The Kepler conjecture is about the best way to pack spheres tightly together. Kepler thought that a cubic close packing, the way oranges get stacked at the grocery store, was best. However, it wasn't until 1997 that this was proved for the 3 dimensional case. (As an interesting aside, for the first time ever Annals of Mathematics gave up trying to review his paper... it was just too hard to verify since it relied on tons of computer calculations.)

The problem also can be solved in higher dimensions as well. Henry Cohn and Noam Elkies have published a new paper that uses linear programming to improve the upper bounds in dimensions 4 through 36. They conjecture that their techniques solves the problem (i.e. the bounds meet) for dimensions 8 and 24.

The reason I bring this up is that linear programming is a technique that is near and dear to my heart, and it is cool to see new uses for it.

(Via a facinating article in the print edition of Nature by Ian Stewart, a math professor at Univ. of Warwick who writes lots of popular math books.)