The symplectic geometry of anti-self-dual Einstein metrics

Project Information

SymplecticEinstein

Grant agreement ID: 646649

DOI

10.3030/646649

Project closed

EC signature date 31 March 2015

Start date 1 September 2015

End date 31 August 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 162 880,00

EU contribution

€ 1 162 880,00

Coordinated by

UNIVERSITE LIBRE DE BRUXELLES
Belgium

Periodic Reporting for period 4 - SymplecticEinstein (The symplectic geometry of anti-self-dual Einstein metrics)

Reporting period: 2020-03-01 to 2021-08-31

Summary of the context and overall objectives of the project

The project "SymplecticEinstein" is founded on a new formulation of Einstein’s equations in dimension 4. This new approach reveals a surprising link between four-dimensional Einstein manifolds and six-dimensional symplectic geometry. The 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.

Solutions to Einstein's field equations are possible models for the universe. Finding solutions is as difficult as it is important. The aim of this project is to exploit the hidden symplectic geometry of these equations which I recently discovered together with co-authors. This confluence of two different geometries makes many new potential techniques available to either side. The overall objective is to both find new solutions to Einstein's equations and better understand the solutions we already have. Moreover, I will use techniques from the study of Einstein metrics to explore the symplectic manifolds which arise this way.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

During this project we have found new examples of solutions to Einstein's equations. They have interesting geometric properties (negative curvature). We have also discovered a new and surprising link bewteen the topology of knots in three-dimensional space and the geometry of minimal surfaces in four-dimensional hyperbolic space.

Our results have been diseminated by publication in scienfitc journals, as is usual for pure mathematics. I have also recorded a short video (in French) explaining some of the ideas behind the project to a general audience. You can watch this on Facebook, or on the webpage of the Fondation ULB: https://fondationulb.be/fr/geometrie.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The two biggest breakthroughs are those mentioned above. The new examples of negatively curved solutions to Einstein's equations answers a question that is at least 40 years old. The unexpected link between the topology of knots and the geometry of minimal surfaces opens new avenues of research which will occupy many researchers in the years to come.

Working on the G2 Laplacian flow applied to hypersymplectic structures