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The hidden geometry of Einstein’s equations – Joël Fine

Toward a new geometric understanding of space-time

Einstein’s equations describe the curvature of space-time. They are essential for a full understanding of the history of our cosmos, but also for practical purposes such as the use of GPS.

Project description in PDF


Born in 1978. PhD in Differential Geometry (Imperial College London) supervised by the fields Medalist Sir Simon Donaldson, Master in Mathematics and placed 3rd in his year(Oxford). Post-doc at the University of Edinburgh. Associate Professor in Mathematics (ULB). Pr. Joël Fine is a specialist in geometry, in particular geometric analysis and the study of Einstein’s equations. He has published in leading mathematical journals such as Duke Mathematical Journal, Journal of Differential Geometry, Journals of the London Mathematical Society, Mathematische Annalen…

Distinguished international invitations include a visiting professorship at the Simons Center for Geometry and Physics (New York) and plenary speaker at the Chern Centennial conference (Mathematical Sciences Research Institute, Berkeley). He is promotor of several research projects, most notably those financed by a prestigious European Research Council Consolidator Grant and an FNRS mandat d’impulsion scientifique.

The project

1 -Einstein’s fundamental equations

Einstein’s equations describe the curvature of time and space. They are of course central to physics, but they also play a critical role in modern mathematics.
The first important question is “on which geometric spaces can we solve Einstein’s equations?” These correspond to possible shapes of the universe.

Despite fantastic advances (most notably Fields Medal winning work of Yau in 1977) we  still know of a very limited number of solutions to these fundamental equations. The second important question is “what do solutions to the equations tell us about the underlying spaces?” This was the theme of Perelman’s recent proof of the Poincaré conjecture, for which he was offered  the Fields Medal in 2006.

Perelman’s work settled many deep problems in three-dimensional geometry. The corresponding questions in dimension four,  the  dimension truly relevant to  Einstein’s theory, remain completely open.

Uncovering the hidden geometry behind Einstein’s equations will potentially open the way to a quantum theory of gravity


My goal is to explore a new, previously hidden, geometry behind Einstein’s equations. My new idea is to rewrite Einstein’s equations a “gauge theory”. In gauge theories, the force which seemingly plays the main role is not the central character. Instead there is a “potential” from which the force is derived and it is the potential that is truly fundamental. In classical physics, the potential cannot be observed directly, only the force can. But in quantum physics, one can see the potential even when there is no force at all (a  phenomenon called the Aharonov-Bohm effect).

Hyperbolic geometry plays a key role in our understanding of 3-dimensional geometry. It is also essential to the discovery by Fine-Panov of the diversity of symplectic Calabi-Yau manifolds, as predicted by string theory

In 2011, I found  a potential for Einstein’s theory. The original “obvious” object can be written in terms of a potential. The potential brings a geometry completely different from the traditional geometry of Einstein’s theory.  This new geometry takes place in a 6-dimensional space, of which 4-dimensional space-time is a shadow. This opens up two exciting possibilities: the hidden 6-dimensional geometry can be used to try and solve Einstein’s equations; it also shows how Einstein’s equations help in understanding four-dimensional spaces, similar to the use of metric geometry in three dimensions, central to Perelman’s work.

This gauge theoretic approach to Einstein’s equations was also discovered independently by Kirill Krasnov. We have since collaborated intensively on our joint theory, greatly benefitting from our complimentary backgrounds. The importance of our research has been recognized by two separate European Research Council grants, mine in mathematics, Krasnov’s in theoretical physics.



The traditional approach to Einstein’s equations is to use metric geometry. Over the past 100 years many results have been proved, but recent progress has been slow as the techniques available have been fully exploited and no new ones discovered.

Gauge theories are used to understand all microscopic phenomenon, such as this proton-proton scattering which occurred at the LHC in Geneva. This project represents one of the first uses of gauge theory to understand the macroscopic nature of gravity and space-time.

The gauge theoretic approach to Einstein’s equations reveals a new geometry, namely symplectic geometry, which we can exploit. In the last thirty years there has been an explosion in the study of symplectic spaces, thanks to the discovery of a whole host of new powerful techniques.

