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 $1 million prize being awarded by the institute to the discoverer(s).
At present, the only Millennium Prize problem to have been solved is the Poincaré conjecture, which was solved by the Russian mathematician Grigori Perelman in 2003.
In dimension 2, a sphere is characterized by the fact that it is the only closed and simply-connected surface. The Poincaré conjecture states that this is also true in dimension 3. It is central to the more general problem of classifying all 3-manifolds. The precise formulation of the conjecture states:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
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 the award. Perelman was officially awarded the Millennium Prize on March 18, 2010, but he also declined that 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 S. Hamilton.
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 and cryptography (see P versus NP problem proof consequences). A common example of a P versus NP problem is the travelling salesman problem.
Most mathematicians and computer scientists expect that P ≠ NP.
The official statement of the problem was given by Stephen Cook.
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.
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.
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.
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.
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.
In the 2017 American film Gifted, a major subplot turns on a mathematician who had made progress on the Navier-Stokes problem before her untimely death. In one scene, a seven-year-old girl is shown a display about the Millennium Problems and expresses a desire to solve one.