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Saturday, August 19, 2006

Daily Kos and Mathematics?

What does the Daily Kos have to do with mathematics? Well, we have a "Science Friday" and this past Friday we had a gem of an article:

The Poincaré Conjecture Hotlist

Fri Aug 18, 2006 at 07:57:01 AM PDT

It's Science Friday!
We are here to blow up the brains of any Republicans who may be sneaking, yes SSSNEAKING Precious, around this board.

It was mentioned earlier, in another post (on cats) that there had been no mention of the Poincaré Conjecture Proved--This Time for Real

In mathematics, a 3-manifold is a 3-dimensional manifold. A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important. For example, lines are one-dimensional, and planes two-dimensional.

In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and a pair of circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus.

Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.

Additional structures are often defined on manifolds. Examples of manifolds with additional structure include differentiable manifolds on which one can do calculus, Riemannian manifolds on which distances and angles can be defined, symplectic manifolds which serve as the phase space in classical mechanics, and the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity.

Russian mathematician Dr. Grigori (Grisha) Perelman of the Steklov Institute of Mathematics (part of the Russian Academy of Sciences in St. Petersburg) gave a series of public lectures at the Massachusetts Institute of Technology last week. These lectures, entitled "Ricci Flow and Geometrization of Three-Manifolds," were presented as part of the Simons Lecture Series at the MIT Department of Mathematics on April 7, 9, and 11. The lectures constituted Perelman's first public discussion of the important mathematical results contained in two preprints, one published in November of last year and the other only last month.

Perelman, who is a well-respected differential geometer, is regarded in the mathematical community as an expert on Ricci flows, which are a technical mathematical construct related to the curvatures of smooth surfaces. Perelman's results are clothed in the parlance of a professional mathematician, in this case using the mathematical dialect of abstract differential geometry. In an unusally explicit statement, Perleman (2003) actually begins his second preprint with the note, "This is a technical paper, which is the continuation of [Perelman 2002]." As a consequence, Perelman's results are not easily accessible to laypeople. The fact that Perelman's preprints are intended only for professional mathematicians is also underscored by the complete absence of a single reference to Poincaré in either paper and by the presence of only a single reference to Thurston's conjecture.

Stripped of their technical detail, Perelman's results appear to prove a very deep theorem in mathematics known as Thurston's geometrization conjecture.


Thurston's conjecture has to do with geometric structures on mathematical objects known as manifolds, and is an extension of the famous Poincaré conjecture. Since Poincaré's conjecture is a special case of Thurston's conjecture, a proof of the latter immediately establishes the former.

Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds.

Before stating Thurston's geometrization conjecture in detail, some background information is useful. Three-dimensional manifolds possess what is known as a standard two-level decomposition. First, there is the connected sum decomposition, which says that every compact three-manifold is the connected sum of a unique collection of prime three-manifolds.

The second decomposition is the Jaco-Shalen-Johannson torus decomposition, which states that irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either "atoroidal" or "Seifert-fibered."

Thurston's conjecture is that, after you split a three-manifold into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries:

1. Euclidean geometry,

2. Hyperbolic geometry,

3. Spherical geometry,

4. The geometry of S^2xR,

5. The geometry of H^2xR,

6. The geometry of the universal cover SL_2R^~ of the Lie group SL_2R,

7. Nil geometry, or

8. Sol geometry.

Here, S^2 is the 2-sphere (in a topologist's sense) and H^2 is the hyperbolic plane. If Thurston's conjecture is true, the truth of the Poincaré conjecture immediately follows. Thurston shared the 1982 Fields Medal for work done in proving that the conjecture held in a subset of these cases.

Six of these geometries are now well understood, and there has been a great deal of progress with hyperbolic geometry (the geometry of constant negative scalar curvature). However, the geometry of constant positive curvature is still poorly understood, and in this geometry, the Thurston elliptization conjecture extends the Poincaré conjecture (Milnor).

Results due to Perelman (2002, 2003) appear to establish the geometrization conjecture, and thus also the Poincaré conjecture. Unlike a number of previous manuscripts attempting to prove the Poincaré conjecture, mathematicians familiar with Perelman's work describe it as well thought-out and expect that it will be difficult to locate any mistakes (Robinson 2003).

In the form originally proposed by Henri Poincaré in 1904 (Poincaré 1953, pp. 486 and 498), Poincaré's conjecture stated that every closed simply connected three-manifold is homeomorphic to the three-sphere. Here, the three-sphere (in a topologist's sense) is simply a generalization of the familiar two-dimensional sphere (i.e., the sphere embedded in usual three-dimensional space and having a two-dimensional surface) to one dimension higher. More colloquially, Poincaré conjectured that the three-sphere is the only possible type of bounded three-dimensional space that contains no holes. This conjecture was subsequently generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere. The generalized statement is now known as the Poincaré conjecture, and it reduces to the original conjecture for n = 3.

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).

Renewed interest in the Poincaré conjecture was kindled among the general public when the Clay Mathematics Institute included the conjecture on its list of million-dollar-prize problems. According to the rules of the Clay Institute, any purported proof must survive two years of academic scrutiny before the prize can be collected. A recent example of a proof that did not survive even this long was a five-page paper presented by M. J. Dunwoody in April 2002 (MathWorld news story, April 18, 2002), which was quickly found to be fundamentally flawed.

Almost exactly a year later, Perelman's results appear to be much more robust. While it will be months before mathematicians can digest and verify the details of the proof, mathematicians familiar with Perelman's work describe it as well thought out and expect that it will prove difficult to locate any significant mistakes.

Clay Mathematics Institute. "The Poincaré Conjecture."

Johnson, G. "A Mathematician's World of Doughnuts and Spheres." The New York Times, April 20, 2003, p. 5.

Perelman, G. "Ricci Flow and Geometrization of Three-Manifolds." Massachusetts Institute of Technology Department of Mathematics Simons Lecture Series.

Perelman, G. "The Entropy Formula for the Ricci Flow and Its Geometric Application." November 11, 2002.

Perelman, G. "Ricci Flow with Surgery on Three-Manifolds." March 10, 2003.

Poincaré, H. Oeuvres de Henri Poincaré, tome VI. Paris: Gauthier-Villars, pp. 486 and 498, 1953.

Robinson, S. "Russian Reports He Has Solved a Celebrated Math Problem." The New York Times, April 15, 2003, p. D3.


Go to the link to see the rest of the article, where the author applies this to politics and to see the comments.


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