Singularity theory
For other mathematical uses, see Mathematical singularity. For non-mathematical uses, see Singularity.In
mathematics,
singularity theory is the study of the failure of
manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor. Probably there will appear a number of
double points, at which the string crosses itself in an approximate 'X' shape. These are the simplest kinds of
singularity. Perhaps the string will also touch itself, coming into
contact with 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'.
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.
Historically, singularities were first noticed in the study of
algebraic curves. The
double point at (0,0) of the curve
y2 =
x3 −
x2and the
cusp there of
y2 =
x3are 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.
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).
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 caused a reaction, in particular on the part of
Vladimir Arnol'd. 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 Arnol'd, 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.
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.
*
Tangent*
Zariski tangent space*
General position*
Contact (mathematics)*
Singular solution*
Folding*
Stratification*
Intersection homology*
Mixed Hodge structure*
Whitney umbrella*
Round function
