SEMINARIUM WYDZIAŁU FIZYKI TECHNICZNEJ I MATEMATYKI STOSOWANEJ | FTiMS - Politechnika Gdańska

Crystallography is a science of determining the arrangement of atoms in crystalline solids. From the mathematical point of view it is a study of tessellations of the Euclidean spaces. In our joint work [1], [2],[3] we have developed some connections between quantitative features of periodic space tessellations and topological invariants of flat manifolds. Following the original work of Coxeter we have constructed and studied certain ``growth'' functions for any given periodic tessellation. Seeing such a tessellation as a CW-complex gives a way to find features of the growth functions which uniquely identify the space group or even tessellation itself only from the numerical invariants. In the ongoing project we investigate relative growth functions which are restricted by either topological paths of rigid frames. This allows one to include defects of the lattice structure or even use some machine learning techniques to recognize the crystallographic group type from the numerical invariants.