The symplectic geometry of anti-self-dual Einstein metrics

Projektinformationen

SymplecticEinstein

ID Finanzhilfevereinbarung: 646649

DOI

10.3030/646649

Projekt abgeschlossen

EK-Unterschriftsdatum 31 März 2015

Startdatum 1 September 2015

Enddatum 31 August 2021

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 162 880,00

EU-Beitrag

€ 1 162 880,00

Koordiniert durch

UNIVERSITE LIBRE DE BRUXELLES
Belgium

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

Berichtszeitraum: 2020-03-01 bis 2021-08-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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