Persistent Homology in Topological Data Analysis

Persistent homology provides a framework for studying the topology of data across multiple scales. Given a point cloud $X \subset \mathbb{R}^n$, we construct a filtration of simplicial complexes ${K_\epsilon}_{\epsilon \geq 0}$ and track how homological features — connected components, loops, voids — appear and disappear as $\epsilon$ varies. The resulting persistence diagram $\text{Dgm}(X)$ is a stable summary of the data's topological structure: the bottleneck distance $d_B(\text{Dgm}(X), \text{Dgm}(Y)) \leq d_H(X, Y)$ ensures robustness to noise. We present recent applications to the study of amorphous materials and neural population geometry.