# 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".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".

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.

## 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

|author=V.I. Arnold

## Singularities in algebraic geometry

|publisher=Springer-Verlag

| publisher = Springer-Verlag

### Algebraic curve singularities

|isbn= 978-3540548119

| isbn = 978-3540548119

|year = 1992

1992年

A curve with double point

|ref =

2012年10月22日

A curve with a cusp

}}

}}

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

|title=Plane Algebraic Curves

$\displaystyle{ y^2 = x^2 + x^3 }$

|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

$\displaystyle{ y^2 = x^3\ }$

|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

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