For other geometic uses, see Singular point of a curve

Singular point of a curve

In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.-Algebraic curves in the plane:...

. For other mathematical uses, see Mathematical singularity

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

. For non-mathematical uses, see Singularity (disambiguation)

The notion of singularity

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, singularity theory is the study of the failure of manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the 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'.

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

3D projection

3D projection is any method of mapping three-dimensional points to a two-dimensional plane. As most current methods for displaying graphical data are based on planar two-dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting.-...

 is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eye

Human eye

The human eye is an organ which reacts to light for several purposes. As a conscious sense organ, the eye allows vision. Rod and cone cells in the retina allow conscious light perception and vision including color differentiation and the perception of depth...

s); in looking at classical statuary the folds

Folding

Fold or folding may refer to:* Paper folding, the art of folding paper* Book folding, in book production* Skin fold, an area of skin that folds* Fold , in the game of poker, to discard one's hand and forfeit interest in the current pot...

 of drapery are amongst the most obvious features. Singularities of this kind include caustic

Caustic (mathematics)

In differential geometry and geometric optics, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the optical concept of caustics...

s, very familiar as the light patterns at the bottom of a swimming pool

Swimming pool

A swimming pool, swimming bath, wading pool, or simply a pool, is a container filled with water intended for swimming or water-based recreation. There are many standard sizes; the largest is the Olympic-size swimming pool...

.

Other ways in which singularities occur is by degeneration

Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....

 of manifold structure. That implies the breakdown of parametrization

Parametrization

Parametrization is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object....

 of points; it is prominent in general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, where a gravitational singularity

Gravitational singularity

A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...

, at which the gravitational field

Gravitational field

The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

 is strong enough to change the very structure of space-time, is identified with a black hole

Black hole

A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...

. In a less dramatic fashion, the presence of symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 can be good cause to consider orbifold

Orbifold

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...

s, which are manifolds that have acquired 'corners' in a process of folding up resembling the creasing of a table napkin.

Algebraic curve singularities

Historically, singularities were first noticed in the study of algebraic curve

Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

s. The double point at (0,0) of the curve


and the cusp

Cusp (singularity)

In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....

 there of


are qualitatively different, as is seen just by sketching. Isaac Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 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

Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...

 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

Singular point of an algebraic variety

In mathematics, a singular point of an algebraic variety V is a point P that is 'special' , in the geometric sense that V is not locally flat there. In the case of an algebraic curve, a plane curve that has a double point, such as the cubic curveexhibits at , cannot simply be parametrized near the...

; that is, to allow higher dimensions.

The general position of singularities in algebraic geometry

Such singularities in algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system

Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

. 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 variety

In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

 at the point. Intensive studies of such singularities led in the end to Heisuke Hironaka

Heisuke Hironaka

is a Japanese mathematician. After completing his undergraduate studies at Kyoto University, he received his Ph.D. from Harvard while under the direction of Oscar Zariski. He won the Fields Medal in 1970....

's fundamental theorem on resolution of singularities

Resolution of singularities

In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V...

 (in birational geometry

Birational geometry

In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...

 in characteristic

Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...

 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

Affine geometry

In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...

 to projective geometry

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

 is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of compactification

Compactification

Compactification may refer to:* Compactification , making a topological space compact* Compactification , the "curling up" of extra dimensions in string theory* Compaction...

, ending up with a compact manifold (for the strong topology, rather than the Zariski topology

Zariski topology

In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...

, that is).

The smooth theory, and catastrophes

At about the same time as Hironaka's work, the catastrophe theory

Catastrophe theory

In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry....

 of René Thom

René Thom

René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...

 was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of Hassler Whitney

Hassler Whitney

Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...

 on critical point

Critical point (mathematics)

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...

s. Roughly speaking, a critical point of a smooth function

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

 is where the level set

Level set

In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....

 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

Catastrophe theory

In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry....

 as propagated by Christopher Zeeman caused a reaction, in particular on the part of Vladimir Arnold

Vladimir Arnold

Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...

. 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

Equivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

s on singular points, and germs

Germ (mathematics)

In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...

. Technically this involves group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

s of Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s on spaces of jet

Jet (mathematics)

In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...

s; in less abstract terms Taylor series

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 are examined up to change of variable, pinning down singularities with enough derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s. Applications, according to Arnold, are to be seen in symplectic geometry, as the geometric form of classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

.

Duality

An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of Poincaré duality

Poincaré duality

In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...

 is also disallowed. A major advance was the introduction of intersection cohomology

Intersection cohomology

In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years.Intersection cohomology was...

, 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

Perverse sheaf

The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space X, which may be a real or complex manifold, or a more general Topologically stratified space, usually singular...

 in homological algebra

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

.

Other possible meanings

The theory mentioned above does not directly relate to the concept of mathematical singularity

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

 as a value at which a function isn't defined. For that, see for example isolated singularity, essential singularity

Essential singularity

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.The category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the...

, removable singularity. The monodromy

Monodromy

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...

 theory of differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking, monodromy studies the way a covering map

Covering map

In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p...

can degenerate, while singularity theory studies the way a manifold can degenerate; and these fields are linked.