Let be the Euclidean space . Let be a subset of . If is a submanifold at there is a local canonical form for at : there is a diffeomorphism from an open neighborhood of into an open set of that takes into the intersection of with a linear subspace of . The set looks like a linear subspace in a neighborhood of .
Let be a smooth vector field on . If does not vanish at a point , there is a local canonical form for at : there is a diffeomorphism from an open neighborhood of into an open set of that takes the restriction of to into a constant vector field on . The vector field looks like a constant vector field in a neighbourhood of .
Singularity theory looks at what happens at the points where is a not a submanifold of or vanishes. At these points things get more complicated. In order to say something we will need some additional hypothesis. We will ask for instance that is defined by polynomial equations and has polynomial coefficients. We can replace the field by an arbitrary field . The theory becomes much simpler if we assume the field is algebraically closed. The first singularities to be studied were the singularities of plane curves. Its natural generalization, the study of isolated singularities of complex analytic hypersurface singularities is the core of Singularity theory.
There is no good concise description of Singularity Theory. We can enumerate some of the objects it studies: algebraic sets, analytic sets, algebraic maps, analytic maps, vector fields, differential forms, Lagrangian varieties... We can enumerate some of the questions that are usually asked: do we have topological [smooth, analytic] canonical forms? can we resolve [reduce] its singularities? what are its deformations? what are the objects that have no deformations [only a finite number of non-isomorphic deformations]?
How singularities may arise
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. That implies the breakdown of parametrization of points; it is prominent in general relativity, where a gravitational singularity, at which the gravitational field is strong enough to change the very structure of space-time, is identified with a black hole. In a less dramatic fashion, 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.
Singularities in algebraic geometry
Algebraic curve singularities
Historically, singularities were first noticed in the study of algebraic curves. The double point at (0,0) of the curve
and the cusp there of
are qualitatively different, as is seen just by sketching. Isaac Newton carried out a detailed study of all cubic curves, the general family to which these examples belong. It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves.
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.
The general position of singularities in algebraic geometry
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. One can say that the extrinsic meaning of a singular point isn't in question; it is just that in intrinsic terms the coordinates in the ambient space don't straightforwardly translate the geometry of the algebraic variety at the point. Intensive studies of such singularities led in the end to Heisuke Hironaka's fundamental theorem on resolution of singularities (in birational geometry in characteristic 0). This means that the simple process of 'lifting' a piece of string off itself, by the 'obvious' use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general collapse (through multiple processes). This result is often implicitly used to extend affine geometry to projective geometry: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of compactification, ending up with a compact manifold (for the strong topology, rather than the Zariski topology, that is).
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, 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.
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. 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. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated as the construction of equivalence relations on singular points, and germs. Technically this involves group actions of Lie groups on spaces of jets; in less abstract terms Taylor series are examined up to change of variable, pinning down singularities with enough derivatives. Applications, according to Arnold, are to be seen in symplectic geometry, as the geometric form of classical mechanics.
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. A major advance was the introduction of intersection cohomology, which arose initially from attempts to restore duality by use of strata. Numerous connections and applications stemmed from the original idea, for example the concept of perverse sheaf in homological algebra.
Other possible meanings
The theory mentioned above does not directly relate to the concept of mathematical singularity as a value at which a function isn't defined. For that, see for example isolated singularity, essential singularity, removable singularity. The monodromy theory of differential equations, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking, monodromy studies the way a covering map can degenerate, while singularity theory studies the way a manifold can degenerate; and these fields are linked.
- V. I. Arnol'd , "Catastrophe theory", Springer Verlag