Local Homology

Local homology, a branch of topological data analysis (TDA), focuses on identifying and characterizing the local topological features within data, often represented as point clouds or simplicial complexes. Current research emphasizes the development and application of persistent homology-based methods, including persistence images and kernels, for classification tasks across diverse fields like medical imaging, program analysis, and robotics. This approach offers powerful tools for analyzing complex datasets by revealing underlying structures and relationships invisible to traditional methods, leading to improved performance in applications ranging from object recognition to brain network analysis.