One of the most fundamental descriptions of shape and structure is via topology - the mathematical description of how many connected components, independent loops and enclosed voids an object has.  Topological Data Analysis uses this fundamental viewpoint to extract signatures of shape from many different types of data, from point clouds to digital images, pairwise distance matrices, and even text documents. This information can give insight into many different problems. 
 
Recent work in this area from my group includes

• Identifying local motifs of quasi-crystalline structure from molecular dynamics simulations
• The correspondence of graph-theoretic max-flow to absolute permeability of porous materials
• Topological structure of call-stacks from software vulnerability research
• Pore-configurations in the crystallization of granular bead packs
• Principle component analysis (PCA) of persistent homology rank functions
• Skeletonisation and partitioning of digital images

Current and upcoming projects include 

• Characterising the distribution of leaf shapes
• Self-assembly of periodic surfaces in molecular dynamics simulations
• A study of lung CT-images to determine signatures of disease progression and
• Mathematical results explaining how persistent homology changes with decreasing voxel resolution in digital images. 

Projects range from data analysis to physical modelling to mathematical derivation of results and will be tailored to suit interested students with physics or mathematics background.