Monday, September 13, 2004
List of unsolved mathematical problems
This article describes currently unsolved problems This is a list of lists of unsolved problems in various subjects:
Unsolved problems in mathematics
Unsolved problems in physics
Unsolved problems in chemistry
Unsolved problems in biology
Unsolved problems in economics
Unsolved problems in governance
Unsolved problems in cognitive science
Unsolved problems in neuroscience
Unsolved problems in computer science
Unsolved problems in software engineering
Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships.
The seven Millennium Prize Problems
The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998 by businessman Landon T. Clay, who financed it, and Harvard mathematician Arthur Jaffe.
The Millennium Prize problems
Clay Mathematics Institute The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998 by businessman Landon T. Clay, who financed it, and Harvard mathematician Arthur Jaffe.
The Millennium Prize problems
1) P versus NP
Computational complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps does it take to solve a problem) and space (how much memory does it take to solve a problem).
In this theory, the class P consists of all those decision problems that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; the class
2) The Hodge Conjecture
The Hodge conjecture is a major unsolved problem of algebraic geometry. It is a conjectural description of the link between the algebraic topology of a non-singular complex algebraic variety, and its geometry as captured by polynomial equations that define sub-varieties. It arose as a result of the work of W. V. D. Hodge, who between 1930 and 1940 enriched the description of De Rham cohomology to include extra structure which is present in the case of algebraic varieties (though not restricted to that case).
3) The Poincaré Conjecture
The Poincaré conjecture is widely considered the most important unsolved problem in topology. It was first formulated by Henri Poincaré in 1904. In 2000 the Clay Mathematics Institute selected the Poincaré conjecture as one of seven Millennium Prize Problems and offered a $1,000,000 prize for its solution. The conjecture states:
Every simply connected closed (i.e. compact and without boundary) 3-manifold is homeomorphic to a 3-sphere.
4) The Riemann Hypothesis
The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of Riemann's zeta function ζ(s). It is one of the most important open problems of contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. In June 2004, Louis De Branges de Bourcia claimed to have proved the Riemann hypothesis but this has not yet been confirmed (see below). Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical.)
5) Yang-Mills Existence and Mass Gap
6) Navier-Stokes Existence and Smoothness In fluid dynamics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. For example: they model weather or the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena.
Momentum Equation in 3 dimensions (assuming and are constant:
7) The Birch and Swinnerton-Dyer Conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function L(E,s) at s = 1. It has been proved only in special cases (2004).
Background In 1922 Louis Mordell proved that the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite sub-set of the rational points on the curve from which all other rational points may be generated.
..... Click the link for more information.
http://encyclopedia.thefreedictionary.com/List%20of%20unsolved%20mathematical%20problems
http://www.claymath.org/millennium/
prize.problems@claymath.org
http://www.simonsingh.net/Mathematics_Links.html
Unsolved problems in mathematics
Unsolved problems in physics
Unsolved problems in chemistry
Unsolved problems in biology
Unsolved problems in economics
Unsolved problems in governance
Unsolved problems in cognitive science
Unsolved problems in neuroscience
Unsolved problems in computer science
Unsolved problems in software engineering
Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships.
The seven Millennium Prize Problems
The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998 by businessman Landon T. Clay, who financed it, and Harvard mathematician Arthur Jaffe.
The Millennium Prize problems
Clay Mathematics Institute The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998 by businessman Landon T. Clay, who financed it, and Harvard mathematician Arthur Jaffe.
The Millennium Prize problems
1) P versus NP
Computational complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps does it take to solve a problem) and space (how much memory does it take to solve a problem).
In this theory, the class P consists of all those decision problems that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; the class
2) The Hodge Conjecture
The Hodge conjecture is a major unsolved problem of algebraic geometry. It is a conjectural description of the link between the algebraic topology of a non-singular complex algebraic variety, and its geometry as captured by polynomial equations that define sub-varieties. It arose as a result of the work of W. V. D. Hodge, who between 1930 and 1940 enriched the description of De Rham cohomology to include extra structure which is present in the case of algebraic varieties (though not restricted to that case).
3) The Poincaré Conjecture
The Poincaré conjecture is widely considered the most important unsolved problem in topology. It was first formulated by Henri Poincaré in 1904. In 2000 the Clay Mathematics Institute selected the Poincaré conjecture as one of seven Millennium Prize Problems and offered a $1,000,000 prize for its solution. The conjecture states:
Every simply connected closed (i.e. compact and without boundary) 3-manifold is homeomorphic to a 3-sphere.
4) The Riemann Hypothesis
The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is a conjecture about the distribution of the zeros of Riemann's zeta function ζ(s). It is one of the most important open problems of contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. In June 2004, Louis De Branges de Bourcia claimed to have proved the Riemann hypothesis but this has not yet been confirmed (see below). Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical.)
5) Yang-Mills Existence and Mass Gap
6) Navier-Stokes Existence and Smoothness In fluid dynamics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. For example: they model weather or the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena.
Momentum Equation in 3 dimensions (assuming and are constant:
7) The Birch and Swinnerton-Dyer Conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function L(E,s) at s = 1. It has been proved only in special cases (2004).
Background In 1922 Louis Mordell proved that the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite sub-set of the rational points on the curve from which all other rational points may be generated.
..... Click the link for more information.
http://encyclopedia.thefreedictionary.com/List%20of%20unsolved%20mathematical%20problems
http://www.claymath.org/millennium/
prize.problems@claymath.org
http://www.simonsingh.net/Mathematics_Links.html
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