Singularity theory

Singularity theory studies spaces that are almost manifolds, 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, dropping it on the floor, and flattening it.

In some places the flat string will cross itself in an approximate “X” shape. The points on the floor where it does this are one kind of singularity, the double point: one 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”.

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 bifurcations 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 matrices depending on parameters to wavefronts.

In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. 

Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features.

Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.

Other ways in which singularities occur is by degeneration of manifold structure. 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.

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, only the stable phenomena 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.

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