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 nondeterministic 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
The Millennium problems 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