Mbour Lectures on Shape Dynamics

This is the course website of a voluntary lecture series on Shape Dynamics given by Lucas Hackl, tutor at AIMS Sénégal.


Shape dynamics is a reformulation of General Relativity and a new approach to Quantum Gravity. It uses methods of Riemannien geometry, symplectic geometry, Dirac analysis of constrained systems, fiber bundles and lie algebras. I will present the definition of shape dynamics using the methods of a linking theory described by H. Gomes und T. Koslowski in arXiv:1101.5974.

Lecture Series

We will meet every Sunday at 11am in the classroom of the African Institute for Mathematical Sciences in Sénégal. Usually, I will give a lecture of about one hour, while using the remaining time until lunch for discussion, examples and explicit calculations in a tutorial style.
IMPORTANT: I moved the second lecture to Sunday 25/11/2012 at 2:30 pm in the morning.

Lecture Notes

I decided to write lecture notes for the lecture series. The notes are work in progress, meaning that I add and improve the content as the course proceeds. Therefore, I am also very happy to get feedback from you and if you have good suggestions I will acknowledge you in the notes. You will always find an up-to-date version here:


The following outline is a first draft. I am happy to receive feedback and can adjust the structure of the course depending on your interests and backgrounds. Let me know!
  1. Basics of Differential Geometry
    1. Manifolds
    2. Tensors
    3. Derivatives
  2. Symplectic Geometry
    1. Symplectic Manifolds
    2. Hamiltonian Vector Field
    3. Poisson Brackets
  3. Constrained Systems
    1. Dirac Procedure
    2. Classification of Constraints
    3. Gauge Symmetry
  4. Linking Theory
    1. General Linking Theory
    2. Phase Space Reduction
    3. Construction Principle
  5. Pseudo-Riemannian Geometry and General Relativity
    1. Connection
    2. Curvature Tensor
    3. Einstein's equations
  6. General Relativity and Shape Dynamics
    1. ADM Phase Space
    2. York Method
    3. Compact Linking Theory
    4. Symmetry Trading


Depending on your interests and enthusiasm I may be able to provide some kind of evaluation: If you do well, you may want to use this for applications and I will happily confirm your performance.