Exact Results in Quantum Theory

The standard model of cosmology suggests that the universe started with a phase of very high density, described by general relativity as a singularity. Quantum gravity attempts to resolve the singularity by a more fundamental, microscopic theory whose equations remain valid and could tell us what happened at or even before the big bang. This talk reviews one approach, loop quantum gravity, and discusses several surprising new results that shed light on the structure of quantum space-time, with implications for mathematical modelling and conceptual questions in cosmology.