Sphere Packings and Discrete Geometry
The Kepler conjecture asks what is the densest packing of congruent balls in three-dimensional Euclidean space. Hales and graduate student Sam Ferguson solved this conjecture in 1998. The proof requires a number of long computer calculations. These include linear programming, computer classification of certain planar graphs, and interval arithmetic calculations.
Another problem in discrete geometry that Hales solved is the honeycomb conjecture, which asserts that the most efficient partition of the plane into equal area cells is the hexagonal honeycomb.