Singularity theory

The presence of symmetry can be good cause to consider orbifolds, which are manifolds that have acquired "corners" in a process of folding up, resembling the creasing of a table napkin.

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.

Such singularities in algebraic geometry are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system.

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.

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.

[2] 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.

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.

Numerous connections and applications stemmed from the original idea, for example the concept of perverse sheaf in homological algebra.