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Asymptotic analysis of repulsive point processes and integrable equations

The 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 f... ver más

31/12/2029

UCLouvain

2M€

Presupuesto del proyecto: 2M€

Líder del proyecto

UNIVERSITE CATHOLIQUE DE LOUVAIN No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Fecha límite participación Sin fecha límite de participación.

Financiación concedida El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2024-10-10

ERC-2023-STG: ERC STARTING GRANTS Scope:Objectives

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1 Participantes

UCLouvain

1.50M€ | Lider

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Duración del proyecto: 62 meses Fecha Inicio: 2024-10-10
Fecha Fin: 2029-12-31

Líder del proyecto

UNIVERSITE CATHOLIQUE DE LOUVAIN No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 2M€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto The 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 time-periodic boundary conditions. One of the main tools we will use is the Deift-Zhou steepest descent method for Riemann-Hilbert (RH) problems. By solving new problems using this approach, this project will contribute to the development of the method itself. Since the range of applicability of RH methods is very broad, these new techniques are likely to have an impact on a wide spectrum of scientific questions.

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