Thursday, 16th June 2005, 4:00 pm

Mathematics Building, Lecture Hall 2

Frank Vallentin

(HU)

"Symmetry in Sphere Packings and Sphere Coverings"

** Abstract: **

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.

Light refreshments will be served in the faculty lounge at 3:30.

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