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Topic: Millennium Prize Problems

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	NationMaster - Encyclopedia: Clay Mathematics Institute (Site not responding. Last check: 2007-10-19)
		In announcing the prize, CMI drew a parallel to Hilbert's problems, which were proposed in 1900, and had a substantial impact on 20th century mathematics.
		The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap.
		The millenium prize problems can be discussed at leisure in more detail in such a new article, whilst this article should keep the current section title 'The Millenium Prize Problems' and briefly discuss the fact that there are these seven problems and their significance, but have a main article link to the MMP article.
	www.nationmaster.com /encyclopedia/Clay-Mathematics-Institute (2337 words)

	prizes
		A $1.0 million prize to the first one to prove or disprove the Riemann Hypothesis.
		Nice explanations of this are given by J C Sprott, and the Chaos Hypertextbook.
		A $1.6 million prize to the first solver of the Eternity Puzzle.
	www.mathpuzzle.com /prizes.html (1111 words)

	Millennium Prize Problems (Site not responding. Last check: 2007-10-19)
		problems gastro problems millennium matrox millennium millennium g450 millennium falcon battlecruiser millennium battlecruiser millennium gold matrox millennium g450 opengl matrox millennium prize pulitzer prize booker prize nobel prize
		Millennium Prize Problems The seven problems proposed by the Clay Mathematics Institute: P versus NP; Hodge Conjecture; Poincaré Conjecture; Riemann Hypothesis; Yang-Mills Existence and Mass Gap; Navier-Stokes Existence and Smoothness; Birch and Swinnerton-Dyer Conjecture.
		The P versus NP Problem Description of the P vs NP question, designated a Clay Mathematics Institute Millennium Prize Problems, by Stephen Cook.
	www.serebella.com /encyclopedia/article-Millennium_Prize_Problems.html (447 words)

	EducationGuardian.co.uk \| comment \| Keith Devlin: The language of life
		Although several of the millennium problems ask you to solve an equation - in two cases equations that come from physics - none of them asks you to calculate a number.
		Using the equation to compute solutions to specific problems serves either to verify that the equation is (or is likely to be) correct, or else to apply the equation for some particular purpose, say building an aircraft or designing a heart-lung machine.
		Another millennium problem is to find a solution to a set of equations formulated by Chen-ning Yang and Robert Mills in 1954 to describe the very nature of matter.
	education.guardian.co.uk /higher/comment/story/0,9828,1303323,00.html (792 words)

	View This: The CMI Millennium Meeting Collection
		Tate was assigned to lecture on the Riemann Hypothesis, the Birch-Swinnerton-Dyer conjecture, and the P vs. NP problem.
		Atiyah was assigned to lecture on the Poincaré conjecture, the Hodge conjecture, the Yang-Mills problem, and the Navier-Stokes problem.
		I think the inclusion of these two problems, a forceful affirmation of the importance of physics-inspired mathematics, may be the most influential aspect of the conference recorded in these videos.
	www.maa.org /reviews/cmivideos.html (1355 words)

	Rules for the CMI Millennium Prize Problems
		If the counter-example shows that the original problem survives after reformulation or elimination of some special case, then the SAB may recommend that a small prize be awarded to the author.
		In the case the P versus NP problem the SAB may recommend the award of the Millennium Prize for deciding the question in either direction.
		With the one exception in the prior paragraph, all decision-making procedures concerning the CMI Millennium Prize Problems are private.
	elib.zib.de /pub/Misc/MillenniumPrizeProblems/www.claymath.org/prize_problems/rules.htm (809 words)

	One Million Bucks for a 100% Solution
		These "Millennium Prize Problems," selected by a blue ribbon committee, are the P versus NP problem, the Riemann hypothesis, the Hodge conjecture, the Poincaré conjecture, the Yang-Mills existence and mass gap, Navier-Stokes existence and smoothness, and the Birch and Swinnerton-Dyer conjecture.
		Prizes show the public that mathematics is far from a closed book, that it is not "all out there simply to be picked up from books or from the Web." They make the public aware of the existence of important unsolved problems and hence of the need for public support and encouragement.
		In its selection of prize problems, with their thrust away from applications, the Clay Institute emphasizes that mathematics has been the Queen, and not merely the handmaiden, of the sciences-and that it ought to remain so.
	www.siam.org /siamnews/10-00/solution.htm (2547 words)

	Millennium Prize Problems Volume
		The prize fund is divided equally among the seven problems.
		The Millennium Prize Problems were selected by the founding Scientific Advisory Board of CMI Ð Alain Connes, Arthur Jaffe, Andrew Wiles, and Edward Witten Ð after consultation with other leading mathematicians.
		The Millennium Prize Problems gives the official description of each of the seven problems and the rules governing the prizes.
	www.claymath.org /publications/Millennium_Problems (161 words)

