11 Singularity theory

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".

Critical point is a wide term used in many branches of mathematics.

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.

In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface.

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.

In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special', in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.

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 futurist and counter-futurist ideas”.

The Swallow's Tail — Series of Catastrophes was Salvador Dalí's last painting. It was completed in May 1983, as the final part of a series based on the mathematical catastrophe theory of René Thom.

In classical algebraic geometry, a tacnode is a kind of singular point of a curve. It is defined as a point where two osculating circles to the curve at that point are tangent. This means that two branches of the curve have ordinary tangency at the double point.

Transhumanism is a philosophical movement that advocates for the transformation of the human condition by developing and making widely available sophisticated technologies able to greatly modify or enhance human intellect and physiology.

In mathematics, the Whitney umbrella is a specific self-intersecting surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line, parallel to the axis of the parabola and lying on its perpendicular bisecting plane.