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| {{dablink|For other geometic uses, see [[Singular point of a curve]]. For other mathematical uses, see [[Mathematical singularity]]. For non-mathematical uses, see [[Singularity (disambiguation)]].}}
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| {{Refimprove|date=July 2011}}
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| {{Tone|date=July 2011}}
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| {{lead rewrite|date=September 2012}}
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| In [[mathematics]], '''singularity theory''' is the study of the failure of [[manifold]] structure.
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| ==Informal discussion==
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| A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor. Probably there will appear a number of [[double point]]s, at which the string crosses itself in an approximate 'χ' shape. These are the simplest kinds of ''singularity''. Perhaps the string will also touch itself, coming into ''contact'' with 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 '_'.
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| ==How singularities may arise==
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| 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]].
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| Other ways in which singularities occur is by [[Degeneracy (mathematics)|degeneration]] of manifold structure. That implies the breakdown of [[parametrization]] of points; it is prominent in [[general relativity]], where a [[gravitational singularity]], at which the [[gravitational field]] is strong enough to change the very structure of [[space-time]], is identified with a [[black hole]]. In a less dramatic fashion, 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]].
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| ==Singularities in algebraic geometry==
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| ===Algebraic curve singularities===
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| [[Image:Cubic with double point.svg|thumb|A curve with double point.]] | |
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| Historically, singularities were first noticed in the study of [[algebraic curve]]s. The ''double point'' at (0,0) of the curve
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| :<math>y^2 = x^2 - x^3\ </math>
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| and the [[cusp (singularity)|cusp]] there of | |
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| :<math>y^2 = x^3\ </math>
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| 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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| ===The general position of singularities in algebraic geometry===
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| 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).
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| ==The smooth theory, and catastrophes==
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| 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.
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| ===Arnold's view===
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| 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> V. I. Arnol'd , "Catastrophe theory", Edition: 3 - 1992, Springer</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]]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]].
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| ===Duality===
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| 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]].
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| ==Other possible meanings==
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| The theory mentioned above does not directly relate to the concept of [[mathematical singularity]] as a value at which a function isn't 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.
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| ==See also==
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| <div style="-moz-column-count:3; column-count:3;">
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| *[[Fractal]]
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| *[[Tangent]]
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| *[[Zariski tangent space]]
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| *[[General position]]
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| *[[Contact (mathematics)]]
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| *[[Singular solution]]
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| *[[Folding]]{{dn|date=January 2014}}
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| *[[Stratification (mathematics)]]
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| *[[Intersection homology]]
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| *[[Mixed Hodge structure]]
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| *[[Whitney umbrella]]
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| *[[Round function]]
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| </div>
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Singularity Theory}}
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| [[Category:Singularity theory| ]]
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