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Uniformity conjecture: diophantine geometry: n/a: Unique games conjecture: number theory: n/a: Vandiver's conjecture: number theory: Ernst Kummer and Harry Vandiver: Virasoro conjecture: algebraic geometry: Miguel Ángel Virasoro: Vizing's conjecture: graph theory: Vadim G. Vizing: Vojta's conjecture: number theory: ⇒abc conjecture: Paul ...
Sometimes, a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the Riemann hypothesis is a conjecture from number theory that — amongst other things — makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann ...
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a vertex) or does not exist (if the lines are parallel). Other types ...
For example, two distinct lines can intersect in no more than one point, intersecting lines form equal opposite angles, and adjacent angles of intersecting lines are supplementary. When a third line is introduced, then there can be properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given two ...
As an example, count the conic sections tangent to five given lines in the projective plane. [4] The conics constitute a projective space of dimension 5, taking their six coefficients as homogeneous coordinates , and five points determine a conic , if the points are in general linear position , as passing through a given point imposes a linear ...
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels.
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry , there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two).
A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers. A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers. Sometimes this condition is included in the definition of a model geometry.