Quantifying Spatial Relationships in Labelled Data with Topology

Topological data analysis (TDA) deals with quantifying the "shape of data" using tools from algebraic topology and computational geometry. In many contexts, data comes equipped with a labelling (for example, cell type annotations in spatial biology), and one is interested in quantifying not just the global structure of the data but the spatial relationships between labelled subsets of the data. I will give a brief introduction to TDA and then talk about chromatic Delaunay filtrations, a recently developed family of computational methods in TDA that can address the problem of quantifying spatial relationships in labelled point cloud datasets.