Local Theory in Delone Sets and Tilings.

27^th January 2026, 14:00
Egon Schulte
Northeastern University, US

Abstract

Local detection of global properties in geometric structures is a usually challenging problem of great interest in crystal chemistry. The classical Local Theorem for Delone Sets locally characterizes the uniformly discrete point sets in Euclidean d-space which are regular systems, that is, orbits under crystallographic groups. This is closely related to the Local Theorem for Tilings which says that a tiling of Euclidean d-space is tile-transitive (isohedral) if and only if the large enough neighborhoods of tiles (coronas) satisfy certain conditions. Both results are of great interest in the modeling of crystals. We discuss old and new results from the local theory of Delone sets and tilings including new developments on the regularity radius of Delone sets. The regularity radius \hat{rho}_d is defined as the smallest positive number rho such that each Delone set with congruent clusters of radius rho is a regular system. We discuss bounds for the regularity radius in terms of the parameters of Delone sets, and offer conjectures that have been verified for some particularly interesting classes of Delone sets. The latter is joint work with Nikolai Dolbilin, Alexey Garber and Marjorie Senechal.