Abstract:
Recent developments concerning oscillatory spacelike singularities in general relativity are taking place on two fronts. The first treats generic singularities in spatially homogeneous cosmology, most notably Bianchi types VIII and IX. The second deals with generic oscillatory singularities in inhomogeneous cosmologies, especially those with two commuting spacelike Killing vectors. This paper describes recent progress in these two areas: in the spatially homogeneous case focus is on mathematically rigorous results, while analytical and numerical results concerning generic behavior and so-called recurring spike formation are the main topic in the inhomogeneous case. Unifying themes are connections between asymptotic behavior, hierarchical structures, and solution generating techniques, which provide hints for a link between the nature of generic singularities and a hierarchy of hidden asymptotic symmetries.

Abstract:
We give a detailed discussion of the use of the (3+1) decomposition and of Bardeen's variables in massive gravity linearized over a Minkowski as well as over a de Sitter background. In Minkowski space the Bardeen "potential" \Phi, that in the massless case is a non-radiative degree of freedom, becomes radiative and describes the helicity-0 component of the massive graviton. Its dynamics is governed by a simple Klein-Gordon action, supplemented by a term (\Box \Phi)^2 if we do not make the Fierz-Pauli tuning of the mass term. In de Sitter the identification of the variable that describes the radiative degree of freedom in the scalar sector is more subtle, and even involves expressions non-local in time. The use of this new variable provides a simple and transparent derivation of the Higuchi bound and of the disappearance of the scalar degree of freedom at a special value of $m_g^2/H^2$. The use of this formalism also allows us to uncover the existence of a hidden gauge symmetry of the massive theory, that becomes manifest only once the non-dynamical components of the metric are integrated out, and that is present both in Minkowski and in de Sitter.

Abstract:
An interpretation of spacelike singularities in string theory uses target space duality to relate the collapsing Schwarzschild geometry near the singularity to an inflationary cosmology in dual variables. An appealing picture thus results whereby gravitational collapse seeds the formation of a new universe.

Abstract:
Time dependent orbifolds with spacelike or null singularities have recently been studied as simple models of cosmological singularities. We show that their apparent simplicity is an illusion: the introduction of a single particle causes the spacetime to collapse to a strong curvature singularity (a Big Crunch), even in regions arbitrarily far from the particle.

Abstract:
Three explicit families of spacelike Zoll surface admitting a Killing field are provided. It allows to prove the existence of spacelike Zoll surface not smoothly conformal to a cover of de Sitter space as well as the existence of Lorentzian M\"obius strips of non constant curvature all of whose spacelike geodesics are closed. Further the conformality problem for spacelike Zoll cylinders is studied.

Abstract:
Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

Abstract:
We present the conformally 1+3 Hubble-normalized field equations together with the general total source equations, and then specialize to a source that consists of perfect fluids with general barotropic equations of state. Motivating, formulating, and assuming certain conjectures, we derive results about how the properties of fluids (equations of state, momenta, angular momenta) and generic spacelike singularities affect each other.

Abstract:
This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have symmetries with hidden extensions, then we further extend these to hidden symmetries of some hyperbolic link complements.

Abstract:
The study of hidden symmetries within Dirac's formalism does not possess a systematic procedure due to the lack of first-class constraints to act as symmetry generators. On the other hand, in the Faddeev-Jackiw approach, gauge and reparametrization symmetries are generated by the null eigenvectors of the sympletic matrix and not by constraints, suggesting the possibility of dealing systematically with hidden symmetries throughout this formalism. It is shown in this paper that indeed hidden symmetries of noninvariant or gauge fixed systems are equally well described by null eigenvectors of the sympletic matrix, just as the explicit invariances. The Faddeev-Jackiw approach therefore provides a systematic algorithm for treating all sorts of symmetries in an unified way. This technique is illustrated here by the SL(2,R) affine Lie algebra of the 2-D induced gravity proposed by Polyakov, which is a hidden symmetry in the canonical approach of constrained systems via Dirac's method, after conformal and reparametrization invariances have been fixed.

Abstract:
We develop an extension of Bohmian mechanics by defining Bohm-like trajectories for quantum particles in a curved background space-time containing a spacelike singularity. As an example of such a metric we use the Schwarzschild metric, which contains two spacelike singularities, one in the past and one in the future. Since the particle world lines are everywhere timelike or lightlike, particles can be annihilated but not created at a future spacelike singularity, and created but not annihilated at a past spacelike singularity. It is argued that in the presence of future (past) spacelike singularities, there is a unique natural Bohm-like evolution law directed to the future (past). This law differs from the one in non-singular space-times mainly in two ways: it involves Fock space since the particle number is not conserved, and the wave function is replaced by a density matrix. In particular, we determine the evolution equation for the density matrix, a pure-to-mixed evolution equation of a quasi-Lindblad form. We have to leave open whether a curvature cut-off needs to be introduced for this equation to be well defined.