Bernhard Brehm: Bianchi VIII and IX cosmologies: Everything we know from an ODE perspective

In this series of talks, we will explore the behavior of Bianchi VIII and IX cosmologies towards the initial singularity from an ODE perspective. These Bianchi VIII and IX solutions describe a homogeneous anisotropic universe, that expands from a chaotic initial singularity. This chaotic attractor (in reversed time) consists of heteroclinic chains; naively one would like to describe it as a non-uniformly hyperbolic system.   We will prove that almost every solution approaches the attractor quasi-exponentially and forms particle horizons. We will give strong heuristic arguments that Baire-generic solutions approach the attractor slower than quasi-exponentially, and fail to form horizons, which contradicts Baire-based versions of the BKL picture (maybe we will even see a proof).   We assume no prior knowledge of the Bianchi system, and only elementary knowledge about dynamical systems.    In the first talk, we will take a survey of the Bianchi system, and progress as far as we get towards the proof of a.e. quasi-exponential convergence. Questions and interruptions during the talks are welcome.​