1.The P versus NP Problem home / millennium prize problems / the P versus NP problem

It is Saturday evening and you arrive at a big party. Feeling shy, you wonder how many people you already know in the room? Your host proposes that you must certainly know Rose, the lady in the corner next to the dessert tray. In a fraction of a second you are able to cast a glance and verify that your host is correct. However, in the absence of such a suggestion, you are obliged to make a tour of the whole room, checking out each person one by one, to see if there is anyone you recognize. This is an example of the general phenomenon that generating a solution to a problem often takes far longer than verifying a given solution is correct. Similarly, if someone tells you that the number 13,717,421 can be written as the product of two smaller numbers, you might not know whether to believe him, but if he tells you that it can be factored as 3607 times 3803 then you can easily check that it is true using a hand calculator. The problem of deciding whether the answer can be quickly checked can really take much longer to solve, no matter how clever a program we write, is considered one of the outstanding problems in logic and computer science. It was formulated by Stephen Cook in 1971.

2.The Hodge Conjecture home / millennium prize problems / the hodge conjecture

In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add pieces that did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces called projective algebraic varieties, the pieces called Hodge cycles are actually (rational linear) combinations of geometric pieces called algebraic cycles.

3.The Poincare Conjecture home / millennium prize problems / the poincare conjecture

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the the surface of the apple is 'simply connected,' but that the surface of the doughnut is not. Poincare, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out be be extraordinarily difficult, and mathematicians have been struggling with it ever since.

4.The Riemann Hypothesis home / millennium prize problems / the riemann hypothesis

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function "z(s)" called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation z(s) = 0 lie on a straight line. This has been checked for the first 1,500,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.

5.Yang-Mills Theory home / millennium prize problems / yang-mills theory

The equations of quantum physics describe the world of elementary particles. Almost fifty years ago, the physicists Yang and Mills discovered a remarkable relationship between geometry and particle physics, embodied in these equations. In so doing, they paved the way to later combination of the laws for electro-magnetic forces with those for strong and weak ones. The predictions culled from these equations describe particles observed at laboratories around the world, including Brookhaven, Stanford, and CERN. However, the gauge theories of Yang and Mills are not known to have solutions compatible with quantum mechanics, nor to describe the particles observed in nature. Despite this, the "mass gap" hypothesis concerning supposed solutions to the equations is taken for granted by most physicists and provides an explanation of why we do not observe "quarks." Solving this mathematical problem requires establishing a mathematical proof of this phenomenon.

6.Navier-Stokes Equations home / millennium prize problems / navier-stokes equations

Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. These and other fluid phenomena are described by the mathematical equations known by the names of the mathematicians Navier and Stokes. Unlike many problems in quantitative science, the solutions to these equations are not known, nor is it even known how to solve these equations. The solution to this problem entails showing the existence and smoothness of solutions to the Navier-Stokes equations.

7.The Birch and Swinnerton-Dyer Conjecture home / millennium prize problems / the birch and swinnerton-dyer conjecture

Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like x^2 + y^2 = z^2 . Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function z(s) near the point s=1. In particular this amazing conjecture asserts that if z(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if z(1) is not equal to 0, then there is only a finite number of such points.