One theme of the project is to exploit these new methods to discover more solutions to Einstein’s equations.

Another theme is to explore this relation in reverse: the metric geometry of Einstein’s equations can be used to prove deep new results about symplectic geometry.

One example of this is the discovery by Dmitri Panov and myself of the existence of a huge number of “symplectic Calabi-Yau manifolds”. The internal consistency of string theory and mirror symmetry predicts the existence of many spaces carrying a special symplectic geometric structure.

Panov and I were the first to discover these spaces. We did so by exploiting the new link between Einstein’s equations in 4-dimensions and symplectic geometry in 6-dimensions. I fully believe that this is the tip of the iceberg, and that many  more important results in symplectic geometry can be proved this way.

Part of the Hopf fibration, a collection of circles filling out three-dimensional space. This is the simplest example of a gauge theory, corresponding to electromagnetic forces on a two-dimensional sphere

The final and most important theme is a combination of the previous two. The two different geometries in play – symplectic and metric – can be combined to create a new collection of techniques going far beyond purely metric or purely symplectic methods. I have made a first step in this direction, showing that symplectic geometry and metric geometry together give limits on the singularities which can occur in space-time (where the curvature becomes infinite, such as at a black hole). There is much more which can be done here: a new fusion of metric and symplectic ideas will vastly improve our understanding of 4 dimensional geometry and has the potential to revolutionise the study of Einstein’s equations.

4 -Some research highlights

A 3-dimensional projection of the skeleton of a 120-cell, the key building block in the work of Fine-Panov, discovering a huge number of symplectic Calabi-Yau manifolds.

Highlights in my track record include the discovery of a vast number of symplectic Calabi-Yau spaces, answering a twenty year old question, inspired by string theory, which had stumped even the Fields Medal winning mathematician Yau himself.

I was also responsible for finding the quantum analogue of the Calabi flow, a special way of deforming space so that the local average of its curvature becomes constant.

This curvature condition is a generalisation of Einstein’s equations and Calabi flow and its quantisation are at the heart of current research in this direction.


Pr. Joël Fine’ research is supported by:

5 -Differential geometry group, ULB mathematics dept

A black hole is an example of a singularity in space-time. The precise nature of singularities is governed by the symplectic geometry, revealed by the gauge theoretic approach.

I am currently the head of a group of six: four postdoctoral researchers and two doctoral students. The international reputation of my work helps me select from the very best mathematicians worldwide.

I received over fifty serious candidates for the last postdoctoral position open in my group, with applicants from universities  such as Harvard, MIT, Stanford, Cambridge, Oxford and Imperial College.

My current team has experts in both metric and symplectic geometry. I am privileged to have one of the bright young stars of Chinese mathematics working as a postdoc in my group, Chengjian Yao. At 19 years old, Yao was fast-tracked into his PhD in the US and he has continued to dazzle ever since.

Unfortunately we are the only team in Belgium working in this area. Because of this, we maintain strong links with the geometry groups in London. London is the European centre of excellence for geometry, and I know many of the people there well from my seven years at Imperial College. We have ongoing collaborations, frequent research visits, and a regular joint seminar.

6 -Requested budget over 4 years

The PhD student will investigate solutions to Einstein’s equations which are asymptotically hyperbolic. These spaces have a special geometry  at infinity. The goal is to understand how this interacts with the newly discovered symplectic geometry in six dimensions.

The postdoc will explore the link between the Gromov- Witten theory in the six-dimensional picture and gauge theory in four dimensions. Gromov-Witten theory is a special way to investigate symplectic spaces, named after the Abel prize winning mathematician Misha Gromov and the Fields Medal winning mathematician and physicist Edward Witten.

The travel budget will be used to hold frequent meetings and research seminars with collaborators and colleagues in London, including Sir Simon Donaldson (Fields medalist), André Neves (winner of the Veblen prize) and Michael Singer (editor of the Journal of the London Mathematical Society). It will also pay for regular meetings with Kirill Krasnov (based in Nottingham, UK) the co-discoveror of the new theory on which this project is based.

Click here to see the full budget

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