Singularity theory

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Singularity theorystudies spaces that are almostmanifolds, 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,droppingit on the floor, and flattening it.

In some places the flatstringwill cross itself in an approximate “X” shape. The points on thefloorwhere it does this are one kind of singularity, the double point: onebitof the floor corresponds tomore than onebit 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 Arnolddefines 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…

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