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

以可操作的形式提取、建模、概念化和可视化驻留在复杂数据集中的信息是现代数据分析的核心主题。本项目将在功能数据的广泛领域接受这些挑战，因为它们渗透到科学和实际应用中。该项目将开发数学方法和计算工具，帮助我们执行这些任务，并建立从功能数据到可操作知识的管道。这些方法的潜在用途有着广阔的前景。例子包括动态社会网络的分析，理解基因表达空间谱的变化，以便研究与健康和发育结果的关联，以及询问微生物组和代谢组网络的探索性研究。该项目将利用拓扑数据分析（TDA）技术进一步开发和扩展持久同源性，并将TDA与概率方法相结合。这将使探索在随机域上定义的函数的结构变化的方法和工具的发展成为可能。该项目将开发拓扑方法、计算工具及其基础，以总结、分析和可视化随机紧凑度量空间、网络和图形上的功能数据，并建立从这些数据到可操作知识的管道。该项目的进展将为发展和进化生物学、医学、社会科学和工程等学科的实际应用和研究开辟许多新的视角。虽然由最先进的方法提供动力，但所产生的工具将易于使用，从而获得广泛的效用。为了加强项目的这一重要方面，通过与领域科学家的跨学科合作，这些方法将在涉及植物形状、岩石微观结构和人类微生物组数据分析的案例研究中进行测试。从这个项目中产生的方法和软件工具将被记录并在线分发，供研究团体使用，进一步扩大对其他学科的影响。该项目还将产生资源来支持若干外联活动。

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.