Title: Collapsing higher order Čech to higher order Delaunay complexes: current results and open questions

Abstract: In TDA, it is common to use simplicial complexes to obtain spaces from finite point sets, with the Čech and Delaunay complexes being two well-known options. These can be extended to higher order versions that offer a more robust approach to handling outliers. In this talk, we introduce the higher order Čech and Delaunay complexes and then discuss ongoing work on how to relate them via simplicial collapses. To achieve this, we allow (higher order) Voronoi regions to grow, forming what we call t-Voronoi regions. This approach leads to an explicit collapsing sequence. However, this is ongoing work and hence there is a catch. We finish pointing out the open questions.