Topological Data Analysis: Developments and Applications

Topological Data Analysis (TDA) and its mainstay computational device, persistent homology (PH), has established a strong track record of providing researchers across the data-driven sciences with new insights and methodologies by characterizing low-dimensional geometric structures in high-dimensional data. When combined with machine learning (ML) methods, PH is valued as a discriminating-feature extraction tool. This work highlights many of the recent successes at the intersection of TDA and ML, introduces some of the foundational mathematics underpinning TDA, and summarizes the efforts to strengthen the bridge between TDA and ML. Thus, this document is a launching point for experimentalists and theoreticians to consider what can be learned from the shape of their data.