Collaborative Research: The Topology of Functional Data on Random Metric Spaces, Graphs, and Graphons

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Extracting, modeling, conceptualizing, and visualizing information residing in complex datasets in actionable form are core themes in modern data analysis. This project will take up these challenges in the broad realm of functional data, as they permeate the sciences and practical applications. The project will develop mathematical methods and computational tools that help us execute these tasks and establish a pipeline from functional data to actionable knowledge. There is a vast landscape of potential uses of these methods. Examples include analyses of dynamical social networks, understanding variation in the spatial profile of gene expression to enable studies of associations with health and developmental outcomes, and exploratory studies that interrogate microbiome and metabolomic networks. The project will leverage techniques from topological data analysis (TDA) to further develop and extend persistent homology and integrate TDA with probabilistic methods. This will enable development of methods and tools for probing structural variation in functions defined on random domains. The project will develop topological methods, computational tools, and their foundations to summarize, analyze, and visualize functional data on random compact metric spaces, networks and graphons, and establish a pipeline from such data to actionable knowledge. The advances resulting from the project will open many new perspectives in practical applications and research in such disciplines as developmental and evolutionary biology, medicine, social sciences, and engineering. Though powered by state-of-the-art methods, the resulting tools will be simple to use so as to attain broad utility. To enhance this important aspect of the project, by way of interdisciplinary collaboration with domain scientists, the methods will be tested on case studies that involve analyses of plant shape, rock micro-structure, and human microbiome data. The methods and software tools resulting from this project will be documented and distributed online for use by the research community further broadening the impact on other disciplines. The project also will generate resources to support several outreach activities.