Topological data analysis provides mathematically rigorous computational tools for quantifying connectivity in geometric data sets. The primary mathematical theory is called persistent homology, it measures topological quantities such as components, loops and higher-dimensional cycles as a function of a geometric parameter. A central lesson from TDA is that topological structure in data can only be robustly quantified by studying how it varies over a sequence of length-scales.
Recent applications of persistent homology in materials science include:
- the characterisation of local configurations of spheres in bead packings, and atomic arrangements in fluids, where it enables a clearer picture of phase transitions;
- analysis of Rayleigh-Benard convection patterns to detect and quantify departures from the Boussinesq approximation;
- generating topologically consistent grain partitions and pore networks from x-ray CT images of porous and granular materials to assist in better modelling of fluid transport.
Further applications in engineering include coverage in sensor networks, robot motion planning, and image processing.
This talk will focus on the interpretation of persistence diagrams in the context of porous and granular materials in order to demonstrate exactly what sort of information can be obtained from TDA and how it can lead to new physical insights.