# Riemann--Hilbert Problems

## TL;DR

This paper introduces the nonlinear steepest descent method for Riemann-Hilbert problems, demonstrating its applications in asymptotic analysis, integrable systems, and random matrix theory.

## Contribution

It provides an accessible introduction to the method and showcases its diverse applications in mathematical physics and analysis.

## Key findings

- Asymptotic analysis of special functions using Riemann-Hilbert problems
- Application to inverse scattering for integrable systems
- Proof of universality in random matrix ensembles

## Abstract

These lectures introduce the method of nonlinear steepest descent for Riemann-Hilbert problems. This method finds use in studying asymptotics associated to a variety of special functions such as the Painlev\'{e} equations and orthogonal polynomials, in solving the inverse scattering problem for certain integrable systems, and in proving universality for certain classes of random matrix ensembles. These lectures highlight a few such applications.