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Topic: Geometrization conjecture

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	PlanetMath: Thurston's geometrization conjecture (Site not responding. Last check: 2007-11-01)
		The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces.
		Grigori Perelman sketched a proof of the geometrization conjecture in 2003 using Ricci flow with surgery, which (as of 2006) appears to be essentially correct.
		A geometric structure on a manifold is an isomorphism of the manifold with
	www.planetmath.org /encyclopedia/ThurstonsGeometrizationConjecture.html (769 words)

	Geometrization conjecture - Biocrawler (Site not responding. Last check: 2007-11-01)
		The geometrization conjecture, also known as Thurston's geometrization conjecture, concerns the geometric structure of compact 3-manifolds.
		It 'includes' other conjectures, such as the Poincaré conjecture and the Thurston elliptization conjecture.
		Grigori Perelman may have now solved the Geometrization conjecture (and thus also the Poincaré Conjecture) and there seems to be a consensus among experts that the proof is correct, at least in the case of 3-manifolds with finite fundamental group.
	www.biocrawler.com /encyclopedia/Geometrization_conjecture (525 words)

	Geometrization conjecture - Wikipedia, the free encyclopedia
		The geometrization conjecture, also known as Thurston's geometrization conjecture, concerns the geometric structure of compact 3-manifolds.
		The Fields Medal was awarded to Thurston in 1982 partially for his proof of the conjecture for Haken manifolds.
		Grigori Perelman has offered a proof of the geometrization conjecture, and there seems to be a consensus among experts that the proof is correct.
	en.wikipedia.org /wiki/Thurston's_Geometrization_Conjecture (536 words)

	Poincaré conjecture
		The conjecture has induced a long list of false proofs, and some of them have led to a better understanding of low-dimensional topology.
		The only parts of the Geometrization Conjecture left to be proven are called the Hyperbolization Conjecture and the Elliptization Conjecture.
		The Elliptization Conjecture states that every closed 3-manifold with finite fundamental group has a spherical geometry, i.e.
	www.ebroadcast.com.au /lookup/encyclopedia/po/Poincare_conjecture.html (430 words)

	Thurston elliptization conjecture: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-01)
		The Elliptization Conjecture is a special case of Thurston's Geometrization Conjecture The geometrization conjecture, also known as thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds....
		Poincaré conjecture In mathematics, the poincaré conjecture is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds....
		Geometrization conjecture The geometrization conjecture, also known as thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds....
	www.absoluteastronomy.com /t/thurston_elliptization_conjecture (672 words)

	Ian's home page
		One of the most important conjectures in 3-manifold topology is the geometrization conjecture of Thurston.
		The beautiful thing about the geometrization conjecture is that it gives a complete invariant, and moreover this invariant is a canonical rigid geometric structure (or decomposition into finitely many such structures), which Thurston compares to a crystal, a unique hardening of this flabby topological object.
		The conjecture also implies the Poincare conjecture, that a simply connected compact 3-manifold is homeomorphic to the 3-sphere, for which the solution is now worth $1,000,000.
	www.math.uic.edu /~agol/blog/030226.html (1292 words)

	Thurston's conjecture
		The Geometrization Conjecture, also known as Thurston's Geometrization Conjecture, concerns the geometric structure of compact 3-dimensional manifolds.
		The case of 3-manifolds that should be spherical has been slower, but provided the spark needed for Richard S. Hamilton[?] to develop his Ricci flow[?].
		Grigori Perelman may have now solved the Geometrization Conjecture (and thus also the Poincaré Conjecture) but because this latter makes Perelman eligible for a million dollar Clay Millennium Prize[?] his work will need to survive two years of systematic scrutiny before the conjecture(s) will be deemed to have been solved.
	ebroadcast.com.au /lookup/encyclopedia/th/Thurston's_conjecture.html (415 words)

	An Intro to Perelman
		This conjecture states that the only compact three dimensional simply connected manifold is a three dimensional sphere.
		The Poincare Conjecture is one of the conjectures on their list of Milleneum Problems and they will soon determine whether they believe it has been proven and grant a one million dollar award to the mathematicians whose work led to the proof.-->
		He was not as geometric in his examination of mathematics as Perelman.
	comet.lehman.cuny.edu /sormani/others/perelman/introperelman.html (1915 words)

	Clay Research Award
		This conjecture posits an essentially geometric necessary and suffcient condition, "Psi", for a pseudo-differential operator of principal type to be locally solvable, i.e., for the equation Pu = f to have local solutions given a finite number of conditions on f.
		The lemma is a conjectured identity between orbital integrals for two groups, e.g., the unitary groups U(n) and U(p)xU(q), where p+q = n.
		For his work in combining analytic power with geometric insight in the field of random walks, percolation, and probability theory in general, especially for formulating stochastic Loewner evolution.
	www.claymath.org /research_award (971 words)

