Conference

We present a new model of inflation which does not rely on fundamental scalar fields.  The theory is a conformally invariant gauge field theory minimally coupled to massless fermions.  At the beginning of inflation, the conformal symmetry is dynamically broken by a BCS condensation of the fermions, leading to a spontaneously violation of conformal symmetry.  A quasi de-Sitter inflationary regime is driven by the interaction between a homogenous plasma dynamo between the gauge, gravitational and condensate field.  Unique observational consequences of this model are twofold : (1) The sourcing of B-mode vorticity fluctuations in upcoming CMB powerspectra. (2) A possible inflationary Baryogenesis mechanism connected to the initial conditions of inflation.

I show how the local Lorentz and/or diffeomorphism invariances may be broken by a varying c, softly or harshly, depending on taste. Regardless of the fundamental implications of such dramas, these  symmetry breakings may be of great practical use in cosmology. They may solve the horizon and flatness problems. A near scale-invariant spectrum of fluctuation may arise, even without inflation. Distinct imprints may be left, teaching us an important lesson: our foundations may be flimsier than we like to think.

New techniques for obtaining the complete set of analytic solutions of the usual cosmological equations continue to shed new light on various aspects of cosmology. This approach, which was developed with a locally Weyl invariant formulation of gravity in 3+1 dimensions, was inspired by the 2T-physics formulation of all physics in 4+2 dimensions. I will first give a review of the general aspects of the analytic cosmological solutions, including recent work with Shi-Hung Chen, Paul Steinhardt and Neil Turok, on geodesic completeness through the singularity and the nature of the Big Crunch/Big Bang transition. Then I will include the full Standard Model, and discuss how the instability of the Higgs potential and the conditions for the electroweak phase transition develop slowly during cosmological evolution of the universe, thus understanding that the electroweak phase transition is not "spontaneous" and the Standard Model is not decoupled from gravity or cosmology. This scenario, which is currently being developed further, is the natural cosmological consequence of coupling the Standard Model to Gravity by imposing Weyl symmetry, a feature that is required if 2T-physics in 4+2 dimensions is considered as the starting point. There are no dimensionful parameters, not even the Newton constant. Emergent dimensionful scales are dynamical during the evolution of the universe.

The Conformal Method (as well as the closely related Conformal Thin Sandwich Method) has proven to be a very useful procedure both for constructing and for parametrizing solutions of the Einstein initial data constraint equations, for initial data sets with constant mean curvature (CMC). Is this true for non CMC data sets as well? After reviewing the CMC results, we discuss what we know and don't know about non CMC initial data sets and the effectiveness of the Conformal Method in handling them.

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.