Title: Asymptotic analysis of repulsive point processes and integrable equations
Abstract: he purpose of this project is to apply and develop robust mathematical methods for solving asymptotic problems on repulsive point processes and partial differential equations. The point processes considered will mostly be taken from the theory of random matrices, such as the eigenvalues of random normal matrices, but we will also consider discrete point processes with a more combinatorial structure, such as lozenge tilings of a hexagon.
These models are used in neural networks, multivariate statistics, nuclear physics and number theory, and therefore have been widely discussed in the physics and mathematics literature. We will investigate asymptotic properties of such processes as the number of points (or eigenvalues, or lozenges) gets large.
All the point processes considered have in common an interesting feature: they are repulsive, in the sense that neighbouring points repel each other. However, in other aspects these processes are very different from one another, and some of them require completely novel techniques. For example, the tiling models are related to non-Hermitian matrix-valued orthogonal polynomials and two-dimensional point processes are out of reach of standard methods when the rotation-invariance is broken.
An important part of the project is to develop novel techniques to analyse these point processes.
The last part of the project focuses on integrable partial differential equations. The objective is to develop a new approach for solving long-standing problems with periodic boundary conditions.