REU Project

Topological data analysis (TDA) provides tools for studying time-varying data, but many existing methods struggle to distinguish data that are isometric at each time but differ qualitatively. Kim and Memoli's spatiotemporal persistent homology addresses this shortcoming but produces multiparameter modules that are computationally challenging. To mitigate this, we introduce dynamic curvature-set persistent homology, extending Gomez and Memoli's curvature-set persistence to dynamic metric spaces. The resulting modules are interval-decomposable and admit efficient algorithms for computing interleaving distances. Our construction is stable with respect to the dynamic Gromov-Hausdorff distance introduced by the same authors, and we implement both the erosion distance $d_E$ (due to Puuska) and a new Hausdorff-inspired distance $d_2$ with cubic and quadratic runtime, respectively. This enables a robust computational pipeline for distinguishing dynamic data, as demonstrated in experiments with the boids model, where we successfully detect parameter changes.