	Encyclopedia Search
		Conjecture left to be proven are called the Hyperbolization...
		Conjecture left to be proven are called the Hyperbolization...a proof of the
		conjecture, carrying out a program outlined earlier by Richard...
	www.encyclopedian.com /search.php?searWords=Geometrization (112 words)

	Is the Poincaré Conjecture Proved At Last?
		Poincaré conjectured, early in the 20th century, that this property in fact characterizes the three-sphere, that is, that any other compact three-dimensional manifold with the loop contracting property would have to be homeomorphic to S³.
		Analogues of the conjecture in dimension five and higher were proved in the 1960s by Smale, Stallings, and Wallace.
		Interest in the conjecture was heightened by the fact that in 2000 the Clay Mathematics Institute made it one of its seven Millennium Problems, offering one million dollars for its solution.
	www.maa.org /news/080806poincare.html (573 words)

	If It Looks Like a Sphere...: Science News Online, June 14, 2003 (Site not responding. Last check: 2007-11-01)
		Poincaré's conjecture is one of the simplest possible questions to ask about three-dimensional spaces, yet it has stumped mathematicians from Poincaré's time to the present.
		Thurston, who proved large portions of his conjecture, was awarded a Fields Medal—mathematics' version of a Nobel prize—in large part for this body of work.
		In the paper, he writes that his work "removes the major stumbling block in Hamilton's approach to geometrization." Although the posted paper makes no reference to the Poincaré conjecture, experts in the field immediately realized what he was driving at.
	www.sciencenews.org /20030614/bob10.asp (2652 words)

	The Poincare conjecture
		The Poincare Conjecture is essentially the first conjecture ever made in topology; it asserts that a 3-dimensional manifold is the same as the 3-dimensional sphere precisely when a certain algebraic condition is satisfied.
		The conjecture was formulated by Poincare around the turn of the 20th century.
		The Generalized Poincare Conjecture states that for every n, an n-dimensional manifold homotopy equivalent to the n-sphere is homeomorphic to the n-sphere.
	www.math.unl.edu /~mbrittenham2/ldt/poincare.html (2822 words)

	Thurston's conjecture Info - Bored Net - Boredom (Site not responding. Last check: 2007-11-01)
		Six of the eight geometries above are now clearly understood and known to correspond to Seifert manifolds and certain torus bundles.
		The case of 3-manifolds that should be spherical has been slower, but provided the spark needed for Richard Hamilton to develop his Ricci flow.
		Grigori Perelman may have now solved the Geometrization Conjecture (and thus also the Poincaré Conjecture) but because this latter makes Perelman eligible for a million dollar Millennium Prize Problems his work will need to survive two years of systematic scrutiny before the conjecture(s) will be deemed to have been solved.
	www.borednet.com /e/n/encyclopedia/t/th/thurston_s_conjecture.html (397 words)

	Scientific.ru » Общий форум
		The geometrization conjecture is a more general statement about three-dimensional 'surfaces', derived in the late 1970s by William Thurston, now at Cornell University in Ithaca, New York.
		Poincaré's unsolved conjecture is a limited case of Thurston's theory.
		Poincarand#233;'s unsolved conjecture is a limited case of Thurston's theory.
	www.scientific.ru /dforum/common/1075340453 (3456 words)

	Crazy Ben’s (mostly) Poetry Blog » 2005 » December (Site not responding. Last check: 2007-11-01)
		The Ricci flow was introduced by Richard Hamilton in 1981 in order to gain insight into the geometrization conjecture of William Thurston, which concerns the topological classification of three-dimensional smooth manifolds.
		In particular, the result of geometrization may be a geometry that is not isotropic.
		Grigori Perelman may have now solved the Geometrization conjecture (and thus also the Poincaré Conjecture) and there seems to be a consensus among experts that the proof is correct, at least in the case of 3-manifolds with finite fundamental group.
	hardisty.blog.usf.edu /2005/12 (3732 words)

	The biggest science breakthrough of the year
		For, although Poincare is generally thought of as one of the greatest mathematicians of all time, his interests were as much in physics as in mathematics, and he came within a whisker of formulating relativity theory before Einstein beat him to the punch.
		In fact, the more general Geometrization Conjecture that Perelman's argument also established tells us we can in principle determine a great deal more about the shape of the universe.
		Their story led off with a conjecture of its own: that the proof of the Poincare conjecture may turn out to be the number 1 math story of the entire 21st century.
	www.maa.org /devlin/devlin_12_06.html (1047 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		My colleague, Dan Burghelea, believes that there is a major development, if not a complete proof of Thurston's Geometrization Conjecture for 3-manifolds (and hence also of the Poincare Conjecture).
		The paper in question is by Grisha Perelman (of the Steklov Institute in St. Petersburg), entitled "The entropy formula for the Ricci flow and its geometric applications", which was deposited in the Math.
		Moreover Perelman refers to another 1999 paper of Hamilton on the latter's program to prove the Geometrization Conjecture and concludes his introduction with the following tantalizing statement: "We have not been able to confirm Hamilton's hope...
	www.lehigh.edu /~dmd1/zf2.txt (273 words)

