The symplectic geometry of anti-self-dual Einstein metrics

Informations projet

SymplecticEinstein

N° de convention de subvention: 646649

DOI

10.3030/646649

Projet clôturé

Date de signature de la CE 31 Mars 2015

Date de début 1 Septembre 2015

Date de fin 31 Août 2021

Financé au titre de

• EXCELLENT SCIENCE - European Research Council (ERC)

Coût total

€ 1 162 880,00

Contribution de l’UE

€ 1 162 880,00

Coordonné par

UNIVERSITE LIBRE DE BRUXELLES
Belgium

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

Période du rapport: 2020-03-01 au 2021-08-31

Résumé du contexte et des objectifs généraux du projet

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.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

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.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

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