Singularity theory

For non-mathematical singularity theories, see singularity.

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 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, the stable phenomena only 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.

Related Topics:
Catastrophe theory - René Thom - Hassler Whitney - Critical point - Smooth function - Level set

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