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The 'Clay Mathematics Institute (CMI)' is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998 by businessman Landon T. Clay, who financed it, and Harvard mathematician Arthur Jaffe, who conceived and implemented its structure and mission. The Institute is run according to a standard structure comprising a board of directors that decides on grant-awarding and research proposals, and a scientific advisory committee that oversees and approves the board's decisions. As of May, 2006, the board is integrated by members of the Clay family (including Landon Clay), whereas the advisory committee is composed of leading authorities in mathematics, namely Sir Andrew Wiles, Yum-Tong Siu, Richard Melrose, Gregory Margulis,and Simon Donaldson

Contents

Millennium Prize Problems

P versus NP

The Hodge conjecture

The Poincaré conjecture

The Riemann hypothesis

Yang-Mills existence and mass gap

Navier-Stokes existence and smoothness

The Birch and Swinnerton-Dyer conjecture

Other activities

References

External links

Millennium Prize Problems

The institute is best known for its establishment on May 24, 2000 of 'Millennium Prize Problems'. These seven problems are considered by CMI to be "important classic questions that have resisted solution over the years". The first person to solve each problem will be awarded $1,000,000 by the CMI. In announcing the prize, CMI drew a parallel to Hilbert's problems, which were proposed in 1900, and had a substantial impact on 20th century mathematics. Of the initial twenty-three Hilbert problems, most of which have been solved, only one (the Riemann hypothesis, formulated in 1859) is one of the seven Millennium Prize Problems.^[1]
For each problem, the Institute had a professional mathematician write up an official statement of the problem which will be the main standard by which a given solution will be measured against.

P versus NP

Main articles: Complexity classes P and 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 ''find'' that solution quickly. This is generally considered the most important open question in theoretical computer science.
The official statement of the problem was given by Stephen Cook.

The Hodge conjecture

Main articles: 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.

The Poincaré conjecture

Main articles: 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 2-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 has been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds. A solution to this conjecture was proposed by Grigori Perelman in 2003; its review was completed in August 2006, and Perelman was awarded the Fields Medal for his solution. Perelman declined the award.^[2]
The official statement of the problem was given by John Milnor.

The Riemann hypothesis

Main articles: Riemann hypothesis

The Riemann hypothesis is that all nontrivial zeros 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.

Yang-Mills existence and mass gap

Main articles: 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 deictic 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.

Navier-Stokes existence and smoothness

Main articles: Navier-Stokes existence and smoothness

The Navier-Stokes equations describe the movement of liquids and gases. 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.

The Birch and Swinnerton-Dyer conjecture

Main articles: 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.

Other activities

Besides the Millennium Prize Problems, the Clay Mathematics Institute also supports mathematics via the awarding of
research fellowships (which range from two to five years, and are aimed at younger mathematicians), as well as shorter-term
scholarships for programs, individual research, and book writing. The Institute also has a yearly Clay Research Award, recognizing major breakthroughs in mathematical research. Finally, the Institute also organizes a number of summer schools, conferences, workshops, public lectures, and outreach activities aimed primarily at junior mathematicians (from the high school to postdoctoral level).

References

1. Arthur Jaffe's first-hand account of how this Millennium Prize came about can be read in The Millennium Grand Challenge in Mathematics
2. Maths genius declines top prize 22 August 2006


★ Keith J. Devlin, ''The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time'', Basic Books (October, 2002), ISBN 0-465-01729-0.

External links

★ The Clay Mathematics Institute

★ The Millennium Grand Challenge in Mathematics

★ The Millennium Prize Problems

★ QEDen: Millennium Problems Wiki