Project description
Advancing understanding of geometric measure theory
Supported by the Marie Skłodowska-Curie Actions programme, the DireStratS project seeks advance understanding of geometric measure theory, addressing problems in both Euclidean and metric spaces. Its first objective focuses on rectifiability and stratification of singular sets, aiming to provide a unifying approach to several known differentiability results, while deliberately attacking others. The second objective centres on Marstrand’s density theory in normed spaces, aiming to extend it to arbitrary normed spaces for deeper understanding of related questions in metric spaces. The final objective addresses minimality of flats among integral current spaces with regard to a generalisation of the fact that a disk is area-minimising among all surfaces with the same boundary.
Objective
DireStratS (Dimension-reduction, Stratification and Slicing in Geometric Measure Theory) is grounded in the area of Geometric Measure Theory (GMT) and involves problems in both Euclidean and metric spaces. I plan to look at the following 3 main objectives:
O1. Rectifiability and stratification of singular sets
O2. Marstrand’s density theorem in normed spaces
O3. Minimality of flats among integral current spaces
The fundamental connection between the objectives is in the methodologies proposed to attack them, namely dimension-reduction, stratification and slicing. Advances in these problems can have an impact beyond the GMT community, especially concerning the structure of singularities, that are ubiquitous in minimal surfaces, material science, free boundary problems, and also hyperbolic conservation laws.
O1 is concerned with the structure of singularities of the set of differentiability of functions, and aims at giving a unifying approach to several known differentiability results, with the intention of attacking also some that are still open. O2 aims at extending Marstrand's density theorem to arbitrary normed spaces, paving the way to a better understanding of related questions (such as Preiss' rectifiability theorem) in metric spaces. O3 is concerned with a generalization (to integral current spaces and an anisotropic setting) of the fact that a disk is area-minimizing among all surfaces with the same boundary.
During the project I will give lectures and hold reading seminars on stratification and the theory of currents, expanding on the relation with the transport equation. I will attend courses offered both by UNIPI and SNS, and the interaction with the communities from both institutions will be fundamental. The fellowship would be a big opportunity to be inserted in the Pisan GMT community, to which I am currently only tangentially related, and to bring new approaches developed from my experience in Warwick and Leipzig.
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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HORIZON.1.2 - Marie Skłodowska-Curie Actions (MSCA)
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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-TMA-MSCA-PF-EF - HORIZON TMA MSCA Postdoctoral Fellowships - European Fellowships
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Call for proposal
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(opens in new window) HORIZON-MSCA-2025-PF
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56126 Pisa
Italy
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