Project description
Resolution of singularities of foliations and differential forms in arbitrary dimensions
Singularities of geometric objects including equations, foliations and morphisms are points where the object behaves in unexpectedly complex ways and they arise across mathematics and physics. Resolving them – replacing them with similar but perfectly ‘smooth’ objects – has provided deeper understanding of singular points in the case of algebraic equations. Extending this to objects in differential geometry and dynamical systems including foliations and differential forms has remained an open challenge since the 19th century, with only fragmentary results in low dimensions. The ERC-funded DiffeRS project aims to apply methods from differential geometry never used in this context to develop a systematic, high-dimensional resolution theory with far-reaching applications in algebraic and differential geometry.
Objective
A singularity of a geometric object, such as an equation, variety, foliation, morphism, etc, is, loosely speaking, a point with non-trivial local behavior. In geometry, analysis, algebra, physics, among other sciences, their occurrence is unavoidable. Resolution of singularities is one of the most successful techniques to study singular points of an algebraic equation. Arguably, Hironaka's proof of the existence of RS for algebraic varieties over a field of characteristic zero stands as one the greatest achievements of algebraic geometry in the previous century.
Extending the technique of resolution of singularities to objects of interest to differential geometry and dynamical systems, such as foliations, differential forms and metrics, has intrigued mathematicians since the 19th century. Nothing short of spectacular applications are anticipated, including to Riemannian geometry, Lipschitz geometry, sub-Riemannian geometry, global analysis, birational geometry, among others. However, the geometry of foliations involve transcendental phenomena and only low dimensional results are known. This seriously limits the potential for applications.
The proposed research will approach resolution of singularities of foliations and differential forms from a new direction, bringing to bear methods from differential geometry which have not been used before in this context. Toward this end, our key goals will be to develop methods of resolution of singularities that would encompass key examples in Lipschitz and Riemannian geometry; and to combine the acquired insight with newly developed methods in birational geometry, to produce a systematic approach to resolution of singularities of foliations and differential forms in arbitrary dimensions and its applications in algebraic and differential geometry. I believe that my preliminary works in this direction amply demonstrates the feasibility and potential of this approach.
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.
This project's classification has been human-validated.
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.
This project's classification has been human-validated.
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)
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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.
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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.
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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-COG
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75006 Paris
France
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