# Category: Singularity theory

Whitney topologies
In mathematics, and especially differential topology, functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mapp
Terminal singularity
No description available.
Elliptic singularity
In algebraic geometry, an elliptic singularity of a surface, introduced by , is a surface singularity such that the arithmetic genus of its local ring is 1.
Sard's theorem
In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set o
Brieskorn–Grothendieck resolution
In mathematics, a Brieskorn–Grothendieck resolution is a resolution conjectured by Alexander Grothendieck, that in particular gives a resolution of the universal deformation of a Kleinian singularity.
Jacobian ideal
In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ.Let denote the ring of smooth functions in variables and a function in the rin
Jet (mathematics)
In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. Although this is the defini
Singular point of an algebraic variety
In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space a
Milnor map
In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press,
Du Bois singularity
In algebraic geometry, Du Bois singularities are singularities of complex varieties studied by . gave the following characterisation of Du Bois singularities. Suppose that is a reduced closed subschem
Whitney umbrella
In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface plac
Local uniformization
In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating roughly that a variety can be desingularized near any valuation, or in other words that the Zariski–R
Milnor number
In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex-valued holomorphic function germ then the Milnor
Nakai conjecture
In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961.It states that if V is a complex algeb
Stratified Morse theory
In mathematics, stratified Morse theory is an analogue to Morse theory for general stratified spaces, originally developed by Mark Goresky and Robert MacPherson. The main point of the theory is to con
Ak singularity
In mathematics, and in particular singularity theory, an Ak singularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold. Let be a
Pseudoisotopy theorem
In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's which refers to the connectivity of a group of diffeomorphisms of a manifold.
Bott–Samelson resolution
In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by in the context of compact Lie groups. The algebraic formulation is inde
Whitney conditions
In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965. A stratification of a topologica
Brieskorn manifold
In mathematics, a Brieskorn manifold or Brieskorn–Phạm manifold, introduced by Egbert Brieskorn , is the intersection of a small sphere around the origin with the singular, complex hypersurface studie
Canonical singularity
In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal mod
A-equivalence
In mathematics, -equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs. Let and be two manifolds, and let be two smooth map germs. We say that and are -equ
Delta invariant
In mathematics, in the theory of algebraic curves, a delta invariant measures the number of double points concentrated at a point. It is a non-negative integer. Delta invariants are discussed in the "
Arnold's spectral sequence
In mathematics, Arnold's spectral sequence (also spelled Arnol'd) is a spectral sequence used in singularity theory and normal form theory as an efficient computational tool for reducing a function to
Du Val singularity
In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on
K-equivalence
In mathematics, -equivalence, or contact equivalence, is an equivalence relation between map germs. It was introduced by John Mather in his seminal work in Singularity theory in the 1960s as a technic
Splitting lemma (functions)
In mathematics, especially in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a ne
The Singularity (film)
The Singularity is a 2012 documentary film about the technological singularity, produced and directed by Doug Wolens. The film has been called "a large-scale achievement in its documentation of futuri
Rational singularity
In mathematics, more particularly in the field of algebraic geometry, a scheme has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a prope
Springer resolution
In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra, or the unipotent elements of a reductive algebraic group, introduced by Tonny
Malgrange preparation theorem
In mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René Thom and proved by B. Malgrange .
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. Bifurcation
Signature defect
In mathematics, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem. introduced the signature defect for the cusp singularities of Hil
Transhumanism
Transhumanism is a philosophical and intellectual movement which advocates the enhancement of the human condition by developing and making widely available sophisticated technologies that can greatly
Cusp (singularity)
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of sin
Tacnode
In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp) is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to
Contact (mathematics)
In mathematics, two functions have a contact of order k if, at a point P, they have the same value and k equal derivatives. This is an equivalence relation, whose equivalence classes are generally cal
Critical point (mathematics)
Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is eith
The Swallow's Tail
The Swallow's Tail — Series of Catastrophes (French: La queue d'aronde — Série des catastrophes) was Salvador Dalí's last painting. It was completed in May 1983, as the final part of a series based on
Conifold
In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a c
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of man
Unfolding (functions)
In mathematics, an unfolding of a smooth real-valued function ƒ on a smooth manifold, is a certain family of functions that includes ƒ.
Generic property
In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functio
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
Nodal surface
In algebraic geometry, a nodal surface is a surface in (usually complex) projective space whose only singularities are nodes. A major problem about them is to find the maximum number of nodes of a nod
Acnode
An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are "isolated point or hermit point". For example the equation has an acnode at the
Ridge detection
In image processing, ridge detection is the attempt, via software, to locate ridges in an image, defined as curves whose points are local maxima of the function, akin to geographical ridges. For a fun
Cerf theory
In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions on a smooth manifold , their generic singularities
Pinch point (mathematics)
In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface. The equation for the surface near a pinch point may be put in the form where [4] denotes terms of degr
Eisenbud–Levine–Khimshiashvili signature formula
In mathematics, and especially differential topology and singularity theory, the Eisenbud–Levine–Khimshiashvili signature formula gives a way of computing the Poincaré–Hopf index of a real, analytic v
Singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity
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. For varieties over
Crepant resolution
In algebraic geometry, a crepant resolution of a singularity is a resolution that does not affect the canonical class of the manifold. The term "crepant" was coined by Miles Reid by removing the prefi
Thom–Mather stratified space
In topology, a branch of mathematics, an abstract stratified space, or a Thom–Mather stratified space is a topological space X that has been decomposed into pieces called strata; these strata are mani