	Millennium Prize Problems
		During the Millennium Meeting held on May 24, 2000 at the Collège de France, Timothy Gowers presented a lecture entitled The Importance of Mathematics, aimed for the general public, while John Tate and Michael Atiyah spoke on the problems.
		The rules for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors.
		Formulated in his 1859 paper, the Riemann hypothesis in effect says that the primes are distributed as regularly as possible given their seemingly random occurrence on the number line.
	www.claymath.org /millennium (307 words)

	Rules for the Christian Remling Millennium Prize Problems (Site not responding. Last check: 2007-10-19)
		The Christian Remling Millennium Prize Problems are problems on one-dimensional Schrödinger operators which are not widely known.
		I may also decide that a particular prize be divided among multiple solvers of a problem or their heirs.
		All decision-making procedures concerning the Christian Remling Millennium Prize Problems are private.
	www.mathematik.uni-osnabrueck.de /staff/phpages/remlingc/mill/rules.htm (454 words)

	Clay Mathematics Institute
		Thus no future civilization could ever hope to build a supercomputer capable of solving the problem by brute force; that is, by checking every possible combination of 100 students.
		In fact, one of the outstanding problems in computer science is determining whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure.
		Problems like the one listed above certainly seem to be of this kind, but so far no one has managed to prove that any of them really are so hard as they appear, i.e., that there really is no feasible way to generate an answer with the help of a computer.
	www.claymath.org /millennium/P_vs_NP (246 words)

	Math Bounties
		The publicity of the awards and problems is valuable because it reminds people that mathematics is an evolving, living discipline.
		Whereas new discoveries in biology or astronomy are often touted in the press, mathematics moves at a slower, and sometimes less accessible pace.
		The goal of the Millennium Prizes, he says, is to attract attention to these problems and entice young people to be interested in solving them.
	www.acfnewsource.org /science/math_bounties.html (617 words)

	Global Futures Studies & Research by the MILLENNIUM PROJECT
		The Millennium Project of the American Council for the United Nations University is a global participatory futures research think tank of futurists, scholars, business planners, and policy makers who work for international organizations, governments, corporations, NGOs, and universities.
		The Millennium Project was selected among the "100 Best Practices" by UN Habitat, best 7 foresight organizations by Battelle Northwest for the U.S. Department of Energy, and among the "Top Picks" by Future Survey, of the World Future Society.
		The purpose of the Millennium Project is to be an international utility to assist in organizing futures research by continuously updating and improving humanity's thinking about the future and making that thinking available for feedback as a geographically and institutionally dispersed think tank.
	acunu.org /millennium (193 words)

	Millennium Math Prize Problems
		In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven “Millennium Prize Problems.” The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years.
		The rules that follow for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors.
		The CMI Millennium Prize Problems are not intended to shape the direction of mathematics in the next century.
	antigravitypower.tripod.com /MathProblems/index.html (732 words)

	Amazon.com: Books: The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time (Site not responding. Last check: 2007-10-19)
		One can of course think of many other problems that fit the stature of the millennium problems, such as the invariant subspace conjecture, or developing a complete mathematical model of the cell, but these seven will no doubt spark the curiosity of a few young persons as they further their studies in mathematics.
		Some of the millennium problems, such as the Riemann hypothesis, the NP problem, the Poincare conjecture, and the Navier-Stokes equations, require only an undergraduate education.
		He also places the problems in their proper historical context (discusses their etiology) and he also looks at some of the ways that the problems have been approached in the past.
	www.amazon.com /exec/obidos/tg/detail/-/0465017290?v=glance (2464 words)

	The CMI Millennium Prize Problems, Winter and Spring 2001
		Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers.
		This is an example of the general phenomenon that generating a solution to a problem often takes far longer than verifying that a given solution is correct.
		Similarly, if someone tells you that the number 13,717,421 can be written as the product of two smaller numbers, you might not know whether to believe him, but if he tells you that it can be factored as 3607 times 3803 then you can easily check that it is true using a hand calculator.
	atlas.math.vanderbilt.edu /~bisch/ucsb/clay_seminars.html (1179 words)

