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第24行： − − 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. − − 本奇点理论研究点和奇点集的一般现象，作为流形(无奇点的空间)可以通过多条路线获得特殊的奇点的概念的一部分。当三维物体被投影到二维空间(例如我们的一只眼睛)时，投影是一种方式，在视觉方面非常明显; 在观看古典雕像时，褶皱的帷幔是最明显的特征之一。奇点包括焦散，非常熟悉的光模式在底部的游泳池。 − − 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. − − 奇异点的其他形式是流形结构的退化。对称性的存在是考虑 orbifolds 的好理由，orbifolds 是在折叠过程中获得“角”的流形，类似于餐巾纸的折痕。 − − − − ==Singularities in algebraic geometry== − − ===Algebraic curve singularities=== − − − − [[File:Cubic with double point.svg|thumb|right|upright=0.75|A curve with double point]] − − A curve with double point − − 双点曲线具有双点的曲线 − − [[File:Cusp.svg|thumb|upright=0.75|right|A curve with a cusp]] − − A curve with a cusp − − 有尖头的曲线 − − − − Historically, singularities were first noticed in the study of [[algebraic curve]]s. The ''double point'' at (0, 0) of the curve − − Historically, singularities were first noticed in the study of algebraic curves. The double point at (0, 0) of the curve − − 从历史上看，奇异点最早是在代数曲线研究中被注意到的。曲线(0,0)处的双点 − − − − :<math>y^2 = x^2 + x^3 </math> − − <math>y^2 = x^2 + x^3 </math> − − 数学 y ^ 2 x ^ 2 + x ^ 3 / math − − − − and the [[cusp (singularity)|cusp]] there of − − and the cusp there of − − 还有这里的尖端 − − − − :<math>y^2 = x^3\ </math> − − <math>y^2 = x^3\ </math> − − 数学 y ^ 2 x ^ 3 / math − − − − are qualitatively different, as is seen just by sketching. [[Isaac Newton]] carried out a detailed study of all [[cubic curve]]s, 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 (mathematics)|multiplicity]] (2 for a double point, 3 for a cusp), in accounting for intersections of curves. − − 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. − − 是截然不同的，就像素描一样。艾萨克 · 牛顿对所有的三次曲线进行了详细的研究，这些曲线就是这些例子所属的一般家族。在 b zout 定理的公式中注意到，为了解释曲线的交点，这样的奇点必须用重数来计算(2为双点，3为尖点)。 − − − − 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. − − 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. − − 那时，定义非奇异的一般概念只需要很短的一步，也就是说，允许更高的维度。 − − − − ===The general position of singularities in algebraic geometry=== − − Such singularities in [[algebraic geometry]] are the easiest in principle to study, since they are defined by [[polynomial equation]]s and therefore in terms of a [[coordinate system]]. One can say that the ''extrinsic'' meaning of a singular point isn't in question; it is just that in ''intrinsic'' terms the coordinates in the ambient space don't straightforwardly translate the geometry of the [[algebraic variety]] at the point. Intensive studies of such singularities led in the end to [[Heisuke Hironaka]]'s fundamental theorem on [[resolution of singularities]] (in [[birational geometry]] in [[characteristic (algebra)|characteristic]] 0). This means that the simple process of "lifting" a piece of string off itself, by the "obvious" use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general ''collapse'' (through multiple processes). This result is often implicitly used to extend [[affine geometry]] to [[projective geometry]]: it is entirely typical for an [[affine variety]] to acquire singular points on the [[hyperplane at infinity]], when its closure in [[projective space]] is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of [[compactification (mathematics)|compactification]], ending up with a ''compact'' manifold (for the strong topology, rather than the [[Zariski topology]], that is). − − Such singularities in algebraic geometry are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system. One can say that the extrinsic meaning of a singular point isn't in question; it is just that in intrinsic terms the coordinates in the ambient space don't straightforwardly translate the geometry of the algebraic variety at the point. Intensive studies of such singularities led in the end to Heisuke Hironaka's fundamental theorem on resolution of singularities (in birational geometry in characteristic 0). This means that the simple process of "lifting" a piece of string off itself, by the "obvious" use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general collapse (through multiple processes). This result is often implicitly used to extend affine geometry to projective geometry: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of compactification, ending up with a compact manifold (for the strong topology, rather than the Zariski topology, that is). − − 这样的奇点在代数几何中原则上是最容易研究的，因为它们是由多项式方程定义的，因此是以坐标系为单位的。