CORDIS
EU research results

CORDIS

English EN

The symplectic geometry of anti-self-dual Einstein metrics

Project information

Grant agreement ID: 646649

Status

Ongoing project

  • Start date

    1 September 2015

  • End date

    31 August 2021

Funded under:

H2020-EU.1.1.

  • Overall budget:

    € 1 162 880

  • EU contribution

    € 1 162 880

Hosted by:

UNIVERSITE LIBRE DE BRUXELLES

Belgium

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.

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

Project information

Grant agreement ID: 646649

Status

Ongoing project

  • Start date

    1 September 2015

  • End date

    31 August 2021

Funded under:

H2020-EU.1.1.

  • Overall budget:

    € 1 162 880

  • EU contribution

    € 1 162 880

Hosted by:

UNIVERSITE LIBRE DE BRUXELLES

Belgium