Editing Singularity theory (section)

{{about|the mathematical discipline|other geometric uses|Singular point of a curve|other mathematical uses|Singularity (mathematics)|non-mathematical uses|Singularity (disambiguation)}}
{{short description|Mathematical theory}}

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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 (mathematics)|singularity]], the double point: one [[neighbourhood (topology)|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".

[[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 ({{langx|ru|{{wt|ru|перестройка}}}}), [[Bifurcation theory|bifurcation]]s 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 [[matrix (mathematics)|matrices]] depending on parameters to wavefronts.<ref>{{cite web|last1=Arnold|first1=V. I.|title=Singularity Theory|url=https://www.newton.ac.uk/event/sgt|website=www.newton.ac.uk|publisher=Isaac Newton Institute for Mathematical Sciences|access-date=31 May 2016|date=2000}}</ref>