Hodge conjecture | smooth solution to the Navier-Stokes equations problem | Poincaré conjecture | P vs. NP problem | Riemann hypothesis | Swinnerton-Dyer conjecture | Yang-Mills existence and mass gap problem (total: 7)

The Hodge conjecture posits that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually rational linear combinations of geometric pieces called algebraic cycles.

The smooth solution to the Navier-Stokes equations problem asks if the Navier-Stokes equations on a 3-dimensional domain Ω in R^3 have a unique smooth solution for all time.

The Poincaré conjecture, now proved, posited that every simply connected closed three-manifold is homeomorphic to the three-sphere.

The P vs. NP problem asks for the determination of whether all NP-problems are actually P-problems.

The Riemann hypothesis posits that the nontrivial zeros of the Riemann zeta function ζ(s) all lie on the critical line Re(s) = 1/2.

The Swinnerton-Dyer conjecture posits that if a given elliptic curve has an infinite number of solutions, then the associated L-series has value 0 at a certain fixed point.

The Yang-Mills existence and mass gap problem asks for a proof that for any compact simple gauge group, a nontrivial quantum Yang-Mills theory exists on R^4 and has a mass gap Δ>0.

The three-sphere is the only type of bounded three-dimensional space possible that contains no holes.

| formal statement Riemann hypothesis | for all _(n, n element Z ∧ n!=0)Re(ρ_n) = 1/2

| formulation date | formulators | status Hodge conjecture | 1950 (75 years ago) | William Vallance Douglas Hodge | open smooth solution to the Navier-Stokes equations problem | | | open Poincaré conjecture | 1904 (121 years ago) | Henri Poincaré | proved P vs. NP problem | | | open Riemann hypothesis | 1859 (166 years ago) | Bernhard Riemann | open Swinnerton-Dyer conjecture | 1960 (65 years ago) | Peter Swinnerton-Dyer | open Yang-Mills existence and mass gap problem | 1954 (71 years ago) | Chen Ning Yang | Robert L. Mills | open | proof date | provers Poincaré conjecture | 2003 (99 years later) (22 years ago) | Grigori Perelman

(du)/(dt) + u · del u = -( del P)/ρ + ν del ^2u

Re(ρ_n) = 1/2

It has been verified that the first 1×10^13 nontrivial zeros of the zeta function lie on the critical line. Conrey (1989) proved that at least 40% of the nontrivial zeros of the zeta function lie on the critical line.

| prizes offered for solution | prizes awarded for solution Hodge conjecture | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. | smooth solution to the Navier-Stokes equations problem | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. | Poincaré conjecture | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. (Refused by solver.) | Perelman shared the 2006 Fields Medal for his proof. (Refused by solver.) P vs. NP problem | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. | Riemann hypothesis | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. | Swinnerton-Dyer conjecture | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. | Yang-Mills existence and mass gap problem | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. |

Millennium Prize problems | prize mathematics problems