I was awarded a 2020 Discovery Early Career Research Award to research theory and applications of topological methods to perform statistical shape analysis. The main tools are the persistent homology transform and the Euler characteristic transform. The goal is to quantitatively compare geometric objects such as a set of bones, tumours, leaves, bird beaks, etc.
Persistent homology transform for modeling shapes and surfaces
K Turner, S Mukherjee, DM Boyer - Information and Inference: A Journal of the IMA, 2014
This paper introduces a completely different approach to statistical shape analysis using an algebraic topology analogs of the Radon transform. It can be computed automatically and has no information loss. These methods have been adapted and applied by other research groups including predicting patient outcomes from shapes of brain tumours.
How many directions determine a shape and other sufficiency results for two topological transforms
J Curry, S Mukherjee, K Turner - arXiv preprint arXiv:1805.09782, 2018