# Definition:Millennium Problems

## Definition

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.

## Also known as

Can also be referred to as the **Millennium Prize problems**.

## Also see

- Results about
**the Millennium Problems**can be found**here**.

## Historical Note

The **Millennium problems** were chosen by the Clay Mathematics Institute and announced on $24$ May $2000$.

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

They were directly inspired by the Hilbert $23$ of David Hilbert from $1900$.

## Sources

- 2002: Keith Devlin:
*The Millennium Problems* - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Millennium Prize problems** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Millennium Prize problems** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Appendix $18$: Millennium Prize problems - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next): Appendix $23$: Millennium Prize problems