Changes

Mathematicians call the ordinary circle the "one sphere," which is written <math>\mathbb{S}^1 \ </math>.  The defining quality of a circle is that it is the set of all points in plane which are the same distant from a specific point.  Any deformation of this circle is a <math>\mathbb{S}^1 \ </math> - an object does not need to be a perfect circle to be called a one sphere by topologists, any closed loop which does not intersect itself is an example of <math>\mathbb{S}^1 \ </math>.

Mathematicians call the ordinary circle the "one sphere," which is written <math>\mathbb{S}^1 \ </math>.  The defining quality of a circle is that it is the set of all points in plane which are the same distant from a specific point.  Any deformation of this circle is a <math>\mathbb{S}^1 \ </math> - an object does not need to be a perfect circle to be called a one sphere by topologists, any closed loop which does not intersect itself is an example of <math>\mathbb{S}^1 \ </math>.

The surface of an ordinary sphere, like the Earth, is called a two sphere, or <math>\mathbb{S}^2 \ </math>.  The defining quality of a sphere is that it is the set of all points in a Euclidean (ie, normal) three dimensional space which are the same distance from a set point.  Note that the interior of a sphere is not <math>\mathbb{S}^2 \ </math>, and that like <math>\mathbb{S}^1 \ </math>, a deflated beach ball stretched into some strange shape is still <math>\mathbb{S}^2 \ </math>, so long as it does not self-intersect.

The surface of an ordinary sphere, like the Earth, is called a two sphere, or <math>\mathbb{S}^2 \ </math>.  The defining quality of a sphere is that it is the set of all points in a Euclidean (i.e., normal) three dimensional space which are the same distance from a set point.  Note that the interior of a sphere is not <math>\mathbb{S}^2 \ </math>, and that like <math>\mathbb{S}^1 \ </math>, a deflated beach ball stretched into some strange shape is still <math>\mathbb{S}^2 \ </math>, so long as it does not self-intersect.

The three sphere, or <math>\mathbb{S}^3 \ </math>, refers to the set of all points in a four dimensional Euclidean space which are the same distance from a single point.  Any deformation of that object is a <math>\mathbb{S}^3 \ </math>.

The three sphere, or <math>\mathbb{S}^3 \ </math>, refers to the set of all points in a four dimensional Euclidean space which are the same distance from a single point.  Any deformation of that object is a <math>\mathbb{S}^3 \ </math>.

===Perelman's Proof===

===Perelman's Proof===

The proof of the conjecture relies on a concept called "Ricci flow," which is time-dependent deformation of a space which expands areas which are curved "negatively" (ie, two particles traveling in the same direction diverge) and contracts area which are curved "positively" (ie, two particles traveling in the same direction converge).

The proof of the conjecture relies on a concept called "Ricci flow," which is time-dependent deformation of a space which expands areas which are curved "negatively" (i.e., two particles traveling in the same direction diverge) and contracts area which are curved "positively" (i.e., two particles traveling in the same direction converge).

Two very important insights were necessary to turn the idea of Ricci flow into a proof of the Poincare conjecture.  First, and this is the lesser of the insights, it was realized that no matter how complicated a manifold might be, the simple knowledge that a region of a three manifold could be bounded by a loop (or sphere) that could be shrunk to a point allows us to know for CERTAIN that Ricci flow on that region will either make it look like a patch of a three sphere after some finite "run time" for the flow, or "pinch" it beyond recognition (a singularity).  This insight is due to Hamilton.

Two very important insights were necessary to turn the idea of Ricci flow into a proof of the Poincare conjecture.  First, and this is the lesser of the insights, it was realized that no matter how complicated a manifold might be, the simple knowledge that a region of a three manifold could be bounded by a loop (or sphere) that could be shrunk to a point allows us to know for CERTAIN that Ricci flow on that region will either make it look like a patch of a three sphere after some finite "run time" for the flow, or "pinch" it beyond recognition (a singularity).  This insight is due to Hamilton.

The effect of the Ricci flow has much in common mathematically with the diffusion of heat in a surface, a concept called [[entropy]]. For example, look at the heat diffusion equation <math>u_t = \Delta u \ </math>.  This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take <math>u = \ln g_{jj} \ </math>, where j is any number less than the dimension of the manifold and no summation is implied on that index, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which create such a metric).

The effect of the Ricci flow has much in common mathematically with the diffusion of heat in a surface, a concept called [[entropy]]. For example, look at the heat diffusion equation <math>u_t = \Delta u \ </math>.  This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take <math>u = \ln g_{jj} \ </math>, where j is any number less than the dimension of the manifold and no summation is implied on that index, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which create such a metric).

Noticing this (and other similarities between the effect of Ricci flow on a manifolds metric and the effect of time on heat dispersion, Perelman formulated the concept of "Perelman entropy."   

Noticing this (and other similarities between the effect of Ricci flow on a manifolds metric and the effect of time on heat dispersion) Perelman formulated the concept of "Perelman entropy."   

[[Category:Mathematics]]

[[Category:Mathematics]]