Millennium Prize Problems

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Millennium Prize Problems

The Millennium Prize Problems are seven problems in mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
that were stated by the Clay Mathematics Institute

Clay Mathematics Institute

The Clay Mathematics Institute is a private, non-profit Foundation , based in Cambridge, Massachusetts, Massachusetts. The Institute is dedicated to increasing and disseminating mathematics knowledge....
in 2000. Currently, six of the problems remain unsolved

Unsolved problems in mathematics

This article lists some List of unsolved problems in mathematics. See individual articles for details and sources....
. A correct solution to each problem results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. Only the Poincaré conjecture

Poincaré conjecture

In mathematics, the Poincar? conjecture is a theorem about the Characterization of the 3-sphere among 3-manifold. It began as a popular, important conjecture, but is now considered a theorem to the satisfaction of the awarders of the Fields medal....
has been solved, but the solver, Grigori Perelman

Grigori Perelman

Grigori Yakovlevich Perelman , born 13 June 1966 in Saint Petersburg, Soviet Union , sometimes known as Grisha Perelman, is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology....
, has not pursued the conditions necessary to claim the prize.

Main article: P = NP problem

The question is whether, for all problems for which a computer can verify a given solution quickly (that is, in polynomial time
Polynomial time

In computational complexity theory, polynomial time refers to the computation time of a problem where the run time, , is no greater than a polynomial function of the problem size, n....
), it can also find that solution quickly.

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The Millennium Prize Problems are seven problems in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
that were stated by the Clay Mathematics Institute
Clay Mathematics Institute

The Clay Mathematics Institute is a private, non-profit Foundation , based in Cambridge, Massachusetts, Massachusetts. The Institute is dedicated to increasing and disseminating mathematics knowledge....
in 2000. Currently, six of the problems remain unsolved
Unsolved problems in mathematics

This article lists some List of unsolved problems in mathematics. See individual articles for details and sources....
. A correct solution to each problem results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. Only the Poincaré conjecture
Poincaré conjecture

In mathematics, the Poincar? conjecture is a theorem about the Characterization of the 3-sphere among 3-manifold. It began as a popular, important conjecture, but is now considered a theorem to the satisfaction of the awarders of the Fields medal....
has been solved, but the solver, Grigori Perelman
Grigori Perelman

Grigori Yakovlevich Perelman , born 13 June 1966 in Saint Petersburg, Soviet Union , sometimes known as Grisha Perelman, is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology....
, has not pursued the conditions necessary to claim the prize.

P versus NP

Main article: P = NP problem
The question is whether, for all problems for which a computer can verify a given solution quickly (that is, in polynomial time
Polynomial time

In computational complexity theory, polynomial time refers to the computation time of a problem where the run time, , is no greater than a polynomial function of the problem size, n....
), it can also find that solution quickly. This is generally considered the most important open question in theoretical computer science
Computation

Computation is a general term for any type of information processing. This includes phenomena ranging from human thinking to calculations with a more narrow meaning....
as it has far-reaching consequences in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, philosophy
Philosophy

Philosophy is the study of general problems concerning matters such as existence, knowledge, truth, beauty, justice, validity, mind, and language....
and cryptography
Cryptography

Cryptography is the practice and study of hiding information. In modern times cryptography is considered a branch of both mathematics and computer science and is affiliated closely with information theory, computer security and engineering....
(see P=NP proof consequences).

The official statement of the problem was given by Stephen Cook
Stephen Cook

Stephen Arthur Cook is a noted computer science.Cook formalised the notion of NP-completeness in a famous 1971 paper "The Complexity of Theorem Proving Procedures", which also contained Cook's theorem, a proof that the boolean satisfiability problem is NP-complete....
.

The Hodge conjecture
The Hodge conjecture is that for projective
Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
algebraic varieties
Algebraic variety

In mathematics, an algebraic variety is essentially a set of points where a polynomial or set of polynomials attain a value of zero. Algebraic varieties are one of the central objects of study in classical algebraic geometry....
, Hodge cycle
Hodge cycle

In mathematics, a Hodge cycle is a particular kind of homology class defined on a complex number algebraic variety V, or more generally on a K?hler manifold....
s are rational linear combination
Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field , with some generalisations given at the end of the article....
s of algebraic cycle
Algebraic cycle

In mathematics, an algebraic cycle on an algebraic variety V is, roughly speaking, a homology class on V that is represented by a linear combination of subvarieties of V....
s.

The official statement of the problem was given by Pierre Deligne
Pierre Deligne

Pierre Ren?, Viscount Deligne is a Belgium mathematician. He is known for fundamental work on the Weil conjectures, leading finally to a complete proof in 1973....
.

