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 natural sciencesmathematicspure mathematicstopologysymplectic topologynatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-CoG-2014 - ERC Consolidator Grant Call for proposal ERC-2014-CoG See other projects for this call Funding Scheme ERC-COG - Consolidator Grant Host institution UNIVERSITE LIBRE DE BRUXELLES Net EU contribution € 1 162 880,00 Address AVENUE FRANKLIN ROOSEVELT 50 1050 Bruxelles / Brussel Belgium See on map Region Région de Bruxelles-Capitale/Brussels Hoofdstedelijk Gewest Région de Bruxelles-Capitale/ Brussels Hoofdstedelijk Gewest Arr. de Bruxelles-Capitale/Arr. Brussel-Hoofdstad Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 162 880,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all UNIVERSITE LIBRE DE BRUXELLES Belgium Net EU contribution € 1 162 880,00 Address AVENUE FRANKLIN ROOSEVELT 50 1050 Bruxelles / Brussel See on map Region Région de Bruxelles-Capitale/Brussels Hoofdstedelijk Gewest Région de Bruxelles-Capitale/ Brussels Hoofdstedelijk Gewest Arr. de Bruxelles-Capitale/Arr. Brussel-Hoofdstad Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 162 880,00