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In the first part of the talk, a brief introduction to general relativity and quantum theory is given. Their independent successes are discussed, as well as the desire and difficulty in merging them, to obtain a unique language to describe the universe. Then I focus on Loop quantum gravity, a particular approach towards this objective, in which a discrete microscopic structure of spacetime is envisaged.

An ingredient in recent discussions of curvature singularity avoidance in quantum gravity is the "inverse scale factor" operator and its generalizations. I describe a general lattice origin of this idea, and show how it applies to the Coulomb singularity in quantum mechanics, and more generally to lattice formulations of quantum gravity. The example also demonstrates that a lattice discretized Schrodinger or Wheeler-DeWitt equation is computationally equivalent to the so called "polymer" quantization derived from loop quantum gravity.

The world at the size of individual atoms obeys very different laws of physics from those we are used to in the everyday world around us. Quantum mechanics rules, allowing atoms to be, in some sense, in more than one place at a time. Researchers all over the world are working to build \"quantum computers\" whose memories manipulate an inherently new type of information, \"quantum information.\" Quantum computers have capabilities impossible for a regular computer, no matter how advanced it may be, including the ability to run new kinds of computer programs capable of rapidly solving certain problems, and to make new highly secure codes for secret communication.