I aim to connect various non-perturbative approaches to quantum gravity, which often involve some form of discrete structures, to more conventional physical theories based on general relativity. At the moment I am focussing on the derivation of classical cosmology from fundamental quantum gravity, various models of quantum cosmology, and the role of Lorentz invariance in canonical gravity.
The most natural place to look for testable predictions from quantum gravity is in cosmology. A major first step towards making the connection to cosmology is to be able to describe quantum gravity states that describe macroscopic, homogeneous or approximately homogeneous universes like the one around us, and to be able to do calculations within quantum gravity to derive the effective dynamics of such states. With Daniele Oriti and Lorenzo Sindoni, I have recently managed to complete these steps in the group field theory approach to quantum gravity. We show that certain states whose structure is analogous to condensate states that appear in Bose-Einstein condensation describe macroscopic spatially homogeneous geometries, and that in a semiclassical regime and in the isotropic case, the dynamics they satisfy is given precisely by the Friedmann equation for pure GR. Extensions of the formalism to matter fields and inhomogeneities are currently investigated.
Another recent interest of mine has been a geometric study of `spontaneous' symmetry breaking in gravity, using Cartan geometry. With Derek Wise, I have given a reformulation of Ashtekar variables where local Lorentz symmetry is broken to a rotational subgroup through local observers that specify a local notion of `time'. We then take this picture further by defining general relativity on `observer space' which include all such local observers simultaneously. In general relativity this space is just a direct product of spacetime with the space of unit timelike vectors at each point, but if proposals such as relative locality point in the right direction it could be a more general space. Observer space provides a very general geometric framework for quantum gravity phenomenology, such as in the study of Lorentz-violating theories, or in Finsler geometry.