The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and 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 Millennium Prize Problems
The institute is best known for its establishment on May 24, 2000 of the 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 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.
The seven Millennium Prize Problems are:
The question is whether there are any problems for which a computer can check an answer quickly, but cannot find the answer quickly. This is generally considered the most important open question in theoretical computer science. See Complexity classes P and NP for a more complete discussion.
The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.
In topology, a sphere with a two-dimensional surface is essentially characterized by the fact it is simply connected. 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.
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.
In physics, quantum Yang-Mills theory describes particles with positive mass having classical waves travelling at the speed of light. This is the mass gap. The problem is to establish the existence of the Yang-Mills theory and a mass gap.
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 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 proved that there is no way to decide whether a given equation even has any solutions.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