Topological data analysis (TDA) provides powerful tools for understanding the structure of complex, high-dimensional data, yet most existing methods focus on points, graphs, or simplicial complexes. In this talk, I will present our recently developed de Rham–Hodge–based frameworks for analyzing data on manifolds. These methods provide effective and efficient ways to capture both the topological and geometric information of data and are well-suited for integration with machine learning tasks. I will demonstrate their usefulness through applications in mathematical biology, including protein–ligand binding affinity prediction, single-cell RNA velocity analysis, medical image classification, and B-factor analysis.