Riemann Hilbert Problems

Abstract

Percy Deift will give two lectures on Riemann Hilbert Problems (RHP's).

In the first lecture he will define what is meant by a RHP and give many examples from pure and applied mathematics of problems whose solutions can be expressed in terms of a RHP.  Of particular relevance to this Conference, he will describe RHP's for orthogonal polynomials, for Painlev'e equations and for Toeplitz determinants. He will also discuss the class of "integrable operators" introduced by Its, Izergin, Korepin and Slavnov which provides a rich source for RHP's.

In the second lecture he will consider RHP's that depend on external parameters and show use the steepest descent method for RHP's introduced by Zhou and Deift in order to solve these RHP's asymptotically when the parameters become large. An example of the application of the methods described in these lectures, would be the derivation of Plancherel-Rotach type asymptotics for orthogonal polynomials with respect to exponential weights.