We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional D4 singularities. We give criteria for these singularities types in terms of differential geometry (Theorem 3.4 and 3.5).

In complex scalar fields, singularities of the phase (optical vortices, wavefront dislocations) are lines in space, or points in the plane, where the wave amplitude vanishes. Phase singularities are illustrated by zeros in edge diffraction and amphidromies in the heights of the tides. In complex vector waves, there are two sorts of polarization singularity. The polarization is purely circular o...

A basic problem in gravitational physics is the resolution of spacetime singularities where general relativity breaks down. The simplest such singularities are conical singularities arising from orbifold identifications of flat space, and the most challenging are spacelike singularities inside black holes (and in cosmology). Topology changing processes also require evolution through classically...

A parallel manipulator is naturally associated with a set of constraint functions defined by its closure constraints. The differential forms arising from these constraint functions completely characterize the geometric properties of the manipulator. In this paper, using the language of differential forms, we provide a thorough geometric study on the various types of singularities of a parallel ...

We consider fourteen kinds of two-dimensional triangle hypersurface singularities, and consider what kinds of combinations of rational double points can appear on small deformation fibers of these triangle singularities. We show that possible combinations can be described by Gabriélov graphs and Dynkin graphs. 1. Review of results by Russian mathematicians In this article we assume that every v...

Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with respect to a given conformal structure. The singularities correspond to the zer...

Chapter 0. Introduction Chapter 1. Basic objects and invariants Chapter 2. Resolution of singularities of monomial basic objects Chapter 3. Key inductive lemma Chapter 4. General basic objects and invariants Chapter 5. Inductive algorithm for resolution of singularities of general basic objects Chapter 6. A more down-to-earth approach to the inductive algorithm Chapter 7. Embedded resolution of...

Abstract. The global structure of 2-dimensional dilaton gravity is studied, attending in particular to black holes and singularities. A gravitational energy is defined and shown to be positive at spatial singularities and negative at temporal singularities. Trapped points are defined, and it is shown that spatial singularities are trapped and temporal singularities are not. Thus a local form of...

In this paper, we study the convergence of Calabi-Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi-Yau manifolds.

In this paper, we prove that singularities of F -injective type are Du Bois. This extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of Frobenius in positive characteristic.