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Equisingularity
In algebraic geometry, an equisingularity is, roughly, a family of singularities that are not non-equivalent and is an important notion in singularity theory. There is no universal definition of equisingularity but Zariki's equisingularity is the most famous one. Zariski's equisingualrity, introduced in 1971 under the name " algebro-geometric equisingularity", gives a stratification that is different from the usual Whitney stratification on a real or complex algebraic variety. See also *stratified space In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset o ... References Further reading *https://mathoverflow.net/questions/299314/a-general-definition-of-an-equisingular-family-of-singular-varieties algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stratified Space
In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a Stratification (mathematics)#In topology, stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat). A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is also an abstract stratified space such as a Thom–Mather stratified space. On a stratified space, a constructible sheaf can be defined as a sheaf that is locally constant sheaf, locally constant on each stratum. Among the several ideals, Grothendieck's ''Esquisse d’un programme'' considers (or proposes) a stratified space with what he calls the tame topology. A stratified space in the sense of Mather Mather gives the following definition of a stratified space. A ''prestratification'' on a topological space ''X'' is a partition of ''X'' into subsets (called strata) such that (a) each stratum is locally closed, (b) it is loca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singularity Theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string. Perhaps the string will also touch itself without crossing, like an underlined "U". This is another kind of singularity. Unlike the double point, it is not ''stable'', in the sense that a small push will lift the bottom of the "U" away from the "underline". Vladimir Arnold defines the main goal of singularity theory as describing how objects depend on parameters, particularly in cases where the properties undergo sudden change under a small variation of the parameters. These ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Stratification
In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965. A stratification of a topological space is a finite filtration by closed subsets ''F''''i'' , such that the difference between successive members ''F''''i'' and ''F''(''i'' − 1) of the filtration is either empty or a smooth submanifold of dimension ''i''. The connected components of the difference ''F''''i'' − ''F''(''i'' − 1) are the strata of dimension ''i''. A stratification is called a Whitney stratification if all pairs of strata satisfy the Whitney conditions A and B, as defined below. The Whitney conditions in R''n'' Let ''X'' and ''Y'' be two disjoint ( locally closed) submanifolds of R''n'', of dimensions ''i'' and ''j''. * ''X'' and ''Y'' satisfy Whitney's condition A if whenever a sequence of points ''x''1, ''x''2, … in ''X'' converges to a point ''y'' in ''Y'', and the sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
