math – update

blogging & searching for true math …

Day #22 : The Millenium´s Nicest Problems

Leave a comment

mp-1

Q: Could you just tell the list of the – solved and unsolved – Millenium Problems??…

A:

The Millennium Prize Problems

In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) established seven Prize Problems. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.
The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collège de France. Three lectures were presented: Timothy Gowers spoke onThe Importance of Mathematics; Michael Atiyah and John Tate spoke on the problems themselves.
The seven Millennium Prize Problems were chosen by the founding Scientific Advisory Board of CMI, which conferred with leading experts worldwide. The focus of the board was on important classic questions that have resisted solution for many years.
Following the decision of the Scientific Advisory Board, the Board of Directors of CMI designated a $7 million prize fund for the solutions to these problems, with $1 million allocated to the solution of each problem.
It is of note that one of the seven Millennium Prize Problems, the Riemann hypothesis, formulated in 1859, also appears in the list of twenty-three problems discussed in the address given in Paris by David Hilbert on August 9, 1900.
The rules for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors. The members of these boards have the responsibility to preserve the nature, the integrity, and the spirit of this prize. See: Millennium Prize Problems – Clay Mathematics Institute of Cambridge

Unsolved Problems

P versus NP

Main article: P versus NP problem

The question is whether or not, 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. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether 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, and to biology, philosophy[4] and cryptography (see P versus NP problem proof consequences). A common example of a P versus NP problem is the travelling salesman problem.

mp-3

Most mathematicians and computer scientists expect that P ≠ NP.[6]

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

Hodge Conjecture

Main article: 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.

Riemann Hypothesis

Main article: 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.

mp-4

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

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.[7]

Navier–Stokes Existence and Smoothness

The Navier–Stokes equations describe the motion of fluids. Although they were first stated in the 19th century, they are still not well-understood. The problem is to make progress towards a mathematical theory that will give insight into these equations.

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

Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer conjecture deals with certain types of equations: 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.[8]

Solved Problem

Poincaré Conjecture

Main article: Poincaré conjecture

In dimension 2, a sphere is characterized by the fact that it is the only closed and simply-connected surface. The Poincaré conjecture states that this is also true in dimension 3. It is central to the more general problem of classifying all 3-manifolds. The precise formulation of the conjecture states:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

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 but he declined the award.[1] Perelman was officially awarded the Millennium Prize on March 18, 2010,[2] but he also declined that award and the associated prize money from the Clay Mathematics Institute. The Interfax news agency quoted Perelman as saying he believed the prize was unfair. Perelman told Interfax he considered his contribution to solving the Poincaré conjecture no greater than that of Columbia University mathematician Richard S. Hamilton.[3]

Source: Millennium Prize Problems – WikipediA

 

 

 

Advertisements

Author: Math - Update

Updating Math In Our Mind & Heart!!...

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s