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Millennium Prize Problems
The seven problems are:The Millennium Prize Problems are seven
problems in mathematics that were stated by theClay Mathematics
Institute in 2000. As of March 2012, six of the problems remain unsolved.
A correct solution to any of the problems results in a US$1,000,000
prize (sometimes called aMillennium Prize) being awarded by the
institute. The Poincaré conjecture, the only Millennium Prize
Problem to be solved so far, was solved by Grigori Perelman,
but he declined the award in 2010.
- P versus NP problem
- Hodge conjecture
- Poincaré conjecture (Solved)
- Riemann hypothesis
- Yang–Mills existence and mass gap
- Navier–Stokes existence and smoothness
- Birch and Swinnerton-Dyer conjecture
Contents |
[edit]P versus NP
The question is whether, for all problems for which a computer
can verify a given solution quickly (that is, in polynomial time),
it can also findthat solution quickly. The former describes the
class of problems termed NP, whilst the latter describes P.
The question is whether or not all problems in NP are also in P.
This is generally considered the most important open question
in theoretical computer science as it has far-reaching consequences in mathematics, biology, philosophy[1] and cryptography
(see P versus NP problem proof consequences).
If the question of whether P=NP were to be answered affirmatively
it would trivialise the rest of the Millennium Prize Problems
(and indeed all but the unprovable propositions in
mathematics)[dubious ] because they would
all have direct solutions easily solvable by a formal system.
- "If P = NP, then the world would be a profoundly different
- place than we usually assume it to be. There would be no
- special value in 'creative leaps,' no fundamental gap
- between solving a problem and recognizing the solution
- once it’s found. Everyone who could appreciate a symphony
- would be Mozart; everyone who could follow a step-by-step
- argument would be Gauss..."
- — Scott Aaronson, MIT
Most mathematicians and computer scientists expect that P≠NP.
The official statement of the problem was given by Stephen Cook.
[edit]The Hodge conjecture
The Hodge conjecture is that for projective algebraic varieties,
Hodge cycles are rational linear combinations of algebraic
cycles.The official statement of the problem was given by
[edit]The Poincaré conjecture (proven)
In topology, a sphere with a two-dimensional surface is
essentially characterized by the fact that it is simply connected.
It is also true that every two-dimensional surface which is both
compact and simply connected is topologically a sphere. The
Poincaré conjecture 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-manifolds.
The official statement of the problem was given by John Milnor.
A proof of this conjecture was given by Grigori Perelman in 2003;
its review was completed in August 2006, and Perelman was
selected to receive the Fields Medal for his solution. Perelman
declined that award.[2] Perelman was officially awarded the
Millennium prize on March 18, 2010.[3] On July 1, 2010, it was
reported that Perelman declined the award and associated
prize money from the Clay Mathematics Institute.[4] In rejecting
the Millennium Prize, Perelman stated that he believed the
decisions by the organized mathematics community to be
unjust and that his contribution to solving the Poincaré
conjecture was no greater than that of Columbia University
mathematician Richard Hamilton (who first suggested a
program for the solution).[5]
[edit]The Riemann hypothesis
The Riemann hypothesis is that all nontrivial zeros of the
analytical continuation of the Riemann zeta function have
a real part of 1/2. A proof or disproof of this would have
far-reaching implications in number theory, especially for
the distribution of prime numbers. This was Hilbert's
eighth problem, and is still considered an important
open problem a century later.
The official statement of the problem was given by
[edit]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 (gluons). However, the
postulated phenomenon of color confinement permits
only bound states of gluons, forming massive particles.
This is the mass gap. Another aspect of confinement is
asymptotic freedom 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 and Edward Witten.
[edit]
The Navier–Stokes equations describe the motion of fluids.
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 insight into these equations.
The official statement of the problem was given by Charles Fefferman.
[edit]The Birch and Swinnerton-Dyer conjecture
The Birch and Swinnerton-Dyer conjecture deals with a certain
type of equation, those defining elliptic curves over the rational
numbers. 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 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.
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