Topological Data Analysis for Matching Homology Classes in Time-Varying Data

This thesis explores Topological Data Analysis (TDA) as a framework to study time-varying datasets, which appear in fields ranging from biology and dynamical systems to social networks and neuroscience. TDA encodes data as point clouds and uses algebraic topology to capture intrinsic shape. Persistent Homology (PH) extracts topological features, identifying homology classes (connected components, loops, and higher-dimensional voids) across multiple scales through filtrations. Each feature is assigned a birth and death parameter, visualized via barcodes or persistence diagrams. PH provides a canonical representation of features and is robust to small perturbations. To extend PH to dynamic datasets, this thesis employs vineyards, which track the evolution of homology classes over time, and Topological Optimal Transport (TpOT), a computationally efficient method for matching topological features across large datasets and multiple time steps. Vineyards preserve interpretability but can be computationally intensive, while TpOT aligns features efficiently. Overall, this work combines algebraic and computational approaches to analyze the dynamics of homology classes over time, balancing theoretical rigor with practical efficiency.