The Outline of Talk:

1. An overview of the Cauchy problem of GR
2. How is the stability of a spacetime typically studied (within the initial value formulation)? Or alternatively, what is meant by a stability analyses in GR?
   • linear stability
     • mode stability
   • non-linear stability
3. Hamiltonian formulation of GR: an overview
4. Classical mechanics via symplectic geometry
5. Stability in GR via symplectic geometry

Roughly, the first lecture and maybe part of the second lecture will suffice to cover points #1 and #2. I will discuss these ideas broadly first, show what is meant by mode stability by working explicitly with the Teukolsky equation (for Kerr), and finally I will pick a solid example for non-linear stability to outline why studying it for arbitrary spacetimes is typically hard. This will be via my personal favourite Lemaitre-Tolman-Bondi dust collapse models. These models are particularly easy to understand since they're analytically soluble and are the perfect setting in which to understand non-linear stability in GR.

Lecture 2, we will continue with the Hamiltonian formulation of GR.

Lecture 3, we will wrap up with the symplectic geometry part. The content of the last lecture will mostly be from http://inspirehep.net/record/1721395 and references therein. Since a symplectic geometric approach to GR doesn't still exist in full, we will discuss what needs to be done still, and what are the likely ways to do it in towards the end of the lecture or in the lecture discussion sessions.