Grant: Discovery Grants Program - Individual

In 1980' Dan Shechtman discovered a solid with a discrete diffraction diagram and five-fold rotation symmetry. For one hundred years only fully periodic crystals showed a discrete diffraction diagram, but such solids cannot have a five-fold rotation symmetry. The newly discovered solids are now called quasicrystals, and Schectman was awarded the 2011 Nobel Prize in Chemistry for their discovery. Quasicrystals are solids with many unique properties, leading to many practical applications and showing great untapped potential for a wide array of applications. The area of mathematics studying mathematical models for quasicrystals is called Aperiodic Order. It lies at the intersection of Harmonic Analysis, Fourier Analysis, Discrete Geometry, Dynamical Systems and Ergodic Theory, and Number Theory. It is a new area of mathematics, which is growing fast and expanding in many directions, and offers a wide variety of projects for HQP training. The goal of this proposal is to gain a better understanding of mathematical diffraction theory and mathematical models for quasicrystals. We will first look at systems with a pure point spectrum, continuing and expanding our previous work in this direction. Then, we will look at systems with a mixed diffraction spectrum, moving into a new but related direction. We will start the project by looking at almost periodic measures with Meyer set support, combining two of our areas of expertise. We plan to investigate the deep connection between almost periodic measures with Meyer set support and cut and project schemes, expanding on our previous work in this direction. As an application, we will investigate the class of maximal density weak model sets, which contains many models of number theoretic origin. We will continue the project by investigating systems with mixed diffraction spectrum. While perfect crystals and quasicrystals have pure point diffraction, with the improvement of the diffraction techniques and tools, physicists realized that even the purest physical crystals do show a small continuous(diffuse) spectrum. The presence of this small diffuse background can be explained by the thermal motion present in solids, slight imperfections in the solid, or both. There is now renewed interest in systems with mixed spectrum, both by mathematicians and physicists. As mentioned above, this project aims to understand better the mathematics behind the physical phenomena of diffraction. Ultimately, we would like to understand how to reconstruct a solid from its diffraction pattern. This problem is important for physics, chemistry, material sciences and pharmacology.