# 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).

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.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Millennium Problems"

The following 7 pages are in this category, out of 7 total.