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

• Faster algorithms for persistent homology computation from point clouds
• Symmetry quantification for two-dimensional shapes
• The extended persistent homology transform
• Identifying local motifs of quasi-crystalline structure from molecular dynamics simulations
• The correspondence of graph-theoretic max-flow to absolute permeability of porous materials
• Pore-configurations in the crystallization of granular bead packs

Current and upcoming projects include 

• Characterising the geometry of the lung airway tree from chest CT images
• Self-assembly of crystalline and disordered phases in molecular dynamics simulations
• Symmetry quantification for three-dimensional shapes 

Projects can be tailored to suit student backgrounds in physics or mathematics.  Potential projects range from data analysis to physical modelling to the mathematical derivation of new methods and results.