Find out why six unsolved maths problems are worth $1,000,000 each!

The Millennium Prizes were set up in 2000 to reward anyone who could solve one of seven unsolved problems in mathematics. So far, only one has been solved. The Poincaré conjecture was solved by Grigori Perelman, but he declined the award in 2010.

#1 P versus NP

The question is whether, 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. The former describes the class of problems termed NP, whilst the latter describes P. The question is whether or not 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^[1] and cryptography (see P versus NP problem proof consequences).

“If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in ‘creative leaps,’ no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss…” — Scott Aaronson, MIT.

Most mathematicians and computer scientists expect that P ≠ NP.

#2 The Hodge conjecture

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

#3 The Poincaré conjecture (proven)

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 question had been solved for all other dimensions. The conjecture is central to the problem of classifying 3-manifolds.

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. Perelman was officially awarded the Millennium Prize on March 18, 2010, 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 Poincare conjecture no greater than that of Columbia University mathematician Richard Hamilton.

#4 The 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.

#5 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 ofcolor 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.

#6 Navier–Stokes existence and smoothness

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

#7 The Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture deals with a certain type of equation, 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.