Abstract: Cut and project point sets are defined by identifying a strip of a fixed n-dimensional lattice (the "cut"), and projecting the lattice points in that strip to a d-dimensional subspace (the "project"). Such sets have a rich history in the study of mathematical models of quasicrystals, and include well known examples such as the Fibonacci chain and vertex sets of Penrose tilings.
In the talk I will present the "space of quasicrystals" as introduced by Jens Marklof and Andreas Strombergsson. Given a cut and project set, vary the n-dimensional lattice according to an SLdR action. This defines a "space of quasicrystals", which following Ratner is shown to have a homogeneous structure. Equidistribution results may be applied to establish generic properties of cut and project sets, such as a Siegel summation formula. I will describe this construction and discuss some properties.