The Millennium Problems

Want to earn a million dollars? There's a way, but you have to be really smart! Just find a solution to one of the Millennium Problems of the Clay Mathematics Institute (CMI), which named the seven problems to celebrate mathematics in the new millennium. The Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each. During the Millennium Meeting held on May 24, 2000 at the College de France, Timothy Gowers presented a lecture entitled The Importance of Mathematics, aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. Up for it?

#1 The Poincaré Conjecture

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, is a special case of Thurston's geometrization conjecture. The latter would give an almost complete understanding of three dimensional manifolds.

But, too bad someone already had dibs on this. After nearly a century of effort by mathematicians, Grigori Perelman sketched a proof of the conjecture in a series of papers made available in 2002 and 2003. The proof followed the program of Richard Hamilton. Several high-profile teams of mathematicians have since verified the correctness of Perelman's proof.

The Poincaré conjecture was, before being proven, one of the most important open questions in topology. It is one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined. Don't worry, there's still hope, the Poincaré conjecture remains the only solved Millennium problem.

#2 The 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.

#3 The Hodge Conjecture

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.

#4 The Navier-Stokes Equation

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.

#5 The 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 (by car), 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 (given the methods I know) find a solution.

#6 The Riemann Hypothesis

Formulated in his 1859 paper, the Riemann hypothesis in effect says that the primes are distributed as regularly as possible given their seemingly random occurrence on the number line. Riemann's work gave an 'explicit' formula for the number of primes less than x in terms of the zeros of the zeta function. The first term is x/log(x). The Riemann hypothesis is equivalent to the assertion that other terms are bounded by a constant times log(x) times the square root of x. The Riemann hypothesis asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.

#7 The Yang-Mills Theory

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.

To know more about these math problems, visit the Clay Mathematics Institute website.