The Millennium Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute (CMI) in 2000. The problems are :
Yang–Mills and Mass Gap
Experiment and computer simulations suggest the existence of a “mass gap” in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.
The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper, it asserts that all the ‘non-obvious’ zeros of the zeta function are complex numbers with real part 1/2.
P vs NP Problem
If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.
This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.
The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.
In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston’s geometrization conjecture. Perelman’s proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.
Birch and Swinnerton-Dyer Conjecture
Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles’ proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three.
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 problem have been solved is the Poincaré conjecture, which was solved by the Russian mathematician Grigori Perelman in 2003. He made a landmark contribution to Riemannian geometry and geometric topology.
In 1994, Perelman proved the soul conjecture. In 2003, he proved (confirmed in 2006) Thurston’s geometrization conjecture. This consequently solved in the affirmative the Poincaré conjecture.
In August 2006, Perelman was offered the Fields Medal for “his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow”, but he declined the award, stating: “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.” On 22 December 2006, the scientific journal Science recognized Perelman’s proof of the Poincaré conjecture as the scientific “Breakthrough of the Year”, the first such recognition in the area of mathematics.
On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. On 1 July 2010, he turned down the prize of one million dollars, saying that he considered the decision of the board of CMI and the award very unfair and that his contribution to solving the Poincaré conjecture was no greater than that of Richard S. Hamilton, the mathematician who pioneered the Ricci flow with the aim of attacking the conjecture. Since then, he has announced he has given up the study of mathematics altogether and has cut off communications with all journalists and nearly all his friends. He also turned down the prestigious prize of the European Mathematical Society.