Homology theories in topological data analysis

At the centre of modern topological data analysis one finds persistent homology, which provides some of the essential computable invariants that have found use in a wide variety of contexts. This project will focus first of all on developing formal properties of persistent homology, starting with the excision property. The aim here is to complete the list of already known formal properties of persistent homology to make it similar to other known homology theories. A second aim of the project is to develop a new form of twisted de Rham-type homology theory for simplicial complexes. This will extend the work of Brodzki, Mukherjee and Gao from the case of graphs to the general case. This will provide a very useful generalisation of the algebraic and geometric structure that is at the centre of synchronisation problems. The project will also investigate appropriate practical applications of these ideas.