Cetraro 02/06/08

These lectures will serve as an introduction to lattices. First of all, we will
introduce lattices by considering the packing and the covering problem on a
Euclidean space. This will lead us to the basic definitions and to some
examples. During the first part, we will work on dual lattices and some
geometric inequalities. Furthermore, we will introduce the link between
lattices and error correcting codes, and discuss ideal lattices over number
fields.

The second part will be used to introduce a lot of well known lattices. First
of all, root systems and root lattices will be studied. Then we will see other
lattices, related or not to root lattices. Some important lattices are for
instance the Leech lattice, the Barnes-Wall lattice or Coxeter lattices.


In the next part, we will introduce some tools and examples, namely,
perfection, eutaxy, theta series, extremality (unimodular and $\ell$-modular
lattices), (enumeration of even unimodular $24$-dimensional lattices), (finding
the closest lattice point), (Voronoi cells).

References:

Conway, J.H. and Sloane, N.J.A.: Sphere-packings, lattices, and groups,
Springer Grundlehren Der Mathematischen Wissenschaften; Vol. 290, 1987.

Martinet, J.: Perfect lattices in Euclidean spaces, Springer-Verlag 2003.