العربية
中国
Français
Deutsch
Ελληνική
हिन्दी
Italiano
日本語
Português
Русский
EspañolHAMILTON-PERELMAN SOLUTION OF THE POINCARé CONJECTURE
(Redirected from Hamilton–Perelman solution of the Poincaré conjecture)
This entry provides a description of the solution of the Poincaré conjecture at a level understandable to the general public. The proof described is that of Grigori Perelman using the Ricci flow developed by Richard Hamilton. Links to other such descriptions are included below.
The Poincaré conjecture says that a 3-dimensional manifold which is compact, has no boundary and is simply connected (so lassos cannot tie around it) is a 3-dimensional sphere. To understand the statement one needs to understand "manifold", "compact", "no boundary", "simply connected" and "3-dimensional sphere" and so all these concepts are described below. Perelman and Hamilton proved the conjecture by deforming the manifold using something called the Ricci flow (which behaves a lot like the heat equation, which describes the diffusion of heat through some kind of body). The Ricci flow causes the manifold to deform towards a rounder shape, except for the possibility that it stretches apart from itself (like hot mozzarella) towards what are known as singularities. They then chop the manifold at the singularities (it is called "surgery") and watch all the separate pieces form into ball-like shapes. Major steps in the proof involved showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and wondering whether the surgery might need to be repeated infinitely many times.
A one dimensional sphere is a circle ''x''2 + ''y''2 = 1. A two-dimensional sphere is the surface of a globe. ''x''2 + ''y''2 + ''z''2 = 1. And a three-dimensional sphere is ''x''2 + ''y''2 + ''z''2 + ''w''2 = 1.
A manifold is space created by gluing together charts (which are pieces of Euclidean space). For example
you could take two 2-dimensional disks and curve them around to hemispheres and then glue them together to make a 2-dimensional
sphere.
You could also build a torus (the surface of a bagel) using a rectangular chart as seen in this image.
You can build a 3-dimensional sphere using a pair of solid 3-dimensional balls.
Other manifolds can be created in similar ways. See manifold for an easy and advanced description. We say a manifold has an edge or a boundary, if one of the charts is not glued to another on all sides. In the Poincare conjecture it is required that there be no edges,
just like in the sphere and the torus.
Manifolds can be warped or distorted using diffeomorphisms.
A compact manifold is bounded and does not extend to infinity. A cylinder and a plane are examples of manifolds which
are not compact. In Poincare's Conjecture it is required that the manifolds be compact. See compact for an advanced definition on the level of upper level mathematics majors..
A manifold is simply connected if any loop drawn on the space can be deformed to a point without leaving the
manifold. An example of a simply connected manifold is a sphere. If you try to wrap a lasso around a sphere it
will slide off. An example of a manifold which is not simply connected is a torus.
One can tie a lasso around a bagel and catch hold of it. Nothing short of untying the lasso or
cutting the bagel will let it loose. See simply connected for an easy and advanced description.
'Putting all these terms together, we can now understand the statement of the Poincare conjecture:'
The Poincaré conjecture says that a 3-dimensional manifold which is compact, has no boundary and is simply connected is a 3-dimensional sphere.
The first step is to deform the manifold using the Ricci flow. The Ricci flow was used by Richard Hamilton
as a way to deform manifolds and he used it to prove that many compact manifolds were diffeomorphic to spheres.
He did not prove they were all diffeomorphic to spheres. The Ricci flow is an imitation of the Heat flow equation
which describes the way heat permeates a room. Like the heat flow, Ricci flow tends towards uniform behavior.
Unlike the heat flow, the Ricci flow could run into singularities and stop functioning. These singularities
are the strands of mozzarella:
Hamilton was able to list a number of possible singularities that could form but he was concerned as
to whether he had found all possible singularities. He wanted to cut the manifold at the singularities
and paste in caps, and then run the Ricci flow again. But he needed to understand the singularities.
