The discovery of quasicrystals - honoured with the 2011 Nobel Prize in Chemistry - came as a surprise to physicists, chemists and materials scientists. Quasicrystals are structures which possess long-range order but lack translational symmetry and have found a wealth of applications, from catalysts to low friction coatings, over to theoretical applications in quantum computing. As with many structures, most physical quasicrystals are not perfect, and exhibit local defects. Thus, good theoretical models for quasicrystals should exhibit features of both long-range order and local disorder. Such models are provided by tiling spaces of random substitutions. Here, we will initiate the development of a much-needed and substantial hierarchical classification of these models.
Topological entropy is an invariant of dynamical systems which quantifies their complexity. Throughout this project, we will develop a robust theory of topological entropy for random substitution tiling spaces, leveraging recent developments for symbolic random substitution systems in one-dimension, combined with the rich theory of deterministic substitution tiling spaces. Moreover, we will consider a novel approach to complexity of such tiling spaces in dimensions two and higher, by quantifying the complexity of lower-dimensional subsystems induced by slicing the original system along a hyperplane. This will provide a method of discerning tilings which have the same topological entropy. Thus, creating a new means of distinguishing and ordering random substitution tiling spaces based on their complexity, and hence, taking the initial steps towards establishing a hierarchical classification.
The themes of this project can be found in many mathematical fields outside of aperiodic order, including dynamical systems, fractal geometry, geometric measure theory, harmonic analysis and number theory. Further, as random substitution tiling spaces provide theoretical models for physical phenomena, such as quasicrystals and fractal percolation, we expect that the techniques we develop will have a lasting impact beyond mathematics.