LAUSR

rotation would give rise to a more general class of retrodictive solutions of the Einstein equations without a Big Bang and black holes without a singularity. Thus one might imagine that the Friedmann–Lemaitre models, or their generalisations to include thermal radiation, were an artificial, non-generic, special case because of their excessively large amount of symmetry. This mathematically extremely ambitious project (proposed first by Landau in 1959) was most energetically pursued in what was then the Soviet Union. Based on pioneering work on perturbations of cosmological models by E. M. Lifshitz, Lifshitz together with I. M. Khalatnikov produced in the early 1960’s some encouraging results in this direction but these hopes were shattered in 1965 when R. Penrose, using sophisticated techniques from differential geometry and topology, established that spacetime singularities were generic in Gravitational Collapse. This lead to a rapid development of the subject by S. W. Hawking, R. P.Geroch,G. F. R. Ellis and others leading to the consensus (documented in full in the first volume by Hawking and Ellis in the series of Cambridge Monographs on Mathematical Physics of which the present volume is one) that spacetime singularities are generic. However, the sophisticated abstract methods needed to establish these results gave little clue as to this nature of this more general class of solutions. This challenge was taken up by V. A. Belinski and Kahlatnikov whomade the simplifying assumption that near the singularity spatial derivatives of densities, etc. could be ignored and as a consequence Einstein’s Partial Differential equations could be reduced to a finite system of ordinary differential equations. This so-called BKL approximation to the Einstein equations represented an enormous simplification but nevertheless the mathematical difficulties remained considerable. By 1970, they were able to establish, within the BKL scheme, that the generic solution was indeed singular and that the singularity was oscillatory in nature. There then began an intense period of research in which the properties of these homogeneous cosmologies came under intense scrutiny. The original problemwas conceived as one inClassical Cosmology but following the lead of C. W. Misner, thesemodels became regarded as finite dimensional approximations to J. A. Wheeler’s infinite dimensional manifold of initial data for the Einstein equations which he called Superspace. Since Superspace arises by casting the Einstein equations into the form of an infinite dimensional Hamiltonian system, they could, at least at a formal level, be quantised. In the homogenous case, what has come to be called MiniSuperspace, this lead to models for Quantum Cosmology with a finite number of degrees of freedom. At present there is no widespread consensus about the quantum theory of gravity but many believe that, at least at suitably low energies, Fermi-Bose or Super-Symmetry and Super-Gravity have an important role to play. It was natural therefore for M. Henneax, N. Nicolai and T. Damour to enquire whether the deep mathematical structure uncovered by BKL persists in Super-Gravity theories, many of which admit a description in terms of extra dimensions. It appears that they do. The aim of this monograph is to review in detail the almost 60-year-old journey described above. I know of no other review of the subject as ambitious as this one and I believe that it can be recommended to all physicists and mathematicians who wish to gain a commanding overview of the subject. Metaphorically so to speak, to share with Cortez, the experience of starring Silent upon a peak in Darien. This monograph is certainly a worthy successor of the many previous ones in the series and in particular of the first.