Related papers: Swallowtails and cone-like singularities on maxfac…

We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…

We study the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union…

On Riemann surfaces $M$, there exists a canonical correspondence between a possibly multivalued function $\Psi_X$ whose differential is single valued ($i.e.$ an additively automorphic singular complex analytic function) and a vector field…

We show that a cyclic quotient surface singularity S can be decomposed, in a precise sense, into a number of elementary T-singularities together with a cyclic quotient surface singularity called the residue of S. A normal surface X with…

We construct partitions of $\mathbb{R}^n$ into three sets $\{\mathscr{X}(1),\mathscr{X}(2),\mathscr{X}(3)\}$ that locally minimize interfacial area among compactly supported volume preserving variations and that blow down at infinity to…

The first examples of exceptional terminal singularities are constructed.

We study the question whether the affine semigroup of integer points in a convex cone can be finitely generated up to symmetries of the cone. We establish general properties of finite generation up to symmetry, and then concentrate on the…

In this article we introduce and analyze in detail singular contact structures, with an emphasis on $b^m$-contact structures, which are tangent to a given smooth hypersurface $Z$ and satisfy certain transversality conditions. These singular…

Let $X$ be surface with isolated singularities in the complex projective space $P^3$ and let denote $Y$ the smooth part of $X$. In this note we discuss some aspects of the topology of such quasi-projective surfaces $Y$: the fundamental…

The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category of_controlled sets_, yielding _simplicial controlled sets_, such that one can functorially produce a singular simplicial…

This article introduces the theory of Veronese polytopes, a broad generalisation of cyclic polytopes. These arise as convex hulls of points on curves with one or more connected components, obtained as the image of the rational normal curve…

A sequence of monoidal transformations is defined, in terms of invariants, for a singular hypersurface embedded in a smooth scheme of positive characteristic. Some examples are added to illustrate the improvement of singularities by this…

We generalise the well-known ``embroidery'' envelopes of chords joining points at angles $t$ and $mt$ of a single circle in several ways. Firstly we allow $m$ to be rational (possibly negative) instead of integral, finding formulas for the…

Singularities of the Poynting vector field at resonant light scattering by nanoparticles are discussed and classified. It is shown that there are two generic types of them, namely (i) the singularities related to the vanishing of the…

The purpose of this article is to examine the possible shapes of type I singularities that form in the mean curvature flow of submanifolds of arbitrary codimension, assuming that the initial submanifold satisfies a particular curvature…

Let $X \subset \mathbb{P}^{n+1}$ be a smooth Fano hypersurface of dimension $n$ and degree $d$. The derived category of coherent sheaves on $X$ contains an interesting subcategory called the Kuznetsov component $\mathcal{A}_X$. We show that…

In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in math.AG/0002083. The result can be applied to show the rigidity of all open…

We shall give useful criteria of lips, beaks and swallowtail singularities of smooth map from the plane into the plane. As an application of criteria, we will discuss the singularities of Cauchy problem of single conservation law.

We give details of a new isolated symplectic singularity found in an affine chart in a crepant partial resolution of $\mathbb{C}^4/G_5$, which is 4-dimensional, isolated, and locally simply-connected. We distinguish the new singularity…

We study generalizations of Lorentzian warped products with one-dimensional base of the form $I\times_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an…