Singularity Theory, including: Contact (mathematics), Catastrophe Theory, Conifold, Sard's Theorem, Singular Point Of An Algebraic Variety, Jet ... Whitney Umbrella, Rational Singularity

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More info: In mathematics, singularity theory is the study of the failure of manifold structure. 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 points, at which the string crosses itself in an approximate 'X' 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'. 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'.