Project description
Answering questions of PDEs and dispersive equations
Differential equations are fundamental to many fields, enabling critical calculations and insights. The study of partial differential equations (PDEs), particularly dispersive equations, has seen increasing focus and interest due to their potential in advancing fields such as quantum field theory. Unfortunately, despite recent key breakthroughs in the theory of singular stochastic parabolic PDEs, many key questions of dispersive equations remain unanswered. The ERC-funded CritPDEsRand project aims to develop mathematical tools and concepts to explore these fundamental problems. Specifically, the project will use advances in probability and the interface of the two fields to deduce critical insights and overcome previous challenges.
Objective
"This proposal is concerned with the study of the dynamics of partial differential equations (PDEs), broadly interpreted, in the presence of randomness, with a particular focus on dispersive equations. This is a young but promising emerging field, and it has deep connections with the more established field of constructive quantum field theory.
In recent years, we have witnessed outstanding advances in the theory of singular stochastic parabolic PDEs, and while several breakthroughs have been obtained for the dispersive counterpart, many fundamental questions are still open. In particular, many of these questions are ""critical"" or ""supercritical"" according to our current understanding. The main goal of this proposal is to develop novel mathematical ideas and tools to break this barrier of criticality, and provide a resolution to these fundamental problems.
Over the last ten years, there has been significant progress at the interface of dispersive PDEs and probability. In my short career,
I have been one of the leading figures of this field and I have achieved significant breakthroughs. Particularly, my works on phase transitions for focusing Gibbs measures, ergodicity results for stochastic wave equations and global well posedness result for fractional NLS in negative regularity have opened the door to new exciting possibilities.
In this proposal,
In this proposal,
1. I will work on the Φ^p_d quantum field theories on R^d, both in the focusing and defocusing regime. This will answer major open questions related to the soliton resolution conjecture and in constructive quantum field theory.
2. I will improve our understanding of how Gaussian measures are transported by the flow of nonlinear PDEs.
3. I will develop a theory of (regular) Lagrangian flows for dispersive PDEs, and use it to break the barrier of criticality for these equations."
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics mathematical analysis differential equations partial differential equations
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Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2025-STG
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EH8 9YL Edinburgh
United Kingdom
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