	Poincaré Conjecture
		In 2000, the Clay Mathematics Institute of Cambridge, Mass., identified seven math problems it deemed the most "important classic questions that have resisted solution over the years." Several of them had in fact resisted solution for more than a century—the Rieman Hypothesis, for example, has confounded mathematicians since its formulation in 1859.
		While a layperson might have a tough time penetrating the quantum physics behind the Yang-Mills and Mass Gap problem, they have no difficulty understanding the meaning of the number 1 followed by 6 zeros and preceded by a dollar sign.
		He believes that the most important thing is that the problem is solved.” The Clay Institute has indicated that it may change the requirements for awarding the prize since Perelman has clearly solved it.
	www.factmonster.com /spot/poincare-conjecture.html (738 words)

	Millennium Prize Problems
		No proposed solution to a CMI Millennium problem may be submitted directly to the Clay Mathematics Institute.
		Under the rules for the prize, any proposed solution must be submitted to a peer-reviewed mathematics journal.
		This book will appear in a series of CMI books co-published in cooperation with the American Mathematical Society, and it will be sold through their standard channels of distribution.
	www.lmm.jussieu.fr /~lagree/Kfe/KfeMATHS/claymath/prize_problems/remarks.htm (84 words)

	Science & Ultimate Reality
		Recently, the Clay Mathematics Institute (www.claymath.org) established a series of $1 million rewards for solutions to a set of seven classic unsolved problems in mathematics, known as the "Millennium Prize Problems." The program seeks to recapitulate the vital stimulus David Hilbert gave to 20th-century mathematics by presenting his famous set of 23 great problems.
		Similarly, the Peter Gruber Foundation Cosmology Prize was established to recognize excellence in the field of cosmology.
		The Cosmology Prize seeks to stimulate forward thinking in physical cosmology with particular interest in recognizing major advances that stimulate new conceptualizations in understanding the nature of the cosmos.
	www.metanexus.net /ultimate_reality/info.htm (789 words)

	Clay Mathematics Institute : Millennium Prize Problems (Site not responding. Last check: 2007-10-19)
		The first person to solve each problem will be awarded $1,000,000 by CMI.
		The question is whether there are any problems for which a computer can check an answer quickly, but cannot find the answer quickly.
		There may be objections to this suggestion arising either of further consideration.
	www.termsdefined.net /mi/millennium-prize-problems.html (618 words)

	TU MAA - Problem of the Month (Site not responding. Last check: 2007-10-19)
		The Clay Institute of Cambridge is offering $1 million a problem for solutions to some of the most challenging mathematics problems ever conceived.
		The millennium math problems represent key issues in mathematics that have hitherto defied proof.
		The contest is inspired by the famous lecture on open mathematical questions by David Hilbert at the turn of the century.
	euler.mcs.utulsa.edu /maa/problems.html (58 words)

	NPR : Wanted: Math Solutions
		Devlin's new book The Millennium Problems is what he calls a "tourist guide." He examines the history and nature of each problem, as well as the challenge it represents to the math world.
		The seven millennium problems were chosen by a group of internationally acclaimed mathematicians organized by the Clay Mathematics Institute of Cambridge, Mass.
		The Millennium Prize Problems from the Clay Mathematics Institute, including a video explaining the problems by Keith Devlin.
	www.npr.org /display_pages/features/feature_906866.html (550 words)

	Manchester Geometry Seminar (Site not responding. Last check: 2007-10-19)
		In this video John Tate describes three of the seven $1,000,000 prize problems, announced by the Clay Institute.
		The problems discussed are: the Riemann Hypothesis, the Conjecture of Birch and Swinnerton-Dyer, and the P versus NP problem.
		These questions are among the most important unsolved problems in analysis, number theory, and theoretical computer science.
	www.ma.umist.ac.uk /tv/Seminar/2003-2004/movie.html (106 words)

	Prizes for solving algebra problems
		After puzzling for years over seven unsolved math problems, a U.S.-based mathematics foundation put the ``Millennium Prize Problems'' challenge to the world via the Internet on Wednesday.
		Experts say solving the problems could lead to breakthroughs in encryption and aerospace -- and open areas of mathematics as yet unimagined.
		The list of problems -- like the choice of Paris for launching the group's challenge -- was inspired by a list presented 100 years ago by German mathematician David Hilbert to the International Congress of Mathematicians meeting in Paris.
	www.algebra.com /algebra/higher/prize/index.epl (638 words)

	AARNEWS - Feb. 2001
		The problem asks one to produce a double-negation-free proof--one that is free of n(n(n(r)), where the function n denotes negation and r is any term--that relies solely on the inference rule condensed detachment and deduces the specific formula t, where t is the following.
		Specifically, in the proof we have for this problem, three or four formulas must be proved, each of which is subsumed by a clause along the way.
		Unification is concerned with the problem of identifying given terms, either syntactically or modulo a given logical theory.
	www-unix.mcs.anl.gov /AAR/issuefeb01 (3371 words)

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