Solution of the Poincaré conjecture

This entry describes the solution of the Poincaré conjecture at a level intended for the general public. The proof described is that of Grigori Perelman using the Ricci flow developed by Richard Hamilton. Links to other expositions for general readers are included below, together with links to the original research papers.

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 similarly to the heat equation that describes the diffusion of heat through an object). The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself (like hot mozzarella) towards what are known as singularities. Perelman and Hamilton then chop the manifold at the singularities (a process called "surgery") causing the separate pieces to form into ball-like shapes. Major steps in the proof involve 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, which can be thought of as the set of points, ("x," "y"), in two dimensions that satisfy the equation "x"² + "y"² = "r" ², where "r" is the radius. A two-dimensional sphere is the surface of a globe, or the set of points, ("x," "y," "z") in three dimensions that satisfy the equation "x"² + "y"² + "z"² = "r" ². And a three-dimensional sphere is the set of points in four dimensions, ("x," "y," "z," "w"), that satisfy the equation "x"² + "y"² + "z"² + "w"² = "r" ².

3-sphere is one dimension higher, the boundary of a four-dimensional ball

What is a "manifold"?

A manifold is a space created by gluing together pieces of Euclidean space, called charts. For exampleyou could take two 2-dimensional disks and curve them around to hemispheres and then glue them together to make a 2-dimensionalsphere.

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: Identify each point of the boundary of the first ball with the corresponding point of the second ball.

Other manifolds can be created in similar ways. See manifold for an easy and advanced description. Manifolds can be warped or distorted using diffeomorphisms.

What does "no boundary" mean?

We say a manifold has an edge or a boundary, if one of the charts is not glued to another on all sides. One of the conditions in the Poincaré conjecture is that there be no edges,just like in the sphere and the torus.

What does "compact" mean?

A compact manifold is bounded and does not extend to infinity. Both an infinitely long cylinder and an infinite plane are examples of manifolds whichare not compact. In Poincaré's Conjecture it is required that the manifolds be compact. See compact for an advanced definition.

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 themanifold. An example of a simply connected manifold is a sphere. If you try to wrap a lasso around a sphere itwill 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 orcutting 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 Poincaré conjecture:

The Poincaré conjecture says that a 3-dimensional manifold which is compact, has no boundary and is simply connected must be a 3-dimensional sphere.

Perelman's proof based on Hamilton's Ricci flow

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 equationwhich 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 singularitieshave been likened to the strands in mozzarella cheese.

Hamilton was able to list a number of possible singularities that could form but he was concerned asto whether he had found all possible singularities. He wanted to cut the manifold at the singularitiesand 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 threedimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by takingcircles stretched along a line.

This was proven using something Perelman called the "Reduced Volume" which is closely related to an eigenvalueof a certain "elliptic equation". Eigenvalues are difficult to describe without calculus but they are partof a famous problem: Can you hear the shape of a drum?. Essentially an eigenvalue is like a note being playedby the manifold. Perelman proved this note goes up as the manifold is deformed by the Ricci flow. This helpedhim 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 essencePerelman 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 withoutboundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands runningbetween them. He cuts the strands and continues deforming the manifold until eventually he is left witha collection of round three dimensional spheres. Then he rebuilds the original manifold by connecting thespheres together with three dimensional cylinders, morphs them into a round shape and sees that, despite allthe 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 thecutting does not progress forever? Perelman proved this using soap films on the manifold and showing thatthe areas of the soap films decreases as the manifold undergoes Ricci flow. Eventually the area is so small thatany 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

