The document discusses the seven Millennium Prize Problems in mathematics. These are seven problems stated by the Clay Mathematics Institute in 2000 that each carry a $1 million prize for a correct solution. Six problems remain unsolved. The problems include the P vs NP problem, Hodge conjecture, Riemann hypothesis, Yang-Mills existence and mass gap, Navier-Stokes existence and smoothness, and the Birch and Swinnerton-Dyer conjecture. The only solved problem was the Poincare conjecture, proven by Grigori Perelman in 2003.

1. Millennium
Prize Problems
Research in English by
Erwin Zarsadias and
Rechelle Pasoc
Submitted to Ms. Rodolpo

2. What are Millennium Prize Problems?
The Millennium Prize Problems are seven
problems in mathematics that were stated by
the Clay Mathematics Institute in 2000. As of
December 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 a Millennium Prize) being awarded by the
institute. The prizes were announced at
a meeting in Paris, held on May 24, 2000 at the
Collège de France.

3. The Seven Problems are:
• 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

4. The Solved Problem (Poincare
Conjecture)
• Poincaré conjecture
• 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. Perelman was officially awarded the Millennium Prize on March
18, 2010.Perelman declined the award and associated prize money from
the Clay Mathematics Institute, without giving any reason to the
Institute

6. The Unsolved Problems
• 1 .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 Enrico Bombieri.

7. www.democraticunderground.com

8. 2 .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 & recent status by Michael R.
Douglas.

9. 3 .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.

10. 4 .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.

11. 5 .P Vs. Np Problem
The question is whether, for all problems for
which an algorithm can verify a given solution quickly
(that is, in polynomial time), an algorithm can
also find that 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 one of the most important open questions
in mathematics and theoretical computer science as it
has far-reaching consequences to other problems
in mathematics, biology,philosophy and cryptography
(see P versus NP problem proof consequences).

13. • "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.

14. 6 .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 Pierre Deligne.

15. Is a pentagon equivalent to a triangle when drawn on a
doughnut in very slippery melted chocolate? Photograph: Matt
Parker

16. REFERENCES
• http://www.ehow.com
• http://personal.cityu.edu.hk
• http://www.jhu.edu
• http://www.claymath.org
• http://en.wikipedia.org

17. Thanks for
Listening!