In LQG we work in the spirit of Antonio Machado: "Traveler, there is no path; Paths are made by walking." I will present a bird's eye view of some of the paths that have emerged since Loops 11 and offer a few suggestions.
I try to make the point about what we know and what we do not yet know about the possibility of writing a quantum theory of gravity.
A regime of "polymer quantum field theory on curved spacetime" should emerge in a low energy approximation of quantum gravity based on LQG ideas. This era should be characterized by a polymer scale, and give modifications to the usual semiclassical approximation. I will describe work on gravitational collapse, cosmology, and statistical mechanics in this setting. Results include models of horizon evaporation, inflation and graceful exit without an inflaton potential, and an indication of dimensional reduction from 4 to 2.5 dimensions.
I will describe a very large class of gauge theories that do not use any external structure such as e.g. a spacetime metric in their construction. When the gauge group is taken to be SL(2) these theories describe interacting gravitons, with GR being just a particular member of a whole family of gravity theories. Taking larger gauge groups one obtains gravity coupled to various matter systems. In particular, I will show how gravity together with Yang-Mills gauge fields arise from one and the same diffeomorphism invariant gauge theory Lagrangian. Finally, I will describe what is known about these theories quantum mechanically.
Several lines of evidence hint that quantum gravity at distances a bit larger than the Planck scale may become effectively two-dimensional. I will summarize the evidence for this "spontaneous dimensional reduction," and suggest a further argument based on the effect of vacuum fluctuations on light cones. If this description proves to be correct, it suggests an interesting relationship between small-scale quantum spacetime and the behavior of cosmologies near a spacelike singularity.