Grigori Perelman examined the singularities and discovered they were very simple manifolds: essentially three
dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking
circles stretched along a line:
This was proven using something Perelman called the "Reduced Volume" which is closely related to an eigenvalue
of a certain "elliptic equation". Eigenvalues are difficult to describe without calculus but they are part
of a famous problem: Can you hear the shape of a drum?. Essentially an eigenvalue is like a note being played
by the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helped
him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the
cigar soliton, which looked like a strand sticking out of a manifold with nothing on the other side. In essence
Perelman showed that all the strands that form can be cut and capped and none stick out on one side only.
Completing the proof, Perelman takes any compact, simply connected, three dimensional manifold without
boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running
between them. He cuts the strands and continues deforming the manifold until eventually he is left with
a collection of round three dimensional spheres. Then he rebuilds the original manifold by connecting the
spheres together with three dimensional cylinders, morphs them into a round shape and sees that, despite all
the initial confusion, the manifold was in fact diffeomorphic to a sphere.
Two immediate questions were then: how can one be sure there aren't infinitely many cuts necessary? That the
cutting does not progress forever? Perelman proved this using soap films on the manifold and showing that
the areas of the soap films decreases as the manifold undergoes Ricci flow. Eventually the area is so small that
any cut after the area is that small can only be chopping off three dimensional spheres and not more complicated pieces.
This is described as a battle with a Hydra in Szpiro's book cited below.
★ Taming the fourth dimension, by B. Schechter, New Scientist, 17 July 2004, Vol 183 No 2456
★ CNN: Russian may have solved great math mystery
★ Major math problem is believed solved, by S. Begley, Wall Street Journal, July 21, 2006 explains the current Millennium Prize situation.
★ The Shapes of Space , by G. P. Collins, Scientific American, 2004 July, pp. 94-103
★ If it looks like a sphere..., by E. Klareich, Science News Online, June 14, 2003, Vol 163, No. 24, p 376.
★ Elusive Proof, Elusive Prover: A New Mathematical Mystery, by Dennis Overbye, New York Times, Science, August 15, 2006.
★ Geometrization of Three Manifolds via the Ricci Flow, by Mike Anderson (SUNY Stony Brook), Notices of the AMS, Vol 51, Number 2, (written for mathematicians)
★ Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective by Terence Tao, unpublished arxiv.org preprint (written for mathemeticians).
★ Perelman's Song, by Tina S. Chang [1], listed on Kasman's Mathematical Fiction website, to appear in Math Horizons.
★ Structures of Three-Manifolds, for the scientifically inclined audience by Shing-Tung Yau (Harvard), June 20, 2006.
★ The Work of Grigori Perelman, talk by John Lott (University of Michigan) International Congress of Mathematicians 2006 Presentation, for mathematicians in all areas, excellent graphics
★ Perelman and the Poincare Conjecture, talk by Christina Sormani (CUNY Graduate Center and Lehman College) presented at Williams College, Wellesley College, Lehman College and Tufts University. Transparencies are posted for public use (same as the graphics above) and a guide for math professors interested in giving a similar talk (recommends studying the resources posted here).
★ Clay Mathematics Institute has a description of the Poincare Conjecture as a Millennium Problem by John Milnor (SUNY Stony Brook).
★ Intro Perelman Website by Christina Sormani, (CUNY Graduate Center and Lehman College) has the graphics included above and was used as a framework for this article and a resource for the initial set of links.
★ Who cares about Poincare?, by Jordan Ellenberg, Slate, August 18, 2006, is for the layman
★ ''Poincare's Prize'', by George Szpiro
★ ''The Poincare Conjecture: In Search of the Shape of the Universe'' by Donal O'Shea.
This entry provides a description of the solution of the Poincaré conjecture at a level understandable to the general public. The proof described is that of Grigori Perelman using the Ricci flow developed by Richard Hamilton. Links to other such descriptions are included below.
