How do you write a conjecture in geometry?

How do you write a conjecture in geometry?

Writing a Conjecture

1. You must notice some kind of pattern or make some kind of observation. For example, you noticed that the list is counting up by 2s.
2. You form a conclusion based on the pattern that you observed, just like you concluded that 14 would be the next number.

What does a conjecture entail?

the formation or expression of an opinion or theory without sufficient evidence for proof. an opinion or theory so formed or expressed; guess; speculation. Obsolete.

What is the solution to the Poincare Conjecture?

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface.

What is the Poincare Conjecture simplified?

Poincaré conjecture, in topology, conjecture—now proven to be a true theorem—that every simply connected, closed, three-dimensional manifold is topologically equivalent to S3, which is a generalization of the ordinary sphere to a higher dimension (in particular, the set of points in four-dimensional space that are …

What is the geometrization conjecture?

Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.

Do Haken manifolds satisfy the geometrization conjecture?

Thurston’s hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print. Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery.

What is the uniformization theorem for three-dimensional surfaces?

It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries ( Euclidean, spherical, or hyperbolic ). In three dimensions, it is not always possible to assign a single geometry to a whole topological space.

Can A Thurston metric be left invariant on a Bianchi group?

Most Thurston geometries can be realized as a left invariant metric on a Bianchi group. However S2 × R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.