Topological Data Analysis
Topological Data Analysis (TDA) provides stable-under-noise methods to quantify geometric properties of topological features such as cycles and holes in unorganised data across all scales. Our group also develops TDA applications to Materials Science (with Materials Innovation Factory at Liverpool and Cambridge Crystallographic Data Centre, UK), Computer Vision (with Microsoft Research at Cambridge, UK), Climate (with the Intel Parallel Computing Centre at the University of Liverpool and Lawrence Berkeley National laboratory, USA). More details including papers, C++ software and blog are on the personal webpage

Mathematical crystallography
Our Topological Data Analysis group develops a new continuous approach to quantify a similarity of solid crystalline materials. The new methods go beyond the past discrete classifications in terms of symmetry groups in order to quickly analyse large datasets of simulated crystals for more efficient crystal structure prediction.

Pure mathematics
Topology (embeddings of graphs and high-dimensional knots), singularities (1-parameter projections of links), non-commutative algebra (exponential equations and braid groups)