John Baez's Description of the Millenium Prize Problems

The following descriptions of the 7 Millenium Prize Problems were taken from John Baez's This Week's Finds in Mathematical Physics - Week 148)

There is a 1 million dollar prize for the solution to each problem offered by the Clay Mathematics Institute. See http://www.claymath.org for more details. Here are John Baez's descriptions of the problems:

1. P = NP?

This is the newest problem on the list and the easiest to explain. An algorithm is "polynomial-time" if the time it takes to run is bounded by some polynomial in the length of the input data. This is a crude but easily understood condition to decide whether an algorithm is fast enough to be worth bothering with. A "nondeterministic polynomial-time" algorithm is one that can check a purported solution to a problem in an amount of time bounded by some polynomial in the input data. All algorithms in P are in NP, but how about the converse? Is P = NP? Stephen Cook posed this problem in 1971 and it's still open. It seems unlikely to be true - a good candidate for a counterexample is the problem of factoring integers - but nobody has proved that it's false. This is the most practical question of the lot, because if the answer were "yes", there's a chance that one could use this result to quickly crack all the current best encryption schemes.

2. The Poincaré conjecture.

Spheres are among the most fundamental topological spaces, but spheres hold many mysteries. For example: is every 3-dimensional manifold with the same homotopy type as a 3-sphere actually homeomorphic to a 3-sphere? Or for short: are homotopy 3-spheres really 3-spheres? Poincaré posed this puzzle in 1904 shortly after he knocked down an easier conjecture of his by finding 3-manifolds with the same homology groups as 3-spheres that weren't really 3-spheres. The higher-dimensional analogues of Poincaré's question have all been settled in the affirmative - Smale, Stallings and Wallace solved it in dimensions 5 and higher, and Freedman later solved the subtler 4-dimensional case - but the 3-dimensional case is still unsolved. This is an excellent illustration of a fact that may seem surprising at first: many problems in topology are toughest in fairly low dimensions! The reason is that there's less "maneuvering room". The last couple decades have seen a burst of new ideas in low-dimensional topology - this has been a theme of This Week's Finds ever since it started - but the Poincaré conjecture remains uncracked.

3. The Birch-Swinnerton-Dyer conjecture.

This is a conjecture about elliptic curves, and indirectly, number theory. For a precise definition of an elliptic curve I'll refer you to "week13" and "week125", but basically, it's a torus-shaped surface described by an algebraic equation like this:

y² = x³ + ax + b

Any elliptic curve is naturally an abelian group, and the points on it with rational coordinates form a finitely generated subgroup. When are there infinitely many such rational points? In 1965, Birch and Swinnerton-Dyer conjectured a criterion involving something called the "L-function" of the elliptic curve. The L-function L(s) is an elegant encoding of how many solutions there are to the above equation modulo p, where p is any prime. The Birch-Swinnerton-Dyer conjecture says that L(1) = 0 if and only if the elliptic curve has infinitely many rational points. More generally, it says that the order of the zero of L(s) at s = 1 equals the rank of the group of rational points on the elliptic curve (that is, the rank of the free abelian summand of this group.) A solution to this conjecture would shed a lot of light on Diophantine equations, one of which goes back to at least the 10th century - namely, the problem of finding which integers appear as the areas of right triangles all of whose sides have lengths equal to rational numbers.

4. The Hodge conjecture.

This question is about algebraic geometry and topology. A "projective nonsingular complex algebraic variety" is basically a compact smooth manifold described by a bunch of homogeneous complex polynomial equations. Such a variety always has even dimension, say 2n. We can take the DeRham cohomology of such a variety and break it up into parts H^p,q labelled by pairs (p,q) of integers between 0 and n, using the fact that every function is a sum of a holomorphic and an antiholomorphic part. Sitting inside the DeRham cohomology is the rational cohomology, The rational guys inside H^p,p are called "Hodge forms". By Poincaré duality any closed analytic subspace of our variety defines a Hodge form - this sort of Hodge form is called an algebraic cycle. The Hodge conjecture, posed in 1950 states: every Hodge form is a rational linear linear combination of algebraic cycles. It's saying that we can concretely realize a bunch of cohomology classes using closed analytic subspaces sitting inside our variety.

5. Existence and mass gap for Yang-Mills theory.

One of the great open problems of modern mathematical physics is whether the Standard Model of particle physics is mathematically consistent. It's not even known whether "pure" Yang-Mills theory - uncoupled to fermions or the Higgs - is a well-defined quantum field theory with reasonable properties. To make this question precise, people have formulated various axioms for a quantum field theory, like the so-called "Haag-Kastler axioms". The job of constructive quantum field theory is to mathematically study questions like whether we can construct Yang-Mills theory in such a way that it satisfies these axioms. But one really wants to know more: at the very least, existence of Yang-Mills theory coupled to fermions, together with a "mass gap" - i.e., a nonzero minimum mass for the particles formed as bound states of the theory (like protons are bound states of quarks).

6. Existence and smoothness for the Navier-Stokes equations.

The Navier-Stokes equations are a set of partial differential equations describing the flow of a viscous incompressible fluid. If you start out with a nice smooth vector field describing the flow of some fluid, it will often get complicated and twisty as turbulence develops. Nobody knows whether the solution exists for all time, or whether it develops singularities and becomes undefined after a while! In fact, numerical evidence hints at the contrary. So one would like to know whether solutions exist for all time and remain smooth - or at least find conditions under which this is the case. Of course, the Navier-Stokes equations are only an approximation to the actual behavior of fluids, since it idealizes them as a continuum when they are actually made of molecules. But it's important to understand whether and how the continuum approximation breaks down as turbulence develops.

7. The Riemann hypothesis.

For Re(s) > 1 the Riemann zeta function is defined by

zeta(s) = 1/1^s + 1/2^s + 1/3^s + ....

But we can extend it by analytic continuation to most of the complex plane - it has a pole at s = 1. The zeta function has a bunch of zeros in the "critical strip" where Re(s) is between 0 and 1. In 1859, Riemann conjectured that all such zeros have real part equal to 1/2. This conjecture has lots of interesting ramifications for things like the distribution of prime numbers. By now, more than a billion zeros in the critical strip have been found to have real part 1/2; it has also been shown that "most" such zeros have this property, but the Riemann hypothesis remains open.

If you solve one of these conjectures and win a million dollars because you read about it here on This Week's Finds, please put me in your will.

Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of Baez's research papers, can be obtained at http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html

A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html