Topological data analysis for multiscale biology

Many processes in the life sciences are inherently multi-scale and dynamic. Spatial structures and patterns vary across levels of organisation, from molecular to multi-cellular to multi-organism. With more sophisticated mechanistic models and data available, quantitative tools are needed to study their evolution in space and time. Topological data analysis (TDA) provides a multi-scale summary of data. We review persistent homology and then highlight applications of single parameter persistent homology and multiparameter persistence to proteins, cancer and spatial transcriptomics. Time permitting, we present in-progress work for quantifying spatio-temporal trajectories, which builds on work by Kim and Memoli, Bubenick, Vipond and Lesnick, Bender and Gäfvert.