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	Reference.com/Encyclopedia/Wolfgang Haken
		Wolfgang Haken (born June 21, 1928) is a mathematician who specializes in topology, in particular 3-manifolds.
		In 1976 together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved one of the most famous problems in mathematics, the four-color theorem.
		Haken has introduced several important ideas, including Haken manifolds, Kneser-Haken finiteness, and an expansion of the work of Kneser into a theory of normal surfaces.
	www.reference.com /browse/wiki/Wolfgang_Haken (184 words)

	Haken manifold - Wikipedia, the free encyclopedia
		In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that contains a two-sided incompressible surface.
		Haken manifolds are named after Wolfgang Haken, who pioneered the use of incompressible surfaces.
		We will consider only the case of orientable Haken manifolds, as this simplifies the discussion; a regular neighborhood of an orientable surface in an orientable 3-manifold is just a "thickened up" version of the surface.
	en.wikipedia.org /wiki/Haken_manifold (500 words)

	Haken manifold at AllExperts
		Haken manifolds are named after Wolfgang Haken, who pioneered the use of incompressible surfaces.
		Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one, but it was left to Jaco and Oertel, almost 20 years later, to show there was an algorithm to determine if a 3-manifold was Haken.
		Friedhelm Waldhausen proved that closed Haken manifolds are topologically rigid: roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism (for the case of boundary, a condition on peripheral structure is needed).
	en.allexperts.com /e/h/ha/haken_manifold.htm (753 words)

	Wolfgang Haken Biography \| World of Mathematics
		Wolfgang Haken was born in Berlin in 1928.
		At this time Haken was still carrying out research in mathematics and his discovery of a mathematical technique for discovering if a knot is knotted or not resulted in an invitation to become a visiting professor at the University of Illinois.
		Haken is still active in his research interests in has maintained links with the University of Illinois where he is emeritus professor.
	www.bookrags.com /biography/wolfgang-haken-wom (243 words)

	[No title]
		Wolfgang Haken was born in Berlin on June 21, 1928.
		Haken discovered a finite procedure for deciding whether a knot is knotted or unknotted.
		Haken was a temporary member of the Institute for Advanced Study in Princeton from 1963 to 1965.
	www.facstaff.bucknell.edu /udaepp/090/w3/ryanp.htm (1535 words)

	Four color theorem - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-01)
		It was not until 1976 that the four-color conjecture was finally proven by Kenneth Appel and Wolfgang Haken at the University of Illinois.
		After many faulty attempts the theorem was proved by Kenneth Appel and Wolfgang Haken, publicly announced 22 July 1976, and published in 1977.
		Appel, Kenneth and Haken, Wolfgang and Koch, John, Every Planar map is Four Colorable, Illinois: Journal of Mathematics: vol.21: pp.439-567, December 1977.
	www.kernersville.us /project/wikipedia/index.php/Four_color_problem (1535 words)

	Wolfgang Haken -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-01)
		Wolfgang Haken (born June 21, 1928) is a (A person skilled in mathematics) mathematician who specialized in (The configuration of a communication network) topology, in particular (Click link for more info and facts about 3-manifold) 3-manifolds.
		Haken has introduced several important ideas, including (Click link for more info and facts about Haken manifolds) Haken manifolds, Kneser-Haken finiteness, and an expansion of the work of (Click link for more info and facts about Kneser) Kneser into a theory of normal surfaces.
		Much of his work has an algorithmic aspect, and he is one of the influential figures in (Click link for more info and facts about algorithmic topology) algorithmic topology.
	www.absoluteastronomy.com /encyclopedia/w/wo/wolfgang_haken.htm (159 words)

	Science News: Knot possible: better ways to tell a tangled circle from a knotted loop - mathematical knot research - ... (Site not responding. Last check: 2007-11-01)
		Haken's algorithm involves not the tangled loop itself but the imagined surface for which the loop serves as a boundary.
		The method that Haken formulated, however, is so complicated that no one has yet been able to write a fully practical computer program to follow the necessary steps and, for a given projection, give a "yes" or "no" answer to the question of whether it's an unknot.
		Departing significantly from Haken's approach, Birman and Hirsch's novel algorithm for identifying unknots is built upon the idea of considering a knot or an unknot as a closed braid.
	www.findarticles.com /p/articles/mi_m1200/is_23_160/ai_81790567 (1531 words)

	Characterisation of the three sphere following Haken (ResearchIndex)
		Haken characterisation of Colin Rourke Colin Rourke The purpose of
		Abstract: this paper is to give an exposition of some work of Wolfgang Haken dating from the 1960's of which details have never been published.
		1 Algebraically trivial decompositions of homotopy 3--spheres (context) - Haken - 1968
	citeseer.ist.psu.edu /293493.html (211 words)

	Andrew Yang
		It was not until 1976 that Wolfgang Haken and Kenneth Appel proved the theorem using considerably more advanced techniques and equipment than their predecessors.
		In the early 1970s, mathematicians Wolfgang Haken and Kenneth Appel began to study Heesch’s work in more depth and realized that there seemed to be certain easily identifiable characteristics of configurations that made them reducible.
		It is possible to check that Appel and Haken’s program, when used by an infallible computer, produces a valid proof, but infallible computers do not exist.
	www.its.caltech.edu /~sciwrite/journal03/yang.html (3235 words)

