After a brief introduction, basic ideas of the quantum Riemannian geometry
underlying loop quantum gravity are summarized. To illustrate physical
ramifications of quantum geometry, the framework is then applied to homogeneous
isotropic cosmology. Quantum geometry effects are shown to replace the big bang
by a big bounce. Thus, quantum physics does not stop at the big-bang
singularity. Rather there is a pre-big-bang branch joined to the current
post-big-bang branch by a `quantum bridge'. Furthermore, thanks to the
background independence of loop quantum gravity, evolution is deterministic
across the bridge.Comment: Minor typos corrected, including a factor of π in the expression
of the critical density. 16 pages, 2 figures. To appear in the Proceedings of
the `Einstein Century' Conference, 15-22 July, Paris, edited by J-M Alimi et
al (American Institute of Physics
