This paper discusses the application of geometry and topology to the analysis of large, high-dimensional, and noisy data sets, particularly in the context of modern scientific and engineering data. The authors highlight the challenges in analyzing such data, including the need for qualitative information, the lack of theoretically justified metrics, the non-natural nature of coordinates, and the importance of summarizing data rather than focusing on individual parameter choices. They argue that topological methods, which are less sensitive to metric choices and coordinate systems, can address these issues effectively. The paper introduces persistent homology as a tool for inferring topological information from data, demonstrating its application to natural image statistics and neuroscience. It also explores the construction of simplicial complexes from data and the use of functoriality to analyze clustering methods. The authors provide examples and theoretical foundations for these techniques, emphasizing the computational aspects and their practical applications.This paper discusses the application of geometry and topology to the analysis of large, high-dimensional, and noisy data sets, particularly in the context of modern scientific and engineering data. The authors highlight the challenges in analyzing such data, including the need for qualitative information, the lack of theoretically justified metrics, the non-natural nature of coordinates, and the importance of summarizing data rather than focusing on individual parameter choices. They argue that topological methods, which are less sensitive to metric choices and coordinate systems, can address these issues effectively. The paper introduces persistent homology as a tool for inferring topological information from data, demonstrating its application to natural image statistics and neuroscience. It also explores the construction of simplicial complexes from data and the use of functoriality to analyze clustering methods. The authors provide examples and theoretical foundations for these techniques, emphasizing the computational aspects and their practical applications.