我们可以说，奇点的外在意义并不是问题，只是在内在条件下，环境空间中的坐标并不能直接转换出代数簇点的几何形状。对这些奇点的深入研究最终导致了广中平祐关于奇点解消的基本定理(特征0中的双有理几何)。这意味着，通过在双点处“明显地”使用交叉来“提起”一根弦本身的简单过程，本质上并不具有误导性: 代数几何的所有奇点都可以作为某种非常普遍的崩塌(通过多个过程)来恢复。这个结果常常隐含地被用来将仿射几何推广到射影几何: 对于一个仿射变化来说，在无穷远处获得超平面上的奇异点是完全典型的，当它的闭包在射影空间时。决议说，这种奇异点可以作为一种(复杂的)紧化来处理，最终得到一个紧致流形(对于强拓扑而言，而不是 Zariski 拓扑)。 − − − − ==The smooth theory and catastrophes== − − At about the same time as Hironaka's work, the [[catastrophe theory]] of [[René Thom]] was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of [[Hassler Whitney]] on [[critical point (mathematics)|critical point]]s. Roughly speaking, a ''critical point'' of a [[smooth function]] is where the [[level set]] develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials. To compensate, only the ''stable'' phenomena are considered. One can argue that in nature, anything destroyed by tiny changes is not going to be observed; the visible ''is'' the stable. Whitney had shown that in low numbers of variables the stable structure of critical points is very restricted, in local terms. Thom built on this, and his own earlier work, to create a ''catastrophe theory'' supposed to account for discontinuous change in nature. − − At about the same time as Hironaka's work, the catastrophe theory of René Thom was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of Hassler Whitney on critical points. Roughly speaking, a critical point of a smooth function is where the level set develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials. To compensate, only the stable phenomena are considered. One can argue that in nature, anything destroyed by tiny changes is not going to be observed; the visible is the stable. Whitney had shown that in low numbers of variables the stable structure of critical points is very restricted, in local terms. Thom built on this, and his own earlier work, to create a catastrophe theory supposed to account for discontinuous change in nature. − − 与此同时，任同的突变理论也受到了极大的关注。这是奇点理论的另一个分支，基于 Hassler Whitney 早期关键点的工作。粗略地说，光滑函数的一个临界点是水平集在几何意义上发展出一个奇点的地方。这个理论通常处理可微函数，而不只是处理多项式。为了补偿，只考虑了稳定现象。人们可以说，在自然界中，任何被微小变化破坏的东西都不会被观察到; 可见的东西就是稳定的东西。惠特尼表明，在低数量的变量临界点的稳定结构是非常有限的，在局部条件。汤姆在此基础上，以及他自己早期的工作，创建了一个灾难理论，假定它能解释自然界中不连续的变化。 − − − − ===Arnold's view=== − − While Thom was an eminent mathematician, the subsequent fashionable nature of elementary [[catastrophe theory]] as propagated by [[Christopher Zeeman]] caused a reaction, in particular on the part of [[Vladimir Arnold]].<ref>{{harvnb|Arnold|1992}}</ref> He may have been largely responsible for applying the term '''''singularity theory''''' to the area including the input from algebraic geometry, as well as that flowing from the work of Whitney, Thom and other authors. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated as the construction of [[equivalence relation]]s on singular points, and [[germ (mathematics)|germs]]. Technically this involves [[Group action (mathematics)|group action]]s of [[Lie group]]s on spaces of [[jet (mathematics)|jet]]s; in less abstract terms [[Taylor series]] are examined up to change of variable, pinning down singularities with enough [[derivative]]s. Applications, according to Arnold, are to be seen in [[symplectic geometry]], as the geometric form of [[classical mechanics]]. − − While Thom was an eminent mathematician, the subsequent fashionable nature of elementary catastrophe theory as propagated by Christopher Zeeman caused a reaction, in particular on the part of Vladimir Arnold. He may have been largely responsible for applying the term singularity theory to the area including the input from algebraic geometry, as well as that flowing from the work of Whitney, Thom and other authors. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated as the construction of equivalence relations on singular points, and germs. Technically this involves group actions of Lie groups on spaces of jets; in less abstract terms Taylor series are examined up to change of variable, pinning down singularities with enough derivatives. Applications, according to Arnold, are to be seen in symplectic geometry, as the geometric form of classical mechanics. − − 虽然汤姆是一位杰出的数学家，但随后由克里斯托弗 · 塞曼传播的基本灾难理论的流行本质引起了反响，特别是弗拉基米尔 · 阿诺德的反响。