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L-types and the covering density problem.
Special Events| Speaker: | Mathieu Dutour Sikiric, Institut Rudjer Boskovic |
| Location: | 2112 MSB |
| Start time: | Wed, Feb 1 2006, 4:10PM |
Description
A lattice is a rank n subgroup of R^n. A covering of R^n is
a family of balls of equal radius such that any point belongs
to at least one ball. The covering density is the average number
of balls to which points of R^n belongs to. Our main purpose is
to minimize the covering density in the lattice case: coverings
defined by balls whose center belong to a lattice.
To any lattice L, one associates a Gram matrix G by taking a
basis of the lattice. This is the key idea of Lattice Theory
allowing to use analytic tools. A Delaunay polytope of a lattice
is the convex hull of points lying on an empty sphere. They
form a normal tessellation of R^n (dual to Voronoi tiling).
The covering density is expressed in terms of maximum
radius of Delaunay polytopes and determinant of the Gram
matrix.
L-type were introduced by Voronoi and are defined as the set
of Gram matrices having the same Delaunay tessellation. This
parameter space, together with a semidefinite programming
program of Vallentin and Schuermann allow us to solve the
lattice covering problem, provided that one knows all L-type
domains. In practice, this is possible only up to dimension 5.
We will present the generalization of L-type theory to lattices
having a fixed symmetry group. This will allow us to find best
known covering in dimension 9-15.
Then, we will consider the following extensions of the theory:
- to the case of describing a single Delaunay polytope in a lattice.
- to the case of several orbits of points under translation in searching for non-lattice coverings.
We will mention in passing many interesting and "record breaking" structures that show up in this work.