In
mathematics, singularity theory studies spaces that are almost
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s, 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
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian anim ...
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
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented a ...
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 (russian: ), 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.
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
eye
Eyes are organs of the visual system. They provide living organisms with vision, the ability to receive and process visual detail, as well as enabling several photo response functions that are independent of vision. Eyes detect light and conv ...
s); 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
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descri ...
of manifold structure. The presence of
symmetry can be good cause to consider
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
D ...
s, 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
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the g ...
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
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
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
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In c ...
; 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 equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equati ...
s 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
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
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
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
: 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
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
{{disambiguation ...
, 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
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integrati ...
on
critical points. Roughly speaking, a ''critical point'' of a
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
is where the
level set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is cal ...
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.
Arnold's view
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 relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
s on singular points, and
germs. Technically this involves
group actions of
Lie groups on spaces of
jet
Jet, Jets, or The Jet(s) may refer to:
Aerospace
* Jet aircraft, an aircraft propelled by jet engines
** Jet airliner
** Jet engine
** Jet fuel
* Jet Airways, an Indian airline
* Wind Jet (ICAO: JET), an Italian airline
* Journey to Enceladus a ...
s; in less abstract terms
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
are examined up to change of variable, pinning down singularities with enough
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s. Applications, according to Arnold, are to be seen in
symplectic geometry, as the geometric form of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
.
Duality
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
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.
For ex ...
as a value at which a function is not defined. For that, see for example
isolated singularity,
essential singularity,
removable singularity. The
monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of '' ...
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.
See also
*
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
*
Zariski tangent space
*
General position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...
*
Contact (mathematics)
*
Singular solution
*
Stratification (mathematics)
*
Intersection homology
*
Mixed Hodge structure
*
Whitney umbrella
*
Round function
*
Victor Goryunov
Notes
References
*
*
{{Manifolds