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:

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.

y^{2} = x^{3} + 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.

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.

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