Clay Mathematics Institute
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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. 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, Simon Donaldson and James Carlson.
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[edit] 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]
[edit] P versus NP
The question is whether there are any problems for which a computer can verify a given solution quickly, but cannot find the solution quickly. This is generally considered the most important open question in theoretical computer science.
[edit] The Hodge conjecture
The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.
[edit] The 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]
[edit] The 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.
[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.
[edit] 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.
[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.
[edit] 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).
[edit] References
- ^ Arthur Jaffe's first-hand account of how this Millennium Prize came about can be read in The Millennium Grand Challenge in Mathematics
- ^ http://news.bbc.co.uk/2/hi/science/nature/5274040.stm
- Keith J. Devlin, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Basic Books (October, 2002), ISBN 0-465-01729-0.
