There is a 1 million dollar prize for the solution to each problem offered by the Clay Mathematics Institute. T his is the newest problem on the list and the easiest to explain. This is a crude but easily understood condition to decide whether an algorithm is fast enough to be worth bothering with. Stephen Cook posed this problem in 1971 and it's still open. The Poincare conjecture.
Spheres are among the most fundamental topological spaces, but spheres hold many mysteries.
This is a conjecture about elliptic curves, and indirectly, number theory. A solution to this conjecture would shed a lot of light on Diophantine equations, one of which goes back to at least the 10th century - namely, the problem of finding which integers appear as the areas of right triangles all of whose sides have lengths equal to rational numbers. The Hodge conjecture. This question is about algebraic geometry and topology. Such a variety always has even dimension, say 2n.