Description
The Poincaré conjecture says that a 3-dimensional manifold which is compact, has no boundary and is simply connected (so lassos cannot tie around it) is a 3-dimensional sphere. To understand the statement one needs to understand "manifold", "compact", "no boundary", "simply connected" and "3-dimensional sphere" and so all these concepts are described below. Perelman and Hamilton proved the conjecture by deforming the manifold using something called the Ricci flow (which behaves a lot like the heat equation, which describes the diffusion of heat through some kind of body). The Ricci flow causes the manifold to deform towards a rounder shape, except for the possibility that it stretches apart from itself (like hot mozzarella) towards what are known as singularities. They then chop the manifold at the singularities (it is called "surgery") and watch all the separate pieces form into ball-like shapes. Major steps in the proof involved showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and wondering whether the surgery might need to be repeated infinitely many times.
Breaking down the key terms
What is a ''3-dimensional sphere''?
A one dimensional sphere is a circle ''x''2 + ''y''2 = 1. A two-dimensional sphere is the surface of a globe. ''x''2 + ''y''2 + ''z''2 = 1. And a three-dimensional sphere is ''x''2 + ''y''2 + ''z''2 + ''w''2 = 1.
Circle is a one dimensional sphere
Surface area is a two dimensional sphere
Common Sphere is also known as a three dimensional sphere
What is a ''manifold''?
A manifold is space created by gluing together charts (which are pieces of Euclidean space). For example
you could take two 2-dimensional disks and curve them around to hemispheres and then glue them together to make a 2-dimensional
sphere.
Hemispheres mapped together to form a Sphere
You could also build a torus (the surface of a bagel) using a rectangular chart as seen in this image.
Progressive build of a torus from a rectangle
You can build a 3-dimensional sphere using a pair of solid 3-dimensional balls.
Two solid balls with flattened edges can form of sphere
Other manifolds can be created in similar ways. See manifold for an easy and advanced description. We say a manifold has an edge or a boundary, if one of the charts is not glued to another on all sides. In the Poincare conjecture it is required that there be no edges,
just like in the sphere and the torus.
Manifolds can be warped or distorted using diffeomorphisms.
What does ''compact'' mean?
A compact manifold is bounded and does not extend to infinity. A cylinder and a plane are examples of manifolds which
are not compact. In Poincare's Conjecture it is required that the manifolds be compact. See compact for an advanced definition on the level of upper level mathematics majors..
What does ''simply connected'' mean?
A manifold is simply connected if any loop drawn on the space can be deformed to a point without leaving the
manifold. An example of a simply connected manifold is a sphere. If you try to wrap a lasso around a sphere it
will slide off. An example of a manifold which is not simply connected is a torus.
One can tie a lasso around a bagel and catch hold of it. Nothing short of untying the lasso or
cutting the bagel will let it loose. See simply connected for an easy and advanced description.
Neither of the colored loops can be contracted to a point without leaving the surface.
'Putting all these terms together, we can now understand the statement of the Poincare conjecture:'
The Poincaré conjecture says that a 3-dimensional manifold which is compact, has no boundary and is simply connected is a 3-dimensional sphere.
The Hamilton-Perelman proof
The first step is to deform the manifold using the Ricci flow. The Ricci flow was used by Richard Hamilton
as a way to deform manifolds and he used it to prove that many compact manifolds were diffeomorphic to spheres.
He did not prove they were all diffeomorphic to spheres. The Ricci flow is an imitation of the Heat flow equation
which describes the way heat permeates a room. Like the heat flow, Ricci flow tends towards uniform behavior.
Unlike the heat flow, the Ricci flow could run into singularities and stop functioning. These singularities
are the strands of mozzarella:
thumb
Hamilton was able to list a number of possible singularities that could form but he was concerned as
to whether he had found all possible singularities. He wanted to cut the manifold at the singularities
and paste in caps, and then run the Ricci flow again. But he needed to understand the singularities.
