Objective
The goal of this research proposal is to advance the calculus of variations of area in codimension higher than one, specifically existence and regularity of its critical points (minimal submanifolds) and properties of its gradient flow (mean curvature flow). These are central objects in mathematics since three centuries and contributed to the birth of geometric analysis, geometric measure theory, and calculus of variations. Their (non-)existence often reveals deep links between small-scale geometry (curvature) and large-scale structure (topology). While the hypersurface case is by now well understood, with several deep results in the last two decades, very little is known in codimension at least two, especially for unstable submanifolds not minimizing area. Several projects will focus on the intimate link between area and some well-known physical energies: phase transitions are understood to give diffuse approximations of hypersurfaces, while vortices in models of superconductivity relate to codimension two submanifolds. An energy proposed by me and D. Stern in this context is the abelian Higgs model, which I plan to use to extend the Lagrangian mean curvature flow past singularities and to relate stability and regularity of minimal submanifolds, which are two long-standing questions in geometric analysis (among other projects), by exploiting the much richer structure given by the PDEs solved by critical points of this energy. I will also look at candidates in codimension three and higher, inspired by energies from gauge theory and others of Ginzburg–Landau type, relating stability and minimality in critical dimension and attacking other basic open questions. Finally, I will also work on another set of projects exploiting parametrized varifolds, a variational object pioneered by me and T. Rivière combining advantages of the parametrized and intrinsic viewpoints, to study Lagrangian surfaces and minimal submanifolds of higher dimension
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 topology
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics pure mathematics mathematical analysis differential equations partial differential equations
- natural sciences physical sciences electromagnetism and electronics superconductivity
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Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
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)
MAIN PROGRAMME
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Topic(s)
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.
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
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.
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
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2024-STG
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Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
20136 Milano
Italy
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.