2025 Theses Doctoral

Persistent Homology of Zig-Zag Families of Filtrations

Saunders, Emily Lauren

This thesis introduces a new variant of multi-parameter persistent homology where one parameter is the standard ℝ, and the other is the zig-zag poset Ƶ. This construction is motivated by the need for a general scale parameter in various applications of zig-zag persistent homology. We begin by outlining the fundamental construction and defining an analog of the interleaving distance for Ƶ x ℝ persistence modules. We then establish stability of the metric with respect to the Gromov-Hausdorff distance and discuss convergence and stability under the topological bootstrapping sampling regime. Finally, we will discuss possibilities for a useful invariant on the space of Ƶ x ℝ persistence modules and explore applications of density sensitive bifiltrations to zig-zag sequences of point clouds.