Advances in Topological Data Analysis

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The overarching aim of this project is to extend the foundations of Topological Data Analysis (TDA) in mathematical statistics and applied probability in order to understand the strengths of the TDA methodology and whether it can enable data scientists to make a more informed decision. From the statistical perspective the TDA methodology can be considered as a generalization of cluster analysis which aims at detecting topological structure in data.The first part of the project targets the asymptotic behavior of Betti curves and of functionals obtained from the latter such as persistence landscapes. Establishing pioneering results will lay the groundwork for the second part of the project which is the study of resampling procedures of TDA based objects such as the bootstrap of Betti curves and a further integration of TDA in classical statistical methods such as change point detection.