TRIPODS: Topology, Geometry, and Data Analysis (TGDA@OSU):Discovering Structure, Shape, and Dynamics in Data

该项目将通过从数学、统计和算法的角度考虑复杂数据的几何和拓扑方面来推进数据分析的方法和理论基础，从而增强计算机科学、数学和统计社区之间的协同作用。此外，该项目将有利于一系列有影响力的科学领域，包括医学、神经解剖学、机器学习、地理信息系统、机械工程设计和政治学。研究成果将通过软件包和教程加以实施和传播，使工业界和学术界从业者能够广泛应用。通过这一项目，专业顾问将为跨学科、本科和研究生教育开发课程。俄亥俄州立大学（OSU）已经有统计学和计算机科学与工程联合开设的数据科学课程，包括最近开设的数据分析本科专业，这为开发新课程提供了一个平台，并为未来的行业领导者提供了参与基础研究的机会。此外，该项目旨在与俄勒冈州立大学的转化数据分析和数学生物科学研究所以及其他内部和外部研究和教育中心建立伙伴关系。还包括为外联和培训目的举办讲习班和暑期学校的计划。在过去的几十年里，为数据科学开发了大量的模型、方法和算法框架。然而，随着数据变得越来越复杂，该领域面临着新的挑战。特别是，非欧几里得性质、高阶连通性、隐藏的全局线索以及数据的动态调节对现有方法提出了进一步的挑战。本项目将探索和利用数据中固有的几何和拓扑结构来解决其中的一些问题。主要目的是发现，建模和揭示信息的形式(i)数据中的结构，（ii）数据的形状，以及（iii）动态底层数据。该项目利用了微分和代数拓扑和几何的数学领域的概念，应用统计学和组合学，以及算法，图论和统计/机器学习的计算领域。几何和拓扑数据分析的研究提出了在有限数据、近似和噪声的背景下重塑和重新研究统计和数学中的经典概念的需要。该项目研究数据背后的显式或隐藏结构，如簇树，这是理解和有效处理数据的基础。此外，pi旨在对全球或本地数据背后的精确形状进行建模，这对于提供可以进行各种统计分析的平台至关重要。具体的例子包括曲面模型的形状空间和系统发育树的树空间。最后，该项目将考虑数据中的动态，其中时间和拓扑/几何特征之间的相互作用可以带来更深入的见解。所有这些领域都不可避免地会被新的应用所丰富。

This project will advance the methodological and theoretical foundations of data analytics by considering the geometric and topological aspects of complex data from mathematical, statistical and algorithmic perspectives, thus enhancing the synergy between the Computer Science, Mathematics, and Statistics communities. Furthermore, this project will benefit a range of impactful scientific areas including medicine, neuronanatomy, machine learning, geographic information systems, mechanical engineering designs, and political science. The research products will be implemented and disseminated through software packages and tutorials, allowing widespread application by industrial and academic practitioners. Through this project, the PIs will develop curricula for cross-disciplinary, undergraduate and graduate education. There is already extant data science curriculum offered jointly between Statistics and Computer Science and Engineering at The Ohio State University (OSU), including the recent Data Analytics undergraduate major, providing a platform to develop new courses and an opportunity to engage future industry leaders in basic research. Additionally, this project aims to develop partnerships with the Translational Data Analytics and the Mathematical Biosciences Institutes at OSU, as well as other internal and external research and education centers. Plans for workshops and summer schools are included for outreach and training purposes.In the past few decades, a large number of models, methods, and algorithmic frameworks have been developed for data science. However, as data become increasingly more complex, the field faces new challenges. In particular, the non-Euclidean nature, the higher order connectivity, the hidden global cues, and the dynamics regulating the data pose further challenges to existing methods. This project will explore and leverage the geometric and topological structures inherent in the data to tackle some of these problems. The main aims are to discover, model and reveal information in the form of (i) structures in data, (ii) shapes from data, and (iii) dynamics underlying data. This project leverages concepts from mathematical areas of differential and algebraic topology and geometry, applied statistics and combinatorics, and computational areas of algorithms, graph theory, and statistical/machine learning. Research in geometric and topological data analysis has brought forth the need to recast and reinvestigate classical concepts in statistics and mathematics in the context of finite data, approximations, and noise. This project investigates explicit or hidden structures behind data, such as cluster trees, which are the basis for understanding and efficient processing of data. Additionally, the PIs aim to model the precise shape behind data globally or locally, which are essential for providing a platform where various statistical analyses can be carried out. Particular examples include the shape space of surface models and the tree space of phylogenetic trees. Finally, this project will consider dynamics in the data, where the interplay between temporal and topological/geometric features can lead to deeper insights. All of these areas will inevitably be enriched by new applications.