他可能在很大程度上负责将奇点理论一词应用到这个领域，包括来自代数几何的投入，以及来自惠特尼、 Thom 和其他作者的作品。他在文中明确表示，他不喜欢过于公开地强调香港的一小部分地区。光滑奇异点的基础工作是构造奇异点和胚上的等价关系。从技术上讲，这涉及到李群在喷注空间上的群作用; 在较抽象的术语中，泰勒级数被用来检验变量的变化，用足够的导数来确定奇点。根据 Arnold 的说法，这些应用可以在辛几何中看到，作为经典力学的几何形式。 − − − − ===Duality=== − − An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of [[Poincaré duality]] is also disallowed. A major advance was the introduction of [[intersection cohomology]], which arose initially from attempts to restore duality by use of strata. Numerous connections and applications stemmed from the original idea, for example the concept of [[perverse sheaf]] in [[homological algebra]]. − − An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of Poincaré duality is also disallowed. A major advance was the introduction of intersection cohomology, which arose initially from attempts to restore duality by use of strata. Numerous connections and applications stemmed from the original idea, for example the concept of perverse sheaf in homological algebra. − − 奇点引起数学问题的一个重要原因是，由于流形结构的失败，庞加莱对偶性的调用也被禁止。一个主要的进步是交集上同调的引入，这最初起源于试图恢复二元性使用的阶层。大量的连接和应用源于最初的想法，例如在20世纪90年代同调代数提出的反常捆的概念。 − − − − ==Other possible meanings== − − The theory mentioned above does not directly relate to the concept of [[mathematical singularity]] as a value at which a function is not defined. For that, see for example [[isolated singularity]], [[essential singularity]], [[removable singularity]]. The [[monodromy]] theory of [[differential equation]]s, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking, ''monodromy'' studies the way a [[covering map]] can degenerate, while ''singularity theory'' studies the way a ''manifold'' can degenerate; and these fields are linked. − − The theory mentioned above does not directly relate to the concept of mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essential singularity, removable singularity. The monodromy theory of differential equations, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking, monodromy studies the way a covering map can degenerate, while singularity theory studies the way a manifold can degenerate; and these fields are linked. − − 上面提到的理论并没有直接涉及到奇点作为一个值的概念，在这个值上函数是没有定义的。关于这一点，可以参考孤立奇点，本质奇点，可去奇点。然而，在复数域上围绕奇点的单调微分方程理论却与几何理论有关。粗略地说，单向性研究覆盖映射退化的方式，而奇点理论研究流形退化的方式; 这些领域是相互联系的。 − − − − ==See also== − − {{div col|colwidth=22em}} − − *[[Tangent]] − − *[[Zariski tangent space]] − − *[[General position]] − − *[[Contact (mathematics)]] − − *[[Singular solution]] − − *[[Stratification (mathematics)]] − − *[[Intersection homology]] − − *[[Mixed Hodge structure]] − − *[[Whitney umbrella]] − − *[[Round function]] − − {{div col end}} − − − − == Notes == − − − − <references/> − − − − ==References== − − {{refbegin}} − − * {{Cite book − |title=Catastrophe Theory+ − 文章标题: 灾难理论 − |author=V.I. Arnold + + − + − | 出版商 Springer-Verlag+ − + − | isbn 978-3540548119 − |year = 1992 第227行：
第57行： − + − 我不会让你失望的+ − }}+ 第241行：
第71行： − * {{Cite book+ + − |title=Plane Algebraic Curves + + − + + − 作者 e。Brieskorn − |author2=H. Knörrer+ − 2 h.Knörrer+ 第265行：
第97行： − |publisher=Birkhauser-Verlag − |year=1986 + + − + − 978-3764317690 − |ref = {{harvid|Brieskorn|Knorrer|1996}} − 我不会让你失望的+ + + 第289行：
第121行： − }} − {{refend}} + − [[Category:Singularity theory| ]]
Moved page from wikipedia:en:Singularity theory (history)
{{about|the mathematical discipline|other geometric uses|Singular point of a curve|other mathematical uses|Singularity (mathematics)|non-mathematical uses|Singularity (disambiguation)}}
{{about|the mathematical discipline|other geometric uses|Singular point of a curve|other mathematical uses|Singularity (mathematics)|non-mathematical uses|Singularity (disambiguation)}}
In [[mathematics]], '''singularity theory''' studies spaces that are almost [[manifold]]s, 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, [[projection (mathematics)|dropping]] it on the floor, and flattening it. In some places the flat [[Jordan curve|string]] will cross itself in an approximate "X" shape. The points on the [[plane (geometry)|floor]] where it does this are one kind of singularity, the double point: one [[neighbourhood (topology)|bit]] of the floor corresponds to [[multimap|more than one]] bit of string. Perhaps the string will also touch itself without crossing, like an underlined "<u>U</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".
