SubsidieMeestersLow Regularity Dynamics via Decorated Trees
This project aims to enhance the resolution of singular SPDEs and dispersive PDEs using decorated trees and Hopf algebraic structures, integrating algebraic tools across various fields.
Projectdetails
Introduction
Low regularity dynamics are used for describing various physical and biological phenomena near criticality. The low regularity comes from singular (random) noise or singular (random) initial value.
Examples of Low Regularity Dynamics
The first example is Stochastic Partial Differential Equations (SPDEs) used for describing random growing interfaces (KPZ equation) and the dynamics of the Euclidean quantum field theory (stochastic quantization).
The second concerns dispersive PDEs with random initial data, which can be used for understanding wave turbulence.
Recent Breakthroughs
A recent breakthrough is the resolution of a large class of singular SPDEs through the theory of Regularity Structures invented by Martin Hairer. Such resolution has been possible thanks to the help of decorated trees and their Hopf algebra structures for organizing different renormalization procedures.
Decorated trees are used for expanding solutions of these dynamics.
Project Aim
The aim of this project is to enlarge the scope of resolution given by decorated trees and their Hopf algebraic structures. One of the main ideas is to develop algebraic tools by means of algebraic deformations.
We want to see the Hopf algebras used for SPDEs as deformations of those used in various fields such as numerical analysis and perturbative quantum field theory. This is crucial to work in interaction with these various fields in order to get the best results for singular SPDEs and dispersive PDEs.
Long-Term Objectives
We will focus on the following long-term objectives:
- Give a notion of existence and uniqueness of quasilinear and dispersive SPDEs.
- Derive a general framework for discrete singular SPDEs.
- Develop algebraic structures for singular SPDEs in connection with numerical analysis, perturbative quantum field theory, and rough paths.
- Use decorated trees for dispersive PDEs with random initial data and derive systematically wave kinetic equations in Wave Turbulence.
- Develop a software platform for decorated trees and their Hopf algebraic structures.
Financiële details & Tijdlijn
Financiële details
| Subsidiebedrag | € 1.498.013 |
| Totale projectbegroting | € 1.498.013 |
Tijdlijn
| Startdatum | 1-9-2023 |
| Einddatum | 31-8-2028 |
| Subsidiejaar | 2023 |
Partners & Locaties
Projectpartners
- UNIVERSITE DE LORRAINEpenvoerder
Land(en)
Vergelijkbare projecten binnen European Research Council
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|---|---|---|---|---|
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Stochastic PDEs and Renormalisation
This project aims to advance the study of singular SPDEs by exploring Gibbs measures, developing quasilinear renormalisation, and improving approximation methods for enhanced convergence.
Geometry, Control and Genericity for Partial Differential Equations
This project aims to analyze the impact of geometric inhomogeneities on dispersive PDE solutions and determine the rarity of pathological behaviors using random initial data theories.
Global Estimates for non-linear stochastic PDEs
This project aims to analyze the global behavior of solutions to non-linear stochastic partial differential equations, enhancing understanding of mathematical physics models through advanced PDE techniques.
Generating Unstable Dynamics in dispersive Hamiltonian fluids
This project seeks to rigorously prove the generation of unstable dynamics in water waves and geophysical fluid equations, focusing on energy cascades, orbital instabilities, and rogue wave formation.
Fluctuations in continuum and conservative stochastic partial differential equations
The project aims to analyze conservative stochastic partial differential equations to uncover universal properties and advance mathematical methods in complex dynamical systems influenced by fluctuations.