Topological Data Analysis: Foundations, Algorithms, and Emerging Applications

Topological data analysis (TDA) has evolved into a flexible and robust paradigm for obtaining qualitative, geometry-inspired insights from high-dimensional, noisy, and complex data. Grounded in algebraic topology, geometry, statistics, and machine learning (ML), TDA provides multiscale descriptions through persistent homology, Mapper (a graph-based method that summarizes the shape of high-dimensional data), and related topological signatures that are often inaccessible to standard linear and metric methods. In recent years, and especially during 2024–2025, TDA has expanded rapidly across science, engineering, biomedical research, and socio-economic studies, while also being integrated with modern learning paradigms such as deep learning (DL) and graph learning. This survey summarizes recent developments in TDA using a carefully selected set of articles, with emphasis on 2024–2025. We first present the mathematical and computational foundations of TDA, covering simplicial complexes, filtrations, persistent homology, the Mapper algorithm, and computational advances such as data simplification, stability, and efficiency. We then review applications in time series and dynamical systems, biomedical imaging and precision medicine, engineering and physical sciences, finance and risk analysis, DL and interpretability, and security and critical infrastructure systems. Throughout, we highlight how TDA can extract informative features, function as a model component, and provide a conceptual lens for studying complex systems. However, the survey also emphasizes recurrent failure patterns: TDA performance is highly sensitive to filtration, embedding, and vectorization choices; aggressive simplification can dilute or remove informa.