  The spatially homogeneous, isotropic Standard Cosmological Model appears to
describe our Universe reasonably well. However, Einstein's equations allow a
much larger class of cosmological solutions. Theorems originally due to Penrose
and Hawking predict that all such models (assuming reasonable matter
properties) will have an initial singularity. The nature of this singularity in
generic cosmologies remains a major open question in general relativity.
Spatially homogeneous but possibly anisotropic cosmologies have two types of
singularities: (1) velocity dominated---(reversing the time direction) the
universe evolves to the singularity with fixed anisotropic collapse rates ; (2)
Mixmaster---the anisotropic collapse rates change in a deterministically
chaotic way. Much less is known about spatially inhomogeneous universes.
Belinskii, Khalatnikov, and Lifshitz (BKL) claimed long ago that a generic
universe would evolve toward the singularity as a different Mixmaster universe
at each spatial point. We shall report on the results of a program to test the
BKL conjecture numerically. Results include a new algorithm to evolve
homogeneous Mixmaster models, demonstration of velocity dominance and
understanding of evolution toward velocity dominance in the plane symmetric
Gowdy universes (spatial dependence in one direction), demonstration of
velocity dominance in polarized U(1) symmetric cosmologies (spatial dependence
in two directions), and exploration of departures from velocity dominance in
generic U(1) universes.
