Global Dynamics, Blow-Up, and Bianchi Cosmology

Abstract

Many central problems in geometry, topology, and mathematical physics lead to questions concerning the long-time dynamics of solutions to ordinary and partial differential equations. Examples range from the Einstein field equations of general relativity to quasilinear reaction-advection-diffusion equations of parabolic type. Specific questions concern the convergence to equilibria, the existence of periodic, homoclinic, and heteroclinic solutions, and the existence and geometric structure of global attractors. On the other hand, many solutions develop singularities in finite time. The singularities have to be analyzed in detail before attempting to extend solutions beyond their singularities, or to understand their geometry in conjunction with globally bounded solutions. In this context we have also aimed at global qualitative descriptions of blow-up and grow-up phenomena.We survey some of our results supported in the framework of the Sonderforschungsbereich 647 "Space -Time -Matter" of the Deutsche Forschungsgemeinschaft.

Bianchi IX cosmologies and the golden ratio

“…The different approaches have utilized both analytic and numerical techniques, but most of these deal with specific rather than general behaviours. More rigorous and more general results, by Ringström [9] and others [10][11][12], have begun to turn beliefs about generic behaviour into facts, particularly with respect to the oscillatory behaviour near the cosmological singularity.…”