# 奇点理论

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".

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 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 situations are called perestroika (模板:Lang-ru), bifurcations or catastrophes. Classifying the types of changes and characterizing sets of parameters which give rise to these changes are some of the main mathematical goals. Singularities can occur in a wide range of mathematical objects, from matrices depending on parameters to wavefronts.[1]

Vladimir Arnold 将奇点理论的主要目标定义为描述物体如何依赖于参数，尤其强调在参数发生微小变化的情况下，参数的性质会发生突然变化。这些情况被称为 perestroika ，意为分叉或灾难。分类变化的类型并确定引起这些变化的参数集是研究奇点时的重要数学目标。奇点可以出现在很多数学对象中，从依赖于参数的矩阵到波前。

## How singularities may arise 奇点理论是怎样产生的

In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.

## Singularities in algebraic 代数学下的奇点

Historically, singularities were first noticed in the study of algebraic curves. Isaac Newton carried out a detailed study of all cubic curves, the general family to which these examples belong. It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point), in accounting for intersections of curves. It was then a short step to define the general notion of a singular point of an algebraic variety.In addition, singularities allow higher dimensions.

This page was moved from wikipedia:en:Singularity theory. Its edit history can be viewed at 奇点理论/edithistory

1. Arnold, V. I. (2000). "Singularity Theory". www.newton.ac.uk. Isaac Newton Institute for Mathematical Sciences. Retrieved 31 May 2016.