Sphere Packing and Lattices

In 1611, Kepler proposed that close packing (either FCC or HCP) 
are the densest possible sphere packing. This conjecture was only 
recently (1998) proved by Hales. A little lesser known conjecture 
is the dodecahedral conjecture which says that the volume of the 
Voronoi polyhedron in a packing of spheres of unit radius is at 
least the volume of a regular dodecahedron with 
inradius 1. This will give an upper bound on the packing density.
I will try to outline the proof of this conjecture.

As an introduction I will give a brief overview about lattices and 
their representations.