Riemann-Hilbert methods in asymptotic analysis

In the past two decades new developments on Riemann-Hilbert problems have lead to important breakthroughs in the analysis of solutions to differential equations such as Painlevé transcendents and for the asymptotic analysis of orthogonal polynomials. An important driving force behind these developments came from random matrix theory, where new tools were needed to analyze the behavior of spectra of large random matrices and matrix integrals. They have also appeared in the study of certain partial differential equations, such as the small-dispersion limit of the Korteweg-De-Vries equation. In the course we will start with the basic theory of Riemann-Hilbert problems and then discuss the Deift/Zhou steepest descent method for both Painlevé transcendents and orthogonal polynomials. After the course, the student should have sufficient skills to independently and efficiently read research papers on the topic.

From the content:

• General theory of Riemann-Hilbert problems
• Monodromy of differential equations and the isomonodromy approach to Painlevé transcendents
• Case study of the Painlevé II equation
• Orthogonal polynomials and discrete Painlevé equations
• Deift/Zhou steepest descent for the Riemann-Hilbert problem for orthogonal polynomials
• Double scaling limits: from discrete to continuous Painlevé.

Recommended reading:

1. A.S. Fokas, A. Its, A.A. Kapaev and V.Y. Novokshenov, Painlevé Transcendents. The Riemann-Hilbert approach. Mathematical Surveys and Monographs, 128. American Mathematical Society, Providence, RI, 2006. xii+553 pp.
2. 2 P.A. Deift,Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. viii+273 pp.

Examination is based on oral presentations.