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The symplectic geometry of anti-self-dual Einstein metrics

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.

Field of science

  • /natural sciences/mathematics/pure mathematics/mathematical analysis/differential equations/partial differential equations
  • /natural sciences/mathematics/pure mathematics/topology/symplectic topology
  • /natural sciences/mathematics/pure mathematics/geometry

Call for proposal

ERC-2014-CoG
See other projects for this call

Funding Scheme

ERC-COG - Consolidator Grant

Host institution

UNIVERSITE LIBRE DE BRUXELLES
Address
Avenue Franklin Roosevelt 50
1050 Bruxelles
Belgium
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 162 880

Beneficiaries (1)

UNIVERSITE LIBRE DE BRUXELLES
Belgium
EU contribution
€ 1 162 880
Address
Avenue Franklin Roosevelt 50
1050 Bruxelles
Activity type
Higher or Secondary Education Establishments