Grant: Discovery Grants Program - Individual

This proposal is in mathematical physics, and it is mainly devoted to the study of properties of mathematical models for physical quasicrystals and their diffraction pattern.Since the discovery of quasicrystals by Dan Shechtman in 1984, a discovery for which he was awarded the Nobel Prize for Chemistry in 2011, physicists and mathematicians have been trying to understand the atomic structure of these fascinating structures. In recent years, the unusual physical properties of quasicrystals have led to many practical applications. Some of the aluminum?lithium alloys used in the aerospace industry contain quasicrystalline precipitates. Also, some alloys of steel containing quasicrystals are used to produce superior surgical and acupuncture needles and dental reamers. With the discovery of more potential applications for quasicrystals, it becomes even more important to understand these solids.In the first project, we propose a new method for creating and studying mathematical models, which generalizes the current cut and project method. The project continues research I have done in the past 10 years, and extends it into a previously unexplored direction. This new method leads to a new way of exploring properties of model sets and Meyer sets and models sets, and introduces a new class of mathematical models with interesting diffraction diagrams.In several successfully completed projects, I studied the diffraction of Meyer sets. I showed that this diffraction is essentially discrete (which is the signature of crystals), and that the diffraction diagram is highly ordered. In the second project, we plan to explore the other direction and seek diffraction conditions, which combined with local properties, would imply the Meyer condition. This is an important problem in our research area, and in this project we propose a new method for solving this question.In the third project, we investigate a very deep connection between almost periodicity and the cut and project formalism. The apparent connection between these two seemingly unrelated concepts was first observed 25 years ago, but it is yet to be explained. In this project, we introduce the new concept of sup almost periodic measures, and investigate the properties of these measures and the connection between this new type of almost periodicity and the cut and project schemes.