The Poincaré conjecture
In topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
with a two-dimensional surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
is essentially characterized
Characterization (mathematics)

In the jargon of mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P....
by the fact that it is simply connected. It is also true that every 2-dimensional surface which is both compact
Compact space

In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact"....
and simply connected is topologically a sphere. The Poincaré conjecture
Poincaré conjecture

In mathematics, the Poincar? conjecture is a theorem about the Characterization of the 3-sphere among 3-manifold. It began as a popular, important conjecture, but is now considered a theorem to the satisfaction of the awarders of the Fields medal....
is that this is also true for spheres with three-dimensional surfaces. The question had long been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifold
3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds....
s.

The official statement of the problem was given by John Milnor
John Milnor

John Willard Milnor is an United States mathematician known for his work in differential topology, K-theory, and dynamical systems, and for his influential books....
.

A proof of this conjecture was given by Grigori Perelman
Grigori Perelman

Grigori Yakovlevich Perelman , born 13 June 1966 in Saint Petersburg, Soviet Union , sometimes known as Grisha Perelman, is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology....
in 2003; its review was completed in August 2006, and Perelman was selected to receive the Fields Medal
Fields Medal

The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of Mathematicians of the International Mathematical Union, a meeting that takes place every four years....
for his solution. Perelman declined the award.

The Riemann hypothesis
The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function
Riemann zeta function

In mathematics, the Riemann zeta function, named after Germany mathematician Bernhard Riemann, is a prominent function of great significance in number theory because of its relation to the prime number theorem....
have a real part of 1/2. A proof or disproof of this would have far-reaching implications in number theory
Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, especially for the distribution of prime number
Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC....
s. This was Hilbert's eighth problem
Hilbert's problems

Hilbert's problems are a list of twenty-three problems in mathematics put forth by Germany mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900....
, and is still considered an important open problem a century later.

The official statement of the problem was given by Enrico Bombieri
Enrico Bombieri

Enrico Bombieri is an Italy mathematician, born in Milan. He is now at the Institute for Advanced Study. He is known for work in number theory, algebraic geometry, and mathematical analysis....
.

Yang–Mills existence and mass gap

In physics, classical Yang-Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (gluon
Gluon

Gluons are elementary particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei....
s). However, the postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the mass gap
Mass gap

In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle....
. Another aspect of confinement is asymptotic freedom
Asymptotic freedom

In physics, asymptotic freedom is the property of some gauge theory in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i.e....
which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap.

The official statement of the problem was given by Arthur Jaffe
Arthur Jaffe

File:Arthur Jaffe.jpgArthur Jaffe is an United States mathematical physicist and a professor at Harvard University. Born on December 22, 1937 he attended Princeton University as an undergraduate obtaining a degree in chemistry, and later Clare College, Cambridge, as a Marshall Scholar, obtaining a degree mathematics....
and Edward Witten
Edward Witten

Edward Witten is an United States theoretical physicist and professor at the Institute for Advanced Study. He is one of the world's leading researchers in superstring theory....
.

Navier–Stokes existence and smoothness
The Navier-Stokes equations
Navier-Stokes equations

The Navier?Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances, that is substances which can flow....
describe the motion of liquid
Liquid

Liquid is one of the principal states of matter. A liquid is a fluid that has the particles loose and can freely form a distinct surface at the boundaries of its bulk material....
s and gas
Gas

In physics, a gas is a state of matter, consisting of a collection of particles without a definite shape or volume that are in more or less random motion....
es. Although they were found in the 19th century, they still are not well understood. The problem is to make progress toward a mathematical theory that will give us insight into these equations.

The official statement of the problem was given by Charles Fefferman
Charles Fefferman

Charles Louis Fefferman is an United States mathematician at Princeton University. His primary field of research is mathematical analysis.A child prodigy, Fefferman entered college by twelve and had written his first scientific paper by the age of 15 in German language....
.

The Birch and Swinnerton-Dyer conjecture
The Birch and Swinnerton-Dyer conjecture deals with a certain type of equation, those defining elliptic curve
Elliptic curve

In mathematics, an elliptic curve is a differentiable manifold, algebraic variety#Projective varieties algebraic curve of genus #Algebraic geometry one, on which there is a specified point O....
s over the rational number
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
s. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. Hilbert's tenth problem
Hilbert's tenth problem

'Hilbert's tenth problem' is the tenth on the list of Hilbert's problems of 1900. Its statement is as follows:Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation...
dealt with a more general type of equation, and in that case it was proven that there is no way to decide whether a given equation even has any solutions.

The official statement of the problem was given by Andrew Wiles
Andrew Wiles

Sir Andrew John Wiles Order of the British Empire Fellow of the Royal Society is a United Kingdom mathematician and a professor at Princeton University, specialising in number theory....
.

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