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Workshop Programme

for period 16 September 2005

Dynamical Systems

16 September 2005

Timetable

Friday 16 September
09:00-10:00 Berger, B (National Science Foundation)
  Numerical studies of expanding $T2$-symmetric cosmologies Sem 1
 

The goal of this talk is to illustrate the value of numerical exploration of Einstein's equations and the synergy with mathematical analysis. The initial discussion will discuss expanding (vacuum) Gowdy spacetimes and the nature of the surprising solutions found by Ringstr\"{o}m. The behavior of the Gowdy solutions will then be compared to that of (vacuum) expanding general $T2$-symmetric spacetimes. In all cases, the behavior is that of gravitational waves of decaying amplitude propagating in an averaged "background" homogeneous cosmology.

 
10:00-11:00 Lim, WC (Dalhousie)
  Dynamical systems approach to inhomogeneous cosmology Sem 1
 

We discuss various aspects of the dynamics of inhomogeneous cosmologies: the approach to the initial singularity, spikes, close-to-FL dynamics and close-to-de Sitter dynamics. In analyzing these phenomena, we use approximation methods of a heuristic nature and numerical simulations.

 
11:30-12:30 Uggla, C (Karlstad)
  What have we learned and what can we learn from spatially homogenous cosmology? Sem 1
 

During the last few decades there has been considerable progress as regards rigorous mathematical results about the dynamics of spatially homogeneous cosmological models. Rather than focusing on particular results the emphasis in this talk will be on the underlying reasons why any results at all have been found. It will be shown that the key reason is associated with a hierarchical structure that arises from space-time symmetry properties, scale transformations, and the state space features these induce.

 
14:30-15:30 Andreasson, H (Chalmers)
  Cosmological solutions of the Einstein-Vlasov system Sem 1
 

I will present some of the global results that have been proved in recent years for the Einstein-Vlasov system in the cosmological case. These results are all obtained under symmetry assumptions and require that spacetime admits an $N$-dimensional symmetry group where $N\geq 2.$ I will mainly focus on the cases $N=2$ and $N=3.$ These situations are rather different since only if $N=2,$ gravitational waves are admitted. The question of global existence is quite well-understood but other important global issues are still open. In the last couple of years studies including a cosmological constant $\Lambda$ or a scalar field have been carried out in cases where $N\geq 3.$ In particular, future geodesic completeness has been proved when $\Lambda>0,$ which is an open problem in the case $\Lambda=0.$

 
16:00-17:00 Andersson, L (A.E.I)
  Future asymptotics of cosmological spacetimes Sem 1
 

I will review some results on the asymptotic behavior of cosmological models in the expanding direction. Among the issues I plan to discuss are nonlinear stability, asymptotic "geometrization", and the influence of matter on asymptotic behavior.

 

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