We review recent work on the existence and nature of cosmological singularities that can be formed during the evolution of generic as well as specific cosmological spacetimes in general relativity. We first discuss necessary and sufficient conditions for the existence of geodesically incomplete spacetimes based on a tensorial analysis of the geodesic equations. We then classify the possible sin...

We introduce a general characterization of sudden cosmological singularities and investigate the classical stability of homogeneous and isotropic cosmological solutions of all curvatures containing these singularities to small scalar, vector, and tensor perturbations using gauge invariant perturbation theory. We establish that sudden singularities at which the scale factor, expansion rate, and ...

The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a Q-Gorenstein sing...

The purpose of this paper is to analyze the singularities of a well known benchmark problem “Andrews’ squeezing mechanism”. We show that for physically relevant parameter values this system admits singularities. The method is based on Gröbner bases computations and ideal decomposition. It is algorithmic and can thus be applied to study constraint singularities which arise in more general situat...

By now, many examples of naked singularities in classical general rela-tivity are known. It may however be that a physical principle over and above the general theory prevents the occurrence of such singularities in nature. Assuming the validity of the Weyl curvature hypothesis, we propose that naked singularities are forbidden by the second law of thermodynamics.

We give general mathematical results concerning oscillating singularities and we study examples of functions composed only of oscillating singularities. These functions are deened by explicit coeecients on an orthonormal wavelet basis. We compute their HH older regularity and oscillation at every point and we deduce their spectrum of oscillating singularities.

The minimal degeneration singularities in the affine Grassmannians of simple simply-laced algebraic groups are determined to be either Kleinian singularities of type A, or closures of minimal orbits in nilpotent cones. The singularities for non-simply-laced types are studied by intersection cohomology and equivariant Chow group methods.

This review updates a previous review article [22]. Numerical exploration of the properties of singularities could, in principle, yield detailed understanding of their nature in physically realistic cases. Examples of numerical investigations into the formation of naked singularities, critical behavior in collapse, passage through the Cauchy horizon, chaos of the Mixmaster singularity, and sing...

§ o. Some terminology and notation . 196 198 I. Applications to the birational theory of surfaces · . . .. . .. .. . . . . .. . . . 199 § I. Birational behavior of rational singularities . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . 199 § 2. Resolution of singularities by quadratic transformations and normalization (method of Zariski) . . . . . . . . . . . . . . ...

The analysis of certain singularities in scalar-tensor gravity contained in a recent paper is completed, and situations are pointed out in which these singularities cannot occur. Recently, Abramo et al. [1] have investigated certain geometric singularities occurring in the gravitational sector of scalar-tensor theories described by the action