	Is math a young man's game? - By Jordan Ellenberg - Slate Magazine
		Perelman claims to have proved Thurston's geometrization conjecture, a daring assertion about three-dimensional spaces that implies, among other things, the truth of the century-old Poincaré conjecture.
		And it's the Poincaré conjecture that, courtesy of the Clay Foundation, carries a million-dollar bounty.
		Poincaré couldn't have made this conjecture absent his years of study of topology, or the earlier theorems he'd carefully proved, or the earlier conjectures on the same theme he'd tried out and found to be false.
	www.slate.com /id/2082960 (1528 words)

	No Title (Site not responding. Last check: 2007-11-01)
		About a year ago the Russian mathematician Grigori Perelman announced a proof of the Geometrization Conjecture.
		This is concerned with the existence of certain geometric structures on 3-dimensional manifolds.
		It is known that the Geometrization Conjecture implies the famous Poincare Conjecture, a central problem in topology, open since 1904, and one of the seven Clay Problems.
	math.boisestate.edu /colloquia/KaiserTalk.html (85 words)

	Dr. William H. Jaco
		This lecture gives a slightly new approach to the classification of 2-manifolds and a new proof of the topological invariance of Euler characteristic for 2-manifolds; we discuss difficulties of extending these methods to the classification of 3-manifolds; and show the impossibility of classifying n-manifolds for larger than 3.
		An algorithm is given for constructing the JSJ-decomposition of a 3-manifold and deriving the Seifert invariants of the Characteristic submanifold.
		This lecture discusses the Geometrization Conjecture, the eight locally homogeneous geometries for 3-manifolds and Perelman's Claim of a solution to the Geometrization Conjecture and its implications.
	www.math.okstate.edu /~jaco/pekinglectures.htm (415 words)

	Citebase - Splitting homomorphisms and the Geometrization Conjecture (Site not responding. Last check: 2007-11-01)
		This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds.
		The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture, and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.
		SPLITTING HOMOMORPHISMS AND THE GEOMETRIZATION CONJECTURE 11 [6] D. Gabai, Convergence groups are Fuchsian groups, Annals of Math.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9906124 (604 words)

	[No title]
		This is a response of mine to a post by Greg Kuperberg on sci.math.research: Dear Greg, My understanding of what he is claiming is that, while he can't yet fully prove Hamilton's conjectures, he can prove enough of them to get the geometrization conjecture.
		Moreover in his abstract he writes: We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.
		Perhaps he meant to use some other word than "eclectic", but if you take him literally, then he is claiming the geometrization conjecture.
	www.lehigh.edu /~dmd1/zf4.txt (243 words)

	ResearchChannel - Perelman's Work on the Thurston's Geometrization Conjecture - 2
		This will be a series of three lectures on Perelman's work, aimed at a general mathematical audience.
		The first lecture will briefly review Thurston's conjecture and its consequences, and then go through an outline of Perelman's argument.
		The second and third lectures will go into somewhat more detail, covering key points in the proof, such as the classification of finite time singularities, and (if time permits) Ricci flow with surgery.
	www.researchchannel.org /prog/displayevent.asp?rid=2980 (107 words)

	UC Davis Math: FALL 1996 NEWSLETTER (Site not responding. Last check: 2007-11-01)
		This observation eventually became the Geometrization Conjecture and the Geometrization Theorem.
		The Conjecture asserts that every closed 3-manifold, after cutting along essential spheres and tori, has one of eight geometries, the richest one being hyperbolic geometry.
		The Geometrization Conjecture reaches even further; it subsumes the Poincare conjecture and would provide a classification of 3-manifolds.
	www.math.ucdavis.edu /research/newsletters/1996 (7803 words)

	Citebase - Thurston's Geometrization Conjecture and cosmological models (Site not responding. Last check: 2007-11-01)
		Motivated by Thurston's Geometrization Conjecture, we give a formulation for constructing spatially compact composite spacetimes as solutions for the Einstein equations.
		Such composite spacetimes are built from the spatially compact locally homogeneous vacuum spacetimes which have two commuting Killing vectors by gluing them through a timelike hypersurface admitting a homogeneous spatial slice spanned by the commuting Killing vectors.
		We consider a parabolic-like systems of differential equations involving geometrical quantities to examine uniformization theorems for two- and three-dimensional closed orientable manifolds.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/0010002 (1041 words)

	Page034
		April 15, 2003: There is a hope that the Poincare conjecture might be
		Determining whether Poincare conjecture is true or false is one of the seven
		conjecture is posted on the internet by Sergey Nikitin at Arizona State University.
	www.math.utoledo.edu /~jevard/Page034.htm (601 words)

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