The symplectic geometry of anti-self-dual Einstein metrics

Información del proyecto

SymplecticEinstein

Identificador del acuerdo de subvención: 646649

DOI

10.3030/646649

Proyecto cerrado

Fecha de la firma de la CE 31 Marzo 2015

Fecha de inicio 1 Septiembre 2015

Fecha de finalización 31 Agosto 2021

Financiado con arreglo a

• EXCELLENT SCIENCE - European Research Council (ERC)

Coste total

€ 1 162 880,00

Aportación de la UE

€ 1 162 880,00

Coordinado por

UNIVERSITE LIBRE DE BRUXELLES
Belgium

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

Período documentado: 2020-03-01 hasta 2021-08-31

Resumen del contexto y de los objetivos generales del proyecto

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.

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

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.

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

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