Millennium problems are seven unsolved problems in mathematics identified around the year 2000 by the Clay Mathematics Institute as the most significant challenges known to modern mathematics. The problems were selected by the Clay Mathematics Institute which offered a million dollars for the resolution of each question. So far, only one has been solved, the Poincare conjecture, by Grigori Perelman posting the solution directly on the internet rather than seeking publication by a peer-reviewed mathematical journal.
The Millennium problems
1. The Poincare conjecture, which states that if certain algebraic structures associated with a space are equivalent to those associated with a kind of sphere, then the space itself must be that kind of sphere.
2. The Riemann hypothesis, which states that a certain function investigated by Bernhard Riemann in the 19th century is only equal to zero for complex numbers with real part equal to 1/2. (The so-called trivial zeroes occur at -2, -4, etc.)
3. P=NP, which states that the amount of time it takes to check a possible answer to a certain kind of problem is equal to the amount of time it takes to find that answer. In August 2010, Vinay Deolalikar posted on the internet a proposed proof that P≠NP.
4. The Hodge conjecture, which identifies algebraic structures on certain kinds of complex manifolds.
5. The existence of infinitely differentiable solutions to the Navier-Stokes equations, which would imply certain results about turbulence, an extremely common but not at all understood phenomenon.
6. The Birch and Swinnerton-Dyer conjecture, which states there is a certain level of "simplicity" in the solutions to certain kinds of rational equations.
7. Investigation of the Yang-Mills existence and gap; specifically, whether Yang-Mills theory predicts particles with negative mass, and whether it contains any contradictions. Demonstrations of either negative-mass predictions or contradictions would provide clues as to how to build a more accurate theory.