What is the Singularity theory?

 Singularity theory explores the realms of spaces that slightly deviate from being manifolds. A practical example is a piece of string, which can be considered a one-dimensional manifold when disregarding its thickness. If the string is crumpled, dropped, and flattened, it will create singularities. These are points where the string crosses itself, forming an approximate "X" shape on the floor, known as a double point - a common type of singularity.

In some cases, the string may touch itself without intersecting, forming an underlined "U" shape. This represents another type of singularity, which is unstable as a minor nudge can separate the bottom of the "U" from the "underline".

Singularity theory, as defined by Vladimir Arnold, aims to illustrate how objects are dependent on parameters, especially in situations where properties experience abrupt changes due to small parameter variations - scenarios known as perestroika bifurcations or catastrophes .

The theory focuses on classifying these changes and identifying parameter sets that trigger these transformations, which is a significant mathematical challenge. Singularities can appear in various mathematical entities, from parameter-dependent matrices to wavefronts.

In singularity theory, the general occurrence of points and sets of singularities is examined. It's rooted in the idea that manifolds (spaces without singularities) can acquire unique, singular points through several methods [3].

One method is projection, which is visually evident when three-dimensional objects are projected into two dimensions - like in our eyes when we view classical statues, where the drapery folds are the most noticeable features [3].

This type of singularity includes caustics, commonly seen as 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.