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Mathematics of CCC: Mathematical Physics with Positive Lambda Abstracts of Talks

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Mathematics of CCC: Mathematical Physics with Positive Lambda Abstracts of Talks Mathematics of CCC: Mathematical Physics with Positive Lambda Abstracts of Talks On CCC's motivations, equations, observational implications, and future research Roger Penrose A central motivation for CCC is the huge but largely ignored issue raised by the 2nd law of thermodynamics, not adequately addressed by any other current cosmological proposal. CCC's equations are driven by consistency with Einstein's Λ-equations, the reciprocal hypothesis connecting the conformal factors before and after crossover, and certain suggested restrictions aimed at cutting down the remaining conformal-factor freedom. The apparently relatively mild remaining freedom seems to be of a character that relates it to the conjectured "inverse Higgs-type mechanism" whereby rest mass is to die away in the extremely remote future of each aeon. The most immediate observational test of CCC is the predicted family of circular features in the CMB temperature variations. Some detailed aspects of this prediction will be pointed out, not all of which have been explored hitherto. One of the most important outstanding problems of CCC is to estimate the strengths, numbers, clustering, etc., of the effects on our CMB of supermassive black- hole encounters in the previous aeon, and to calculate these effects on our CMB in order to see whether the observed scale invariance, and deviations from it, etc., are obtained. Positive Lambda, the 2nd law and observations Vahe Gurzadyan Positive Lambda and the 2nd law of thermodynamics, the basic properties of the universe are deeply linked to the Conformal Cyclic Cosmology of Penrose. CCC is predicting definite observable features in the cosmic microwave background radiation and we will discuss the relevant evidence of non-Gaussianity in the cosmic microwave background radiation temperature maps. The appearance of the thermodynamical arrow for a quantum system-bath closed system requires both de-correlated initial conditions and no-memory Markovian dynamics as necessary conditions. The emergence of the arrow for the system evolving according to non-unitary dynamics due to the presence of the bath, then, is a result of limited observability, and the arrow in the observable universe as determined by the Lambda sector acting as a bath. The lensing by the inhomogeneities, voids in the large scale matter distribution can also lead to observational consequences. The equations of CCC Paul Tod A good deal is known about the asymptotics of solutions of the Einstein equations with various matter models and a positive cosmological constant. This material should both motivate and constrain proposals for a conformal extension through the future null infinity of one aeon, identified with the big bang of the next aeon. We review this material and possibilities for the extension. Twistors, tractors, and conformally variant operators Mike Eastwood I'll give an overview of the theory of conformally invariant operators, especially in dimension four where local twistors can be used to effect the "curved translation principle." More generally, the tractor bundle and its connection (constructed from LeBrun's conformal-to-Einstein operator) are centre stage and closely related to the conformal boundaries arising in CCC. I'll survey this landscape starting from scratch (partially in preparation for Rod Gover's following talk on the geometry at infinity). Klein, Poincaré, and geometry at infinity Rod Gover Conformal compactification, as originally defined by Penrose, has long been recognised as an effective geometric framework for relating conformal geometry, and associated field theories "at infinity", to the asymptotic phenomena of an interior (pseudo-)Riemannian geometry of one higher dimension. This notion of compactification can be recast in a way that nicely links to certain conformally invariant PDE and, for example, holonomy reductions of the conformal Cartan and tractor bundles. These links yield a host of applications. This point of view also leads naturally to other notions of geometric compactification that can be expected to be similarly useful. For manifolds M with a complete affine connection ∇, I will define a class of compactification based around projective geometry (that is the geodesic path structure of ∇). This applies to pesudo- Riemannian geometry via the Levi-Civita connection. Gravity in twistor space Tim Adamo We will discuss how twistor theory can be used to study general relativity in the presence of a cosmological constant. After a quick review of some essential facts about twistor theory, we will show how conformal gravity—a theory with fourth-order equations of motion—can be formulated in terms of an action functional on twistor space, and then reduced to apply to general relativity. We also describe how this new action can be used for computations of gravitational observables in (anti-)de Sitter space, as well as its relationship to recent results on the classical S-matrix of Einstein (super-)gravity. On the total mass of closed universes with positive cosmological constant Laszlo Szabados The total mass, the Witten type gauge conditions and the spectral properties of the Sen-Witten and the 3-surface twistor operators in closed universes are investigated. It has been proven that a recently suggested expression M for the total mass density of closed universes is vanishing if, and only if, the spacetime is flat with toroidal spatial topology; it coincides with the first eigenvalue of the Sen-Witten operator; and it is vanishing if, and only if, Witten's gauge condition admits a non- trivial solution. In the present talk we extend the total mass above to closed universes with positive cosmological constant. Assuming that the matter fields satisfy the dominant energy condition, it is shown that the cosmological constant provides a sharp lower bound for the total mass density, and that the total mass density takes this as its minimum value if, and only if, the spacetime is locally isometric with the de Sitter spacetime. Indications that gravity is essentially a conformal theory George Ellis This talk will consider the way that use of the Trace Free Einstein Equations (TFE) both solves the vacuum energy problem, and is compatible with inflationary theory in the early universe. It will carry on to consider how this supports the idea that gravity is essentially a conformal interaction, mediated by a spin 2 graviton; this view must necessarily lead to the TFE, despite various derivations that claim to lead to the EFE. From a covariant viewpoint, the Weyl tensor is the essential carrier of gravitational information, with the matter tensor modolating how this happens. Somehow the essential nature of matter is conformal..
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