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 Discussion Response Post to: Toughest Problems in The world(must have a look)
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boywholived (460)

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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

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 mathematicsbiologyphilosophy[1] and cryptography

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.

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

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]

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

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

Navier–Stokes existence and smoothness

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.

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.

Only two things are infinite, the universe and human stupidity, and I'm not sure about the former.
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