SymplecticEinstein
Financiado
The symplectic geometry of anti self dual Einstein metrics
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-di...
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.
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Duración del proyecto: 77 meses
Fecha Inicio: 2015-03-31
Fecha Fin: 2021-08-31
Fecha Fin: 2021-08-31
Línea de financiación: concedida
El organismo H2020 notifico la concesión del proyecto el día 2021-08-31
Línea de financiación objetivo
El proyecto se financió a través de la siguiente ayuda:
ERC-CoG-2014:
ERC Consolidator Grant
Cerrada
hace 10 años
Presupuesto
El presupuesto total del proyecto asciende a
1M€
Líder del proyecto
UNIVERSITE LIBRE DE BRUXELLES
No se ha especificado una descripción o un objeto social para esta compañía.
Perfil tecnológico
TRL
4-5
Perfil tecnológico
TRL
4-5
382.13K€ |
Participante