Abstracts Mathematics

The Persistent Topology of Geometric Filtrations

by Qingsong Wang

We study the theoretical foundation of the persistent topology of the geometric filtrations in Topological Data Analysis (TDA), such as Vietoris – Rips simplicial complexes, Vietoris – Rips metric thickenings.We introduce a $\ell_p$-relaxation to the Vietoris – Rips metric thickening where $p=\infty$ recovers the usual Vietoris – Rips metric thickening.We prove a stability theorem for the persistent homology of $\ell_p$ relaxed metric thickenings, which is novel even in the case $p=\infty$.The stability theorem then can be employed to show that the filtrations by Vietoris – Rips simplicial complexes and Vietoris – Rips metric thickenings have the same persistent diagram.Therefore, we can employ measure-theoretical methods to study the Vietoris – Rips complex.Some recent study also suggests that the persistent homology of Vietoris – Rips simplicial complex changes when the scale passes the diameter of some extremal configuration of the diameter functional.As an example, we study the extremal configurations on spheres.We implemented the diameter gradient flow and obtained nontrivial extremal configurations on $\Sp^2$ and $\Sp^3$.We find a natural condition for metric spaces that will guarantee the vanishing of the persistence diagram of Vietoris – Rips filtration over certain dimensions. We also demonstrate by a non-collapsing result that the persistent features can be utilized to obtain a quantitive lower bound for the Gromov – Hausdorff distance between Riemannian manifolds.