Crystallographic and quasicrystallographic groups

Valery A. Churkin,
IM SB RAS, Novosibirsk, Russia.

This course of lectures consists of the following parts.
1. The historical review and main results for classical crystallographic groups of translations of Euclidean plain and space.
2. Penrose tiling, quasicrystals and quasicrystallographic groups of translations of an Euclidean space.
3. Connection with crystallographic groups of translations of a pseudo-Euclidean space.
4. An example of a quasicrystallographic group of translations of the Minkowski plane without nondegenerate invariant symmetric bilinear form on its quasilattice as an abelian group, so nonisomorphic to a crystallographic group in a pseudo-Euclidean space.
5. Restricted Bieberbach theorem for pseudo-Euclidean spaces of index (the maximal dimension of a totally isotropic subspace) not greater than 2. Counter examples for restricted Bieberbach theorem for spaces of index greater than 2: groups with two distinct subgroups, that can be realized as lattices of translations under distinct representations in groups of translations of pseudo-Euclidean spaces (pseudo-Euclidean lattices).
6. Theorem on possible values of coranks of intersections for two distinct pseudo-Euclidean lattices in the lattices for a crystallographic group. A crystallographic group with two pseudo-Euclidean lattices in a 6-dimensional space of type (3,3): the description of all its lattices and automorphisms, uniqueness theorem. A crystallographic group with exactly 2n pseudo-Euclidean lattices.

On practical seminars geometrical objects and corresponding groups will be constructed.

This course will consist of eight lectures.