* [http://www.newscientist.com/article.ns?id=mg18324565.000 Taming the fourth dimension] , by B. Schechter, New Scientist, 17 July 2004, Vol 183 No 2456
* [http://www.cnn.com/2004/US/West/01/07/math.mystery.ap/index.html CNN: Russian may have solved great math mystery]
* [http://www.azcentral.com/offbeat/articles/0721wsj-science-journal21-ON.html Major math problem is believed solved] , by S. Begley, Wall Street Journal, July 21, 2006 explains the current Millennium Prize situation.
* [http://www.scientificamericandigital.com/index.cfm?fa=Products.ViewIssuePreview&ARTICLEID_CHAR=1550EBB0-2B35-221B-6A293BA37A12BAFF The Shapes of Space] , by G. P. Collins, Scientific American, 2004 July, pp. 94-103
* [http://www.sciencenews.org/articles/20030614/bob10.asp If it looks like a sphere...] , by E. Klarreich, Science News Online, June 14, 2003, Vol 163, No. 24, p 376.
* [http://www.nytimes.com/2006/08/15/science/15math.html?pagewanted=1&_r=1 Elusive Proof, Elusive Prover: A New Mathematical Mystery] , by Dennis Overbye, New York Times, Science, August 15, 2006.
* [http://www.ams.org/notices/200402/fea-anderson.pdf 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)
* [http://arxiv.org/abs/math.DG/0610903 Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective] by Terence Tao, unpublished arxiv.org preprint (written for mathematicians).

Fiction

* [http://math.cofc.edu/kasman/MATHFICT/mfview.php?callnumber=mf550 Perelman's Song] , by Tina S. Chang [http://blog.360.yahoo.com/blog-5CcC9sQlcqhUDN1P5SbXoWa5x.0-?cq=1&p=4] , listed on [http://math.cofc.edu/kasman/MATHFICT/default.html Kasman's Mathematical Fiction website] , to appear in [http://www.maa.org/mathhorizons/ Math Horizons] .

Lectures

* [http://arxiv.org/abs/math.DG/0607821 Structures of Three-Manifolds] , for the scientifically inclined audience by Shing-Tung Yau (Harvard), June 20, 2006.
* [http://www.math.lsa.umich.edu/~lott/seminarnotes.html 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
* [http://comet.lehman.cuny.edu/sormani/others/perelman/introperelman.html 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

* [http://www.claymath.org/millennium/Poincare_Conjecture/ Clay Mathematics Institute] has a description of the Poincaré Conjecture as a Millennium Problem by John Milnor (SUNY Stony Brook).
* [http://comet.lehman.cuny.edu/sormani/others/perelman/introperelman.html 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.
* [http://www.slate.com/id/2147954 Who cares about Poincare?] , by Jordan Ellenberg, [http://www.slate.com Slate] , August 18, 2006, is for the layman
* [http://www.gang.umass.edu/~kusner/other/3mfd.html A Bit of Cosmic Background] , by Robert Kusner, UMass Math Dept Newsletter 2007.
* [http://terrytao.wordpress.com/category/teaching/285g-poincare-conjecture/ Lectures on Perelman's proof] by T. Tao.

Research papers

See History of Perelman's proof and elaborations for detailed chronology.

Perelman's papers

Perelman's original papers containing the proof:
*cite journal|author=Perelman, Grisha|title=The entropy formula for the Ricci flow and its geometric applications|id = arxiv|archive=math.DG|id=0211159 | date = November 11 2002
*cite journal|author=Perelman, Grisha|title=Ricci flow with surgery on three-manifolds|id=arxiv|archive=math.DG|id=0303109 | date = March 10 2003
*cite journal|author=Perelman, Grisha|title=Finite extinction time for the solutions to the Ricci flow on certain three-manifolds|id=arxiv|archive=math.DG|id=0307245|date = July 17 2003

Detailed

* Bruce Kleiner, John Lott "Notes on Perelman's papers" arxiv|archive=math|id=0605667
* | date = December 3 2006

* John W. Morgan, Gang Tian "Ricci Flow and the Poincaré Conjecture" arxiv|archive=math|id=0607607:Detailed proof, expanding Perelman's papers.

Books for the general audience viewed favorably by mathematicians

* "Poincaré's Prize", by George Szpiro
* "The Poincaré Conjecture: In Search of the Shape of the Universe" by Donal O'Shea.