Abstract
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.