	Four color theorem
		It is obvious that three colors are completely inadequate, and mathematicians were able to prove that five colors were sufficient to color a map.
		However, it was not until 1977 that the conjecture was finally proven by Ken Appel[?] and Wolfgang Haken[?].
		Appel, K. and Haken, W., Every Planar Map is Four-Colorable.
	www.ebroadcast.com.au /lookup/encyclopedia/fo/Four_color_theorem.html (544 words)

	Math Pioneers
		Ken Appel and colleague Wolfang Haken proved tha any map can be drawn using just four colors in one of the most famous mathematical proofs.
		But in the early '70s, Appel and Wolfgang Haken decided to use a computer to prove the four-color theorem, which was developed centuries ago by mapmakers trying to minimize the number of inks they had to use.
		As is often the case in high-level mathematics, finding a solution required an oblique approach: Appel and Haken used the classic method of "proof by contradiction." First, they assumed the existence of a map that required more than four colors.
	www.unhmagazine.unh.edu /sp02/mathpioneers.html (924 words)

	Newsletter Item (Site not responding. Last check: 2007-11-01)
		In the second lecture, Wolfgang Haken (Illinois) and Kenneth Appel (New Hampshire) presented some delightful reminiscences of Heinrich Heesch and outlined his pioneering contributions to the problem.
		Although their work was based on Heesch’s work, they eventually headed in a different direction, seeking unavoidable sets of ‘likely-to-be-reducible’ configurations, rather than producing reducible configurations by the thousands and then trying to package them into unavoidable sets.
		Appel and Haken outlined the challenges posed by their computer-assisted attack on the problem, and how they overcame them.
	www.lms.ac.uk /newsletter/0212/articles.html (1045 words)

	About "Maps of Many Colors" (Site not responding. Last check: 2007-11-01)
		The four-color problem has intrigued and stumped professional and amateur mathematicians alike ever since it was first proposed in 1852 by a British graduate student, Francis Guthrie, in a letter to his younger brother.
		In 1976, Kenneth Appel and Wolfgang Haken of the University of Illinois announced that they had finally proved the four-color theorem...
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/view/4964.html (154 words)

	Four color theorem - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-01)
		However, he was never able to produce a proof.
		If A consisted of three regions, six or more colors might be required; one can construct maps that require an arbitrarily high number of colors.
		The proof of the Four Color Theorem is not simple; it involved more than 1,000 hours of computing time on a 1977 computer checking more than 100,000 particular cases.
	www.sevenhills.us /project/wikipedia/index.php/Four_color_theorem (1535 words)

	Amazon.ca: Books: Four Colors Suffice: How the Map Problem Was Solved (Site not responding. Last check: 2007-11-01)
		It was only when the problem was reduced to a set of special cases that could be examined by a computer that a conclusive "proof" was finally derived by a computer in 1976.
		Wolfgang Haken worked on the theorem, and was told by computer experts that his ideas could not be programmed, but programmer Kenneth Appel disagreed.
		One example: when Haken and Appel needed referees to check their paper, one of them was a mathematician who was bitterly disappointed that his own proof had not scooped them.
	www.amazon.ca /exec/obidos/ASIN/0691120234 (1984 words)

	Mathematical Programming Society Prizes
		While research work in these areas is usually not far removed from practical applications, the judging of papers will be based on their mathematical quality and significance.
		The first awards of the Prize were made at the Tenth International Symposium on Mathematical Programming in 1979: to Richard M. Appel and Wolfgang Haken (proof of the four color theorem); to Richard M. Karp (computational complexity of combinatorial problems); and to Paul D. Seymour (matroids and the max-flow min-cut property).
		Kenneth Appel and Wolfgang Haken, "Every planar map is four-colorable, Part I: Discharging", Illinois J. Math 21 (1977), 429-490.
	www.mathprog.org /prz/fulkerson.htm (1120 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Recently, Haken's algorithm has been analyzed--and, incidentally, > explicated--by Joel Hass of the University of California at Davis, > Jeffrey C. Lagarias of AT&T Laboratories and [Nicholas] Pippenger.
		> Haken gave a procedure for determining whether or not the surface is a disk.
		It is interesting to note that Haken's algorithm only recognizes the unknot.
	www.math.niu.edu /~rusin/known-math/97/unknot.progs (502 words)

	Lippold Haken's Family Pictures
		My parents, Anna-Irmgard and Wolfgang, soon emigrated to the United States because my father was offered a math professorship at the University of Illinois in Urbana-Champaign.
		Carl Francis was born in 1995, Dawn Marie in 1997, James Mark in 2000, and Paul Wolfgang in 2003.
		Lippold on a Saturday Hike with Carl in the backpack, and Dawn in the sling.
	www.cerlsoundgroup.org /HakenFamily (1064 words)

