Objective
This project is founded on a new formulation of Einstein's equations in dimension 4, which I developed together with my co-authors. This new approach reveals a surprising link between four-dimensional Einstein manifolds and six-dimensional symplectic geometry. My project will exploit this interplay in both directions: using Riemannian geometry to prove results about symplectic manifolds and using symplectic geometry to prove results about Reimannian manifolds.
Our new idea is to rewrite Einstein's equations using the language of gauge theory. The fundamental objects are no longer Riemannian metrics, but instead certain connections over a 4-manifold M. A connection A defines a metric g_A via its curvature, analogous to the relationship between the electromagnetic potential and field in Maxwell's theory. The total volume of (M,g_A) is an action S(A) for the theory, whose critical points give Einstein metrics. At the same time, the connection A also determines a symplectic structure \omega_A on an associated 6-manifold Z which fibres over M.
My project has two main goals. The first is to classify the symplectic manifolds which arise this way. Classification of general symplectic 6-manifolds is beyond current techniques of symplectic geometry, making my aims here very ambitious. My second goal is to provide an existence theory both for anti-self-dual Poincaré--Einstein metrics and for minimal surfaces in such manifolds. Again, my aims here go decisively beyond the state of the art. In all of these situations, a fundamental problem is the formation of singularities in degenerating families. What makes new progress possible is the fresh input coming from the symplectic manifold Z. I will combine this with techniques from Riemannian geometry and gauge theory to control the singularities which can occur.
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: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
- natural sciences mathematics pure mathematics topology symplectic topology
- natural sciences mathematics pure mathematics geometry
- 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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H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC)
MAIN PROGRAMME
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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
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.
ERC-COG - Consolidator Grant
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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-2014-CoG
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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.
1050 Bruxelles / Brussel
Belgium
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