7 problems · $7 million · 6 not yet awarded

The Million-Dollar Problems

In 2000 the Clay Mathematics Institute put a $1 million prize on each of seven problems. One has been solved. Six are still open. Play with each one before you meet its equation.

1 Claimed, under review Navier–Stokes existence and smoothness Does a smooth, calm flow of fluid always stay smooth, or can it tie itself into an infinitely violent knot in a finite time? 18 min · Play and read →
2 Unsolved P vs NP If a computer can quickly check an answer, can it always quickly find one too? 18 min · Play and read →
3 Unsolved The Riemann Hypothesis Do all the nontrivial zeros of the Riemann zeta function lie on the single line with real part one half? 18 min · Play and read →
4 Unsolved Yang–Mills existence and the mass gap Can we build the quantum theory of the strong nuclear force as exact mathematics, and prove that its lightest particle has a mass greater than zero? 18 min · Play and read →
5 Unsolved The Birch and Swinnerton-Dyer conjecture Can you tell whether a curve has infinitely many fraction points just by counting its solutions in clock arithmetic? 18 min · Play and read →
6 Unsolved The Hodge conjecture On a shape cut out by polynomial equations, is every hole that passes Hodge's "looks algebraic" test really traced out by smaller shapes that are also cut out by polynomial equations? 20 min · Play and read →
7 Solved The Poincaré conjecture If every loop in a closed three-dimensional space can be shrunk to a point, must that space be the 3-sphere? 17 min · Play and read →