# 奇点理论

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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__".模板:Huh 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, 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".

在数学中，奇点理论研究的空间几乎是流形，但不完全是。如果忽略弦的厚度，弦可以作为一维流形的例子。一个奇点可以通过把它团起来，扔在地板上，然后把它压扁来形成。在某些地方，扁平的字符串会以近似“ x”的形状交叉自身。地板上的这些点是一种奇点，双点: 地板的一个位相当于一个以上的字符串。也许字符串也会在没有交叉的情况下接触自己，就像“ < u > u ”下划线那样。这是另一种奇点。与双点不同，它是不稳定的，在某种意义上说，一个小的推动将提升底部的“ u”远离“下划线”。

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 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 (}}), 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.

将奇点理论的主要目标定义为描述物体如何依赖于参数，特别是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 (}) ，分叉或灾难。对变化的类型进行分类并确定引起这些变化的参数集是一些主要的数学目标。奇异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

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.

Other ways in which singularities occur is by degeneration of manifold structure. The presence of symmetry can be good cause to consider orbifolds, which are manifolds that have acquired "corners" in a process of folding up, resembling the creasing of a table napkin.

|title=Catastrophe Theory

巨灾理论

|author=V.I. Arnold

作者: v.i。阿诺德

## Singularities in algebraic geometry

|publisher=Springer-Verlag

| publisher = Springer-Verlag

### Algebraic curve singularities

|isbn= 978-3540548119

| isbn = 978-3540548119

|year = 1992

1992年

|ref =

2012年10月22日

}}

}}

Historically, singularities were first noticed in the study of algebraic curves. The *double point* at (0, 0) of the curve

|title=Plane Algebraic Curves

平面代数曲线

- [math]\displaystyle{ y^2 = x^2 + x^3 }[/math]

|author=E. Brieskorn

| author = e.Brieskorn

|author2=H. Knörrer

2 = h.Knörrer

and the cusp there of

|publisher=Birkhauser-Verlag

|publisher=Birkhauser-Verlag

|year=1986

1986年

- [math]\displaystyle{ y^2 = x^3\ }[/math]

|isbn= 978-3764317690

| isbn = 978-3764317690

|ref =

2012年10月22日

are qualitatively different, as is seen just by sketching. 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, 3 for a cusp), 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; that is, to allow higher dimensions.

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

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