Monday, June 10

The second Mathematical Crystallography minisymposium was entitled Beyond Classical Crystal Symmetry.

Massimo Nespolo talked about A Stroll Along Structural Paths: Symmetry, Pseudo-Symmetry and Their Exploitation to Understand and Design Structures. Equivalence relations among crystal structures, e.g. isostructure or isomorphous crystals define properties that may be preserved in phase transitions. This results in a classification using archetypic aristotypes that represent families of lower symmetry hettotypes.

Uwe Grimm talked about Recent Advances in Mathematical Diffraction Theory. A diffraction pattern will be a sum of discrete Bragg peaks, a diffuse scattering, and an absolutely continuous background. (For a crystal with atoms at lattice points, the diffraction consists solely of the reciprocal lattice of the given crystal lattice, perhaps represented as a Dirac comb.) But for quasicrystals, one may use a salami slice from a higher dimensional lattice, and a more complex structure. Less orderly structures will have diffusion and background components in their diffraction.

Marjorie Senechal talked about Periodicity, Aperiodicity and Prehistory. 270 years since Rene Just Hauy’s birth, his model of crystals as periodic structures of arrayed “nucleal molecules” is no longer the consensus. Quasicrystals – some of which were portrayed as aperiodic tilings, some of which are systems of clusters (even icosahedral clusters) – made Hauy’s model “wobble”. With the International Year of Crystallography upon us, what is the role of mathematics in crystallography? What are the major problems in mathematical crystallography? E.g., infinite structures, diffuse diffractino, self-assembly, folding, dense packing.

Peter Zeiner talked about Coincidence Site Lattices and Well-rounded Sublattices in the Plane. There are several kinds of nice lattices. For example, a lattice in d-space is well-rounded if its vectors of minimal length span d-space; dense sphere packings induce well-rounded lattices. So what lattices have well-rounded sub-lattices? There is a necessary and sufficient condition for a lattice in the plane to have a well-rounded sub-lattice, and there are methods for enumerating them. Other issues and other kinds of niceness were described.

Then we went to lunch at Imperial Inn.