# Millennium Prize Problems

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. The problems are Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Poincaré conjecture, Riemann hypothesis, and Yang–Mills existence and mass gap. A correct solution to any of the problems results in a US \$1M prize (sometimes called a Millennium Prize) being awarded by the institute. The only solved problem is the Poincaré conjecture, which was solved by Grigori Perelman in 2003.

## Solved problem

### Poincaré conjecture

Main article: Poincaré conjecture

In topology, a sphere with a two-dimensional surface is characterized by the fact that it is compact and simply connected. The Poincaré conjecture is that this is also true in one higher dimension. The problem is to establish the truth value for this conjecture. The truth value had been established for the analogue conjecture for all other dimensionalities. The conjecture is central to the problem of classifying 3-manifolds.

The official statement of the problem was given by John Milnor.

A proof of this conjecture was given by Grigori Perelman in 2003; its review was completed in August 2006, and Perelman was selected to receive the Fields Medal for his solution but he declined that award.[1] Perelman was officially awarded the Millennium Prize on March 18, 2010,[2] but he also declined the award and the associated prize money from the Clay Mathematics Institute. The Interfax news agency quoted Perelman as saying he believed the prize was unfair. Perelman told Interfax he considered his contribution to solving the Poincaré conjecture no greater than that of Columbia University mathematician Richard Hamilton.[3]

## Unsolved problems

### P versus NP

Main article: P versus NP problem

The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can also find that solution quickly. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether all problems in NP are also in P. This is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in mathematics, and to biology, philosophy[4] and cryptography (see P versus NP problem proof consequences).

Most mathematicians and computer scientists expect that P ≠ NP.[6]

The official statement of the problem was given by Stephen Cook.

### Hodge conjecture

Main article: Hodge conjecture

The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

The official statement of the problem was given by Pierre Deligne.

### Riemann hypothesis

Main article: Riemann hypothesis

The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2. A proof or disproof of this would have far-reaching implications in number theory, especially for the distribution of prime numbers. This was Hilbert's eighth problem, and is still considered an important open problem a century later.

The official statement of the problem was given by Enrico Bombieri.

### Yang–Mills existence and mass gap

In physics, classical Yang–Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (gluons). However, the postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the mass gap. Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap.

The official statement of the problem was given by Arthur Jaffe and Edward Witten.[7]

### Navier–Stokes existence and smoothness

The Navier–Stokes equations describe the motion of fluids. Although they were first stated in the 19th century, they are still not well-understood. The problem is to make progress towards a mathematical theory that will give insight into these equations.

The official statement of the problem was given by Charles Fefferman.

### Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture deals with certain types of equations; those defining elliptic curves over the rational numbers. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. Hilbert's tenth problem dealt with a more general type of equation, and in that case it was proven that there is no way to decide whether a given equation even has any solutions.

The official statement of the problem was given by Andrew Wiles.[8]