The Seven Millennium Prize Problems
In 2000, the Clay Mathematics Institute designated seven unsolved problems in mathematics, each carrying a $1,000,000 prize for a correct solution. One has been solved.
P vs NP Problem
Is every problem whose solution can be quickly verified also quickly solvable?
Riemann Hypothesis
Do all non-trivial zeros of the Riemann zeta function have real part equal to 1/2?
Yang–Mills Existence and Mass Gap
Prove that quantum Yang–Mills theory exists and has a positive mass gap.
Navier–Stokes Existence and Smoothness
Do smooth, globally defined solutions exist for the 3D incompressible Navier–Stokes equations?
Hodge Conjecture
Are certain cohomology classes on smooth projective varieties representable by algebraic cycles?
Birch and Swinnerton-Dyer Conjecture
Is the rank of an elliptic curve determined by the behavior of its L-function at s = 1?
Poincaré Conjecture
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.