Crystallography is the study of structures that repeat periodically in space. It is a system for describing and encoding the arrangement of atoms in crystalline materials, and is a fundamental description of structure in condensed matter. The recent explosion in the chemical synthesis of metal-organic frameworks (MOFs) has raised a number of questions that require a detailed mathematical understanding of graph theory in the context of crystallography. This field is referred to as topological crystallography. A metal-organic framework is conveniently abstracted as a graph where the vertices represent a metal complex with fixed local symmetry and bonding sites, and the edges represent organic ligands. One of the important properties of MOFs for applications is their porosity at a nanometer length scale. This makes them potentially able to store small molecules (e.g. CO2 or H2) or to act as efficient catalysts due to a large internal surface area. The question of what framework structures will lead to large and stable porosity is therefore a pressingly important one.
Mathematically, this is related to the question of what is a “natural” tiling of space associated with a periodic framework. This might be a tiling by solid 3-dimensional blocks, but a highly porous framework is often better modelled as sitting on a periodic surface embedded in space. This project will examine the deep mathematical questions behind this idea to clarify connections between spatial and surface tilings, and answer outstanding questions such as why two of the simplest periodic frameworks (dia and srs) appear not to have simple surface embeddings.
The project will draw on the mathematical fields of graph theory, discrete symmetry groups, discrete and differential geometry, and algebraic topology.