused on the case of orthogonal polynomials, or rather denominators of Padé approximations (namely the so-called “non-hermitean” orthogonality). Of special interest are the cases of “semiclassical” orthogonal polynomials introduced by Shohat, Maroni, Marcellán, Rocha, and their asymptotics. The method combines input from potential theory as well as analysis, and is able to provide “strong-asymptotic results” for orthogonal polynomials, in principle allowing for a complete asymptotic expansion for large degree valid point-wise in the plane.

Time permitting, we will touch upon applications of the method beyond orthogonal polynomials and to the newly introduced notion of Padé approximations on higher genus Riemann surfaces.

A rough breakdown of the course is as follows:

• Padé approximations and Orthogonal Polynomials;
• Generalities about Riemann–Hilbert problems (RHPs); formulation of Padé approximation problems in terms of RHPs.
• Large degree asymptotic analysis:
• Elements of potential theory and the “g-function” mechanism;
• Reduction of the RHP to a “small–norm” problem;
• Construction of “parametrices”; approximate solutions;
• Some aspects of geometry of Riemann surfaces needed for the construction of parametrices;
• Padé approximants on Riemann surfaces (time permitting)
• Other applications of the method and interesting problems (time permitting)

The method is seeing continued developments and new applications to this day, with an ever growing literature; the first instances where the method was developed are:

– P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Bull. Amer. Math. Soc. (N.S.), 26(1):119–123, 1992.
– P. Deift, “Orthogonal Polynomials and Random Matrices: a Riemann–Hilbert approach”; Courant Institute Lectures, (’98);
– P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math., 52(11):1335–1425, 1999.