Millennium Prize Problems


The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute on May 24, 2000. The problems are the 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 1 million prize being awarded by the institute to the discoverer.
To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture, which was solved in 2003 by the Russian mathematician Grigori Perelman. He declined the prize money.

Solved problem

Poincaré conjecture

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:
A proof of this conjecture was given by Grigori Perelman in 2003, based on work by Richard Hamilton; 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 Hamilton.

Unsolved problems

P versus NP

The question is whether or not, for all problems for which an algorithm can verify a given solution quickly, 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. A common example of an NP problem not known to be in P is the Boolean satisfiability problem.
Most mathematicians and computer scientists expect that P ≠ NP; however, it remains unproven.
The official statement of the problem was given by Stephen Cook.

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

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. 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, and are one of the pillars of fluid mechanics. However, theoretical understanding of their solutions is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, the general solution for which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.
Even basic properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass. This is called the Navier–Stokes existence and smoothness problem.
The problem is to make progress towards a mathematical theory that will give insight into these equations, by proving either that smooth, globally defined solutions exist that meet certain conditions, or that they do not always exist and the equations break down.
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.