Grigori Perelman examined the singularities and discovered they were very simple manifolds: essentially three
dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking
circles stretched along a line:
thumb
This was proven using something Perelman called the "Reduced Volume" which is closely related to an eigenvalue
of a certain "elliptic equation". Eigenvalues are difficult to describe without calculus but they are part
of a famous problem: Can you hear the shape of a drum?. Essentially an eigenvalue is like a note being played
by the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helped
him eliminate some of the more troublesome singularities that had concerned Hamilton, particularly the
cigar soliton, which looked like a strand sticking out of a manifold with nothing on the other side. In essence
Perelman showed that all the strands that form can be cut and capped and none stick out on one side only.
Completing the proof, Perelman takes any compact, simply connected, three dimensional manifold without
boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running
between them. He cuts the strands and continues deforming the manifold until eventually he is left with
a collection of round three dimensional spheres. Then he rebuilds the original manifold by connecting the
spheres together with three dimensional cylinders, morphs them into a round shape and sees that, despite all
the initial confusion, the manifold was in fact diffeomorphic to a sphere.
thumb
Two immediate questions were then: how can one be sure there aren't infinitely many cuts necessary? That the
cutting does not progress forever? Perelman proved this using soap films on the manifold and showing that
the areas of the soap films decreases as the manifold undergoes Ricci flow. Eventually the area is so small that
any cut after the area is that small can only be chopping off three dimensional spheres and not more complicated pieces.
This is described as a battle with a Hydra in Szpiro's book cited below.
External links
Articles
★ Taming the fourth dimension, by B. Schechter, New Scientist, 17 July 2004, Vol 183 No 2456
★ CNN: Russian may have solved great math mystery
★ Major math problem is believed solved, by S. Begley, Wall Street Journal, July 21, 2006 explains the current Millennium Prize situation.
★ The Shapes of Space , by G. P. Collins, Scientific American, 2004 July, pp. 94-103
★ If it looks like a sphere..., by E. Klareich, Science News Online, June 14, 2003, Vol 163, No. 24, p 376.
★ Elusive Proof, Elusive Prover: A New Mathematical Mystery, by Dennis Overbye, New York Times, Science, August 15, 2006.
★ Geometrization of Three Manifolds via the Ricci Flow, by Mike Anderson (SUNY Stony Brook), Notices of the AMS, Vol 51, Number 2, (written for mathematicians)
★ Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective by Terence Tao, unpublished arxiv.org preprint (written for mathemeticians).
Fiction
★ Perelman's Song, by Tina S. Chang [1], listed on Kasman's Mathematical Fiction website, to appear in Math Horizons.
Lectures
★ Structures of Three-Manifolds, for the scientifically inclined audience by Shing-Tung Yau (Harvard), June 20, 2006.
★ The Work of Grigori Perelman, talk by John Lott (University of Michigan) International Congress of Mathematicians 2006 Presentation, for mathematicians in all areas, excellent graphics
★ Perelman and the Poincare Conjecture, talk by Christina Sormani (CUNY Graduate Center and Lehman College) presented at Williams College, Wellesley College, Lehman College and Tufts University. Transparencies are posted for public use (same as the graphics above) and a guide for math professors interested in giving a similar talk (recommends studying the resources posted here).
Websites
★ Clay Mathematics Institute has a description of the Poincare Conjecture as a Millennium Problem by John Milnor (SUNY Stony Brook).
★ Intro Perelman Website by Christina Sormani, (CUNY Graduate Center and Lehman College) has the graphics included above and was used as a framework for this article and a resource for the initial set of links.
★ Who cares about Poincare?, by Jordan Ellenberg, Slate, August 18, 2006, is for the layman
Books for the general audience viewed favorably by mathematicians
★ ''Poincare's Prize'', by George Szpiro
★ ''The Poincare Conjecture: In Search of the Shape of the Universe'' by Donal O'Shea.
This article provided by Wikipedia. To edit the contents of this article, click here for original source.
psst.. try this: add to faves
