Concept
Singularity theory
Related concepts (14)
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 either not differentiable or the derivative is equal to zero. When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero. Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient is undefined or is equal to zero.
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 theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide.
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 manifolds and are required to fit together in a certain way. Thom–Mather stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of Whitney. They were introduced by René Thom, who showed that every Whitney stratified space was also a topologically stratified space, with the same strata.
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 fields of characteristic 0 this was proved in Hironaka (1964), while for varieties over fields of characteristic p it is an open problem in dimensions at least 4. Originally the problem of resolution of singularities was to find a nonsingular model for the function field of a variety X, in other words a complete non-singular variety X′ with the same function field.
Intersection homology
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 used to prove the Kazhdan–Lusztig conjectures and the Riemann–Hilbert correspondence. It is closely related to L2 cohomology.
Surface (mathematics)
In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. The exact definition of a surface may depend on the context.
René Thom
René Frédéric Thom (ʁəne tɔm; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. 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 founder of catastrophe theory (later developed by Christopher Zeeman). René Thom grow up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940.
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 singular point of a curve. For a plane curve defined by an analytic, parametric equation a cusp is a point where both derivatives of f and g are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope ).
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 definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions. This article first explores the notion of a jet of a real valued function in one real variable, followed by a discussion of generalizations to several real variables.
Singularity (mathematics)
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 example, the function has a singularity at , where the value of the function is not defined, as involving a division by zero. The absolute value function also has a singularity at , since it is not differentiable there. The algebraic curve defined by in the coordinate system has a singularity (called a cusp) at .