Dynamics near Space-like Singularities and Quantum Bounces*

The Belinkskii, Khalatnikov and Lifshitz conjecture says that as one approaches space-like singularities in general relativity, ``time derivatives dominate over spatial derivatives'' so that the dynamics at any spatial point is well captured by an ordinary differential equation. This talk will review results from a formulation of this conjecture motivated by Hamiltonian ``connection dynamics'' both in classical general relativity and loop quantum cosmology.