This course is intended as an introduction to Riemann-Hilbert techniques in Asymptotic analysis. In the past two decades, these techniques have lead to important breakthroughs in the understanding of, for example. orthogonal polynomials, Painlevé transcendents as well as long time behavior of solutions to non-linear PDEs such as the NLS and KdV equations. We start the course with some general theory, but quickly we take a very concrete approach and show how this method works for three special examples: The Strong Szego Limit Theorem for Toeplitz determinants, (asymptotic) properties of solution to the Painleve II equation and asymptotics of orthogonal polynomials. All three examples are motivated by applications to Random Matrix Theory, which we will also briefly discuss.

Lecture notes

• Lecture 1
• Lecture 2
• Lecture 3
• Lecture notes for lectures 4 and 5 were ditributed.
• Lecture 6
• Lecture 7
• Lecture 8
• Lecture 9
• Lecture 10

For more background see:

• P. Deift, Integrable operators.
• P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, vol. 3, New York, American Mathematical Society 2000
• A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, and A. S. Fokas, Painlevé Transcendents: The Riemann Problem Method, RKhD, Moscow (2005); English transl.: A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Yu. Novokshenov Painlevé Transcendents: The Riemann-Hilbert Approach (Math. Surv. Monogr., Vol. 128), Vol. 128, Amer. Math. Soc., Providence, R. I. (2006).
• A. Kuijlaars, Riemann-Hilbert analysis for Orthogonal Polynomials, In: Orthogonal Polynomials and Special Functions, Editors: E. Koelink and W. Van Assche. Lecture notes in Mathematics 1817, Springer.

The examination for this course consists of presentations by students

• April 5
  • Universality of the double scaling limit in random matrix models
    Sergi Arias Garcia and Phillipe Moreillon
  • Pole free solution for the second member in the PI hierarchy
    Samuel From and Julian Mauersberger
• April 12
  • Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight
    Simon Larson and Aron Wenmann
  • Correlations of the characteristic polynomials in the Gaussian Unitary Ensemble or a singular Hankel determinant
    Lukas Schoug and Scott Mason