	The Straight Dope: Did they finally solve the four-color map problem? (Site not responding. Last check: 2007-11-01)
		I don't know that I would call 1976 "recently," but yes, the four-color map problem was solved (more or less) using a computer by two prairie geniuses at the University of Illinois at Champaign-Urbana, Wolfgang Haken and Kenneth Appel.
		The four-color map problem, as all mathematically hip personages know, is to determine whether there is any map that requires the use of more than four different colors if you want to avoid having adjacent regions be the same color.
		Haken and Appel proved that (as was widely suspected) four colors are all you ever need.
	www.straightdope.com /classics/a1_126b.html (329 words)

	Homework 4 - ECS 110
		In 1976 Kenneth Appel and Wolfgang Haken proved one of the most celebrated mathematical results of the century: that every planar map can be four-colored.
		Suppose you draw a map on a piece of paper (or a sphere), making a bunch of regions (e.g., states), each of which is a single connected piece.
		Appel and Haken's proof actually gives an algorithm--a terribly complicated one--which runs in polynomial time and manages to color any planar map with just four colors.
	www.cs.ucdavis.edu /~rogaway/classes/110/spring97/hw4/hw4.html (736 words)

	Math G Mission College Santa Clara
		The proof achieved by Kenneth Appel and Wolfgang Haken based their methods of reducibility on the Kempe chains.
		Appel and Hakenís work is referred to as a ìproof,î and the Four Color Conjecture is now noted as the Four Color Theorem, denoting some finality in the problem.
		According to the summary of the Robertson, Seymore, Thomas and Sanders proof, not even the part of the Appel-Haken proof which is supposedly ìhand-checkableî had ever been verified in its entirety due to its complexity and the tediousness of the task.
	www.missioncollege.org /Depts/Math/beard2.htm (2429 words)

	Amazon.com: Four Colors Suffice: How the Map Problem Was Solved: Books: Robin Wilson (Site not responding. Last check: 2007-11-01)
		Wolfgang Haken worked on the theorem, and was told by computer experts that his ideas could not be programmed, but programmer Kenneth Appel disagreed.
		One example: when Haken and Appel needed referees to check their paper, one of them was a mathematician who was bitterly disappointed that his own proof had not scooped them.
		It was only when the problem was reduced to a set of special cases that could be examined by a computer that a conclusive "proof" was finally derived by a computer in 1976.
	www.amazon.com /Four-Colors-Suffice-Problem-Solved/dp/0691115338 (2920 words)

	EducationGuardian.co.uk \| eG weekly \| Devilish digits
		The four-color map problem was finally solved, by Wolfgang Haken and Kenneth Appel, in 1976.
		Haken and Appel became instantly famous among mathematicians.
		But little academic or public acclaim came to Faid, perhaps because no one had previously realised that the identity of the Antichrist was a mathematical problem.
	education.guardian.co.uk /egweekly/story/0,5500,1213378,00.html (422 words)

	Last doubts removed about the proof of the Four Color Theorem
		In 1976, two mathematicians at the University of Illinois, Kenneth Appel and Wolfgang Haken, announced that they had solved the problem.
		Appel and Haken had to identify and examine around 1500 different ways that a node could be appropriately removed and show that any minimal counterexample network must contain at least one node of one of those 1500 kinds.
		Appel and Haken started their computer-assisted investigation in 1972 and four years later they had their answer.
	www.maa.org /devlin/devlin_01_05.html (1470 words)

	FOUR COLOURS SUFFICE - - Penguin Books
		Kenneth Appel and Wolfgang Haken's eventual and hard-won solution, which involved 1,200 hours of computer time, nearly 10 years' work, and their five children to help finish it, was greeted with great enthusiasm, scepticism and concern about the use of the computer.
		He will be exploring the four-colour problem at Gresham College, London on Monday 21st October, The Mathematics Institute, Oxford on Tuesday 22nd October and at The London Mathematical Society on Wednesday 23rd October - the 150th anniversary of the problem.
		Wolfgang Haken and Kenneth Appel will be in the UK in the week of publication.
	www.penguin.ca /nf/Book/BookDisplay/0,,9780713996708,00.html (593 words)

	[No title]
		Wolfgang Haken and Kenneth Appel The solution provided by Wolfgang Haken and Kenneth Appel in 1976 was controversial because they used a computer to generate and color all of the possible types of maps.
		Haken and Appel’s Solution In 1976, Wolfgang Haken and Kenneth Appel used a computer program to undertake proving the Four Color Theorem.
		After over one hundred years, Wolfgang Haken and Kenneth Appel contributed a precise resolution to the matter, albeit not everyone accepts the solution provided by a computer program.
	comp.uark.edu /~zkali/Fourcolortheorem.doc (3360 words)

	BBC - Radio 4 - Another 5 Numbers - The Number Four
		Then in 1976, Americans Kenneth Appel and Wolfgang Haken postulated a ground-breaking theory, based on an elaboration of Kempe's work.
		It was the first major work of its kind to be proved using a computer, something that the old school mathematicians of the time found controversial.
		Independent verification and the test of time have convinced most sceptics that Appel and Haken did crack it.
	www.bbc.co.uk /radio4/science/another51.shtml (576 words)

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