The Early Universe in Loop Quantum Cosmology Martin Bojowald 10 pages, 3 figures, plenary talk at VI Mexican School on Gravitation and Mathematical Physics, Nov 21-27, 2004 http://arxiv.org/abs/gr-qc/0503020 "Loop quantum cosmology applies techniques derived for a background independent quantization of general relativity to cosmological situations and draws conclusions for the very early universe. Direct implications for the singularity problem as well as phenomenology in the context of inflation or bouncing universes result, which will be reviewed here. The discussion focuses on recent new results for structure formation and generalizations of the methods." Bojowald's papers usually worth looking at. Every 6 months to a year, he sums up the current status and recent advances in Loop Quantum Cosmology. his last survey was about one year ago. Here is a sample exerpt from page 5 (mu is the discrete internal time parameter that arises in quantum cosmology): "The evolution dictated by this difference equation in internal time mu does not stop at any finite value of mu. In particular, we can uniquely evolve initial values for the wave function through the classical singularity situated at mu = 0. Thus, there is no singularity where energy densities would diverge or the evolution would stop. This comes about as a consequence of the basic loop properties: the discreteness of spatial geometry leads to finite operators for the inverse volume as well as evolution in discrete internal time. Both properties enter in the demonstration of singularity free evolution. Physically, this means that around the classical singularity continuous space-time and with it the classical theory dissolve. Discrete quantum geometry, on the other hand, still makes sense and allows us to evolve to the other side of the classical singularity."