{{expert|date=July 2020}}
In [[mathematics]], '''singularity theory''' studies spaces that are almost [[manifold]]s, 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, [[projection (mathematics)|dropping]] it on the floor, and flattening it. In some places the flat [[Jordan curve|string]] will cross itself in an approximate "X" shape. The points on the [[plane (geometry)|floor]] where it does this are one kind of singularity, the double point: one [[neighbourhood (topology)|bit]] of the floor corresponds to [[multimap|more than one]] bit of string. Perhaps the string will also touch itself without crossing, like an underlined "<u>U</u>".{{huh|the underlined U I see doesn't touch itself; they're separated|date=July 2020}} 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>U</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, 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>U</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，就像一个带下划线的“ u / u”。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”。
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 </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 (}}), 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.
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。对引起这些变化的变化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。
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。
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. [[3D projection|Projection]] is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our [[human eye|eye]]s); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include [[caustic (mathematics)|caustic]]s, very familiar as the light patterns at the bottom of a swimming pool.
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. [[3D projection|Projection]] is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our [[human eye|eye]]s); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include [[caustic (mathematics)|caustic]]s, very familiar as the light patterns at the bottom of a swimming pool.
Other ways in which singularities occur is by [[Degeneracy (mathematics)|degeneration]] of manifold structure. The presence of [[symmetry]] can be good cause to consider [[orbifold]]s, which are manifolds that have acquired "corners" in a process of folding up, resembling the creasing of a table napkin.
Other ways in which singularities occur is by [[Degeneracy (mathematics)|degeneration]] of manifold structure. The presence of [[symmetry]] can be good cause to consider [[orbifold]]s, which are manifolds that have acquired "corners" in a process of folding up, resembling the creasing of a table napkin.
|title=Catastrophe Theory
|title=Catastrophe Theory
巨灾理论
|author=V.I. Arnold
|author=V.I. Arnold
作者: v.i。阿诺德
作者: v.i。阿诺德
==Singularities in algebraic geometry==
|publisher=Springer-Verlag
|publisher=Springer-Verlag
|publisher=Springer-Verlag
| publisher = Springer-Verlag
===Algebraic curve singularities===
|isbn= 978-3540548119
|isbn= 978-3540548119
|isbn= 978-3540548119
| isbn = 978-3540548119
|year = 1992
|year = 1992
1992年
1992年
|ref = {{harvid|Arnold|1992}}
[[File:Cubic with double point.svg|thumb|right|upright=0.75|A curve with double point]]
|ref =
|ref =
2012年10月22日
[[File:Cusp.svg|thumb|upright=0.75|right|A curve with a cusp]]
}}
}}
Historically, singularities were first noticed in the study of [[algebraic curve]]s. The ''double point'' at (0, 0) of the curve
|title=Plane Algebraic Curves
|title=Plane Algebraic Curves
平面代数曲线
平面代数曲线
:<math>y^2 = x^2 + x^3 </math>
|author=E. Brieskorn
|author=E. Brieskorn
|author=E. Brieskorn
| author = e.Brieskorn
|author2=H. Knörrer
|author2=H. Knörrer
2 = h.Knörrer
and the [[cusp (singularity)|cusp]] there of
|publisher=Birkhauser-Verlag
|publisher=Birkhauser-Verlag
|publisher=Birkhauser-Verlag
|publisher=Birkhauser-Verlag
|year=1986
|year=1986
1986年
1986年
:<math>y^2 = x^3\ </math>
|isbn= 978-3764317690
|isbn= 978-3764317690
|isbn= 978-3764317690
| isbn = 978-3764317690
|ref =
|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 curve]]s, 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 (mathematics)|multiplicity]] (2 for a double point, 3 for a cusp), in accounting for intersections of curves.
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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.
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