Riemann–Hilbert methods in partial differential equations, integrable probability, and random matrix theory

In the 1850´s, Riemann considered the problem of determining an analytic function in a domain of the complex plane from certain relations between the real and imaginary parts of its boundary values. Half a century later, Hilbert analyzed a more general class of boundary value problems for analytic functions.

The class of problems whose solution can be extracted by relating them to a so-called Riemann–Hilbert problem for an analytic function has been increasing ever since Hilbert´s work and has virtually exploded in recent decades.

The purpose of this project is to use Riemann–Hilbert methods in new ways to solve several open problems in partial differential equations, integrable probability, and random matrix theory. More precisely, the project involves the following four objectives:1.

Derive long-time asymptotics for the "bad" Boussinesq equation, which is of fundamental importance in fluid mechanics.2. Develop new tools for the study of integrable evolution equations on the circle.3. Study statistical properties of lozenge tilings of a hexagon.4. Derive asymptotics for structured determinants relevant in random matrix theory.