"Symmetry in Sphere Packings and Sphere Coverings"
The sphere packing problem asks for the most efficient way to pack equal non-overlapping spheres into d-dimensional Euclidean space whereas the sphere covering problem asks for the most efficient way to cover d-dimensional Euclidean space by equal overlapping spheres.
In this talk I will consider efficient sphere packings and coverings coming from low dimensional lattices. The collection of good lattices appears to be a zoo with many sporadic and exceptional creatures. I will give a tour through this zoo using symmetry as an organizing principle.
The intuitive notion that a highly symmetric lattice gives an efficient sphere packing is beautifully captured in the notion of strongly perfect lattices introduced by B.B. Venkov ten years ago. A strongly perfect lattice is a lattice whose shortest vectors form a spherical 4-design. All strongly perfect lattices give local optimal lattice sphere packings and many prominent lattices (the 4-dimensional root lattice D_4, the 8-dimensional root lattice E_8 and the Leech lattice in dimension 24) are strongly perfect. There is hope that a classification of all strongly perfect lattices up to dimension 24 is possible. I will give a short introduction to Venkov's theory.
For the sphere covering problem the situation is different. Highly symmetric lattices do not necessarily give efficient coverings. We propose a dichotomy principle instead: a highly symmetric lattice either gives a local optimum or a "local pessimum" for the covering density function. Although a precise statement of this principle is not yet established we give some examples: the lattices D_4 and E_8 give local pessima whereas the lattice A_2 and the Leech lattice give local optima. I will give a report on what are the best known sphere coverings in dimensions 1,...,24.