Category:Millennium Problems

This category contains results about Millennium Problems.

The Millennium problems are a collection of seven mathematical problems stated by the Clay Mathematics Institute on $24$ May $2000$.

Each carries a prize of $\$1 \, 000 \, 000$ (US dollars).

Six of them remain unsolved.

They are as follows:

P versus NP

The class of problems for which an algorithm can find a solution in polynomial time is termed $P$.

The class of problems for which an algorithm can verify a solution in polynomial time is termed $NP$.

The $P$ versus $NP$ question is:

Are all problems in $NP$ also in $P$?

The Hodge Conjecture

It is conjectured that:

For projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

The Poincaré Conjecture

This is the only one of the seven to be solved so far:

Let $\Sigma^m$ be a smooth $m$-manifold.

Let $\Sigma^m$ satisfy:

$H_0 \struct {\Sigma; \Z} = 0$

and:

$H_m \struct {\Sigma; \Z} = \Z$

Then $\Sigma^m$ is homeomorphic to the $m$-sphere $\Bbb S^m$.

The Riemann Hypothesis

All the nontrivial zeroes of the analytic continuation of the Riemann zeta function $\zeta$ have a real part equal to $\dfrac 1 2$.

Yang-Mills Existence and Mass Gap

To establish rigorously:

the Yang-Mills quantum theory

the mass of the least massive particle of the force field is strictly positive

(that is, the mass of each type of elementary particle is bounded below by a strictly positive value).

Navier-Stokes Existence and Smoothness

It has not yet been proven that the Navier-Stokes equations:

always exist in ordinary $3$-dimensional space

if they do exist, they do not contain any singular points.

The Birch and Swinnerton-Dyer Conjecture

When the solution to a Diophantine equation in polynomials are the points of an Abelian variety, the order of the group of rational points is related to the behavior of an associated $\zeta$ (zeta) function $\map \zeta s$ near $s = 1$.

In particular:

if $\map \zeta 1 = 0$ then there is an infinite set of rational points

if $\map \zeta 1 \ne 0$ then there is a finite set of rational points.