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Evidence from several approaches to quantum gravity hints at the possibility that spacetime undergoes a "spontaneous dimensional reduction" at very short distances.  If this is the case, the small scale universe might be described by a theory with two-dimensional conformal symmetry.  I will summarize the evidence for dimensional reduction and indicate a tentative path towards using this conformal invariance to explore quantum gravity.

In 2-dim it is known that a unitary, well defined quantum field theory, if scale invariant must also be invariant under conformal transformations. Whether this is also true in dimensions higher than two has been an open question for decades. We have discovered renomalization group flows in 4-epsilon dimensions corresponding to scale but not conformal invariant theories. The flows correspond to limit cycles or ergodic behavior,  neither of which had been reported in relativistic quantum field theories either.   There seems to be a deep connection between scale without conformal invariance and this type of renormalization group  behavior. We will present these results and list some of open questions, including the possibility of such behavior in integral dimensions.

I shall describe Relationalism, especially in the Leibniz-Mach-Barbour sense of the word and my variations on that theme. My presentation shall give five extensions to Barbour's work: (more or less) phase space, categorization, subsystems analysis, quantization, and physics as a propositional logic (`questions about physical systems'). I shall also briefly explain how some of Crane and Rovelli's ideas do fit within this scheme, whilst others are at odds with the LMB scheme, leaving one choosing options rather thanjust considering unions.   I shall also present how scale-invariant and scaled relational particle models (the latter originally discovered by Barbour and Bertotti in 1982) can, in dimension 1 and 2, which suffice to toy-model many midisuperspace aspects of GR, be very generally solved at the following levels. 1) configuration space geometry following my fortuitous connection with Kendall's work in the statistical theory of shape involving the self-same space of shapes, and then the cone over this in the scaled case. 2) Conserved quantities and classical equations of motion. 3) Quantum equations of motion and their solutions. 4) Parallels of many Problem of Time strategies. I view this second paragraph as relevant not only by 4) but more widely by how it is a model of quantum background independence (BI), with BI being argued to be the other half to 'relativistic gravitation' in that gestalt entity known as General Relativity.

I will start by showing that gravity, with positive cosmological constant in 2+1 dimensions, can be formulated as a theory of dynamic conformal spatial geometry. Exploiting the isomorphism between the isometry group of de Sitter space in D+1 dimensions and the conformal group in D dimensions, I will reinterpret the Chern--Simons formulation of 2+1 gravity as a gauge theory of a conformal connection. In Cartan's generalization of geometry, this connection represents an evolving spatial geometry locally modeled off the conformal sphere. After a suitable phase space reduction, we obtain shape dynamics. This remodeling explains, in 2+1 dimensions, the remarkable success of the York procedure for solving the initial value problem of general relativity and the uniqueness of the shape dynamics Hamiltonian. I will finish by speculating about possible connections between this work and the general shape dynamics program with holographic renormalization, AdS/CFT, and Horava gravity.

I review the best-matching construction,  and the striking properties of a Jacobi-type action first introduced by Baierelein, Sharp and Wheeler. The simplest theories compatible with such an action principle must have a universal light-cone and gauge symmetry. I also describe the implementation of three-dimensional conformal symmetries on the basis of the BSW action, which gives a first-principles derivation of York's solution of the initial value problem in General Relativity.

Shape Dynamics first arose as a theory of particle interactions formulated without any of Newton's absolute structures. Its fundamental arena is shape space, which is obtained by quotienting Newton's kinematic framework with respect to translations, rotations and dilatations. This leads to a universe defined purely intrinsically in relational terms. It is then postulated that a dynamical history is determined by the specification in shape space of an initial shape and an associated rate of change of shape. There is a very natural way to create a theory that meets such a requirement. It fully implements Mach's principle and shows how time and local inertial frames are determined by the universe as whole. If the same principles are applied to a spatially closed universe in which geometry is dynamical, they lead rather surprisingly to a theory that, modulo some caveats, is dynamically equivalent to general relativity but dual to it in that refoliation invariance is traded for three-dimensional conformal invariance. This shows that there is a hidden three-dimensional conformal symmetry within general relativity. It is in fact what underlies York's crucial method of solution of the initial-value problem in general relativity. It is also remarkable that, as in York's work, shape dynamics inescapably introduces a mathematically distinguished notion of absolute simultaneity, the desirability of which has been found in two currently popular approaches to quantum gravity: causal dynamical triangulations and Horava gravity. I aim to express the key ideas and techniques of shape dynamics as simply as possible.

The tremendous empirical success of Einstein's relativity has pushed Aether into a chapter in the history books, for nearly a century. However, a phenomenologically consistent quantum mechanical treatment of gravity has motivated a revival of aether, by taming its UV-divergences, or the cosmological constant problem. Here I will outline the phenomenological implications of a physically motivated aether model. I will also discuss how aether could potentially evade traditional tests of Lorentz violation, through strong coupling. 

This talk reviews the idea of quantum gravity with anisotropic scaling, and presents a scenario in which this theory of gravity is coupled to matter, described by the standard model or beyond.This "multicritical universe" scenario predicts systematic, energy-dependent, calculable Lorentz-violating corrections to the relativistic dispersion relations of matter.