Related papers: Uniqueness properties for spherical varieties

The paper is devoted to metric properties of singularities. We investigate the relations among topology, metric properties and smoothness. In particular, we present some higher dimensional analogous of Mumford's theorem on smoothness of…

We review the basic definitions and properties concerning smooth structures, convenient spaces, diffeological spaces and tangent structures. The relation betwen them is described. A tangent structure is constructed for each pre-convenient…

A class $\mathfrak{M}$ of coarse spaces is called a variety if $\mathfrak{M}$ is closed under formation of subspaces, coarse images and products. We classify the varieties of coarse spaces and, in particular, show that if a variety…

Diffeology extends differential geometry to spaces beyond smooth manifolds. This paper explores diffeology's key features and illustrates its utility with examples including singular and quotient spaces, and applications in symplectic…

We study over a number field, the iterates of automorphisms of the affine space. More precisely, we are interested in the periodic and non-periodic points; for the former the questions are similar to the ones about torsion points on abelian…

If a real value invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension on the manifold, then the invariant is completely determined by Euler characteristics of the…

A singular foliation is a partition of a manifold into leaves of perhaps varying dimension. Stefan and Sussmann carried out fundamental work on singular foliations in the 1970s. We survey their contributions, show how diffeological objects…

We characterize singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric…

We study curve singularities in a smooth surface relative to a smooth boundary curve. We consider the semiuniversal deformations and equisingular deformations of curves with a fixed local intersection number $w$ with the boundary, and prove…

A stratified space is a kind of topological space together with a partition into smooth manifolds. These kinds of spaces naturally arise in the study of singular algebraic varieties, symplectic reduction, and differentiable stacks. In this…

We prove Barth-type connectedness results for low-codimension smooth subvarieties with good numerical properties inside certain "easy" ambient spaces (such as homogeneous varieties, or spherical varieties). The argument employs some basics…

We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the…

This is a survey of recent and classical results concerning various types of homogeneity, such as n-homogeneity, discrete homogeneity, and countable dense homogeneity. Some new results are also presented, and several problems are posed.

We define the notion of 1-affineness for a prestack, and prove an array of results that establish 1-affineness of certain types of prestacks.

We construct spherical wavelets based on approximate identities that are directional, i.e. not rotation-invariant, and have an adaptive angular selectivity. The problem of how to find a proper representation of distinct kinds of details of…

In this paper, we use (bi)semicosimplicial language to study the classical problem of infinitesimal deformations of a closed subscheme in a fixed smooth variety, defined over an algebraically closed field of characteristic 0. In particular,…

We give a criterion for a group homomorphism on a valued abelian group to be surjective and to preserve spherical completeness. We apply this to give a criterion for the existence of integration on a valued differential field. Further, we…

For curves singularities the dimension of smoothing components in the deformation space is an invariant of the singularity, but in general the deformation space has components of different dimensions. We are interested in the question what…

In this note we propose the generalization of the notion of a holomorphic contact structure on a manifold (smooth variety) to varieties with rational singularities and prove basic properties of such objects. Natural examples of singular…

In this paper we consider convex improper affine maps of the 3-dimensional affine space and classify their singularities. The main tool developed is a generating family with properties that closely resembles the area function for non-convex…