Collaborative Research: CDS&E-MSS: Exact Homological Algebra for Computational Topology

合作研究：CDS

• 批准号: 1854703
• 负责人: Lori Ziegelmeier
• 金额: $ 12.98万
• 依托单位: Macalester College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854703&HistoricalAwards=false
• 关键词: Collaborative; Research; CDS&E; MSS; Exact

A central problem in data-driven scientific inquiry is how to interpret structures in large data sets uncovered by modern tools. The field of topological data analysis provides a potential solution via the language of homology, which encodes features of interest as cycles. These, in principle, can be located and understood as generators, which reveal explicit structure in the original data. However, fundamental mathematical and computational challenges have restricted most topological analyses to the study of persistence diagrams, numerical summaries that omit generators and, thus, dramatically limit modeling power and explainability. This project draws on diverse ideas from the mathematical domains of algebraic topology, numerical linear algebra, category and order-lattice theory, computation, and combinatorics, and from the scientific and engineering domains of biological aggregations, brain, and medical imaging. It provides ample opportunities for training mathematical scientists for the mastery of these tools, and for developing new, exploratory methods in STEM teaching and learning.The ExHACT project will provide the tools needed to realize the full modeling and explanatory capability of generating cycles by creating a unified theoretical and computational tool set for persistent homological algebra. Recent results in the fields of matroid theory and exact categories (from which the project draws its name) developed by one of the PIs provide the foundation for efficiently performing the necessary computations using well-understood matrix manipulations. The PIs will capitalize on this new opportunity by developing theoretical and computational tools for the study of persistent generators, induced homomorphisms of persistence modules, exact and spectral sequences, and relative persistent homology, among other methods. They will augment this computational core with data visualization capabilities to facilitate graphical exploration of homological data in an intuitive fashion for scientists without extensive mathematical background, and provide new tools for existing research groups that currently apply topological methods in materials science, neuroscience, biochemistry, and biological aggregations. ExHACT will also enable custom functionality and workflows to be built by more experienced users, providing a stable community platform for the development of new methodologies in topological data analysis. All software functionality will be extensively documented, including both technical specifications and detailed use cases, in order to make a full suite of computation and visualization capabilities accessible to a broad audience.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

数据驱动的科学询问中的一个核心问题是如何解释现代工具发现的大数据集中的结构。拓扑数据分析的领域通过同源语言提供了潜在的解决方案，该语言将感兴趣的特征描述为周期。这些原则上可以找到并理解为发电机，它们在原始数据中揭示了明确的结构。然而，基本的数学和计算挑战已将大多数拓扑分析限制在持续图的研究，数值摘要的研究，这些摘要省略了发电机，从而极大地限制了建模能力和解释性。该项目借鉴了代数拓扑，数值线性代数，类别和秩序理论，计算和组合学以及生物聚集，大脑和医学成像的科学和工程领域的数学领域，数值线性代数，类别和秩序理论，计算和组合学理论。它为培训数学科学家的掌握这些工具提供了足够的机会，并在STEM教学和学习中开发了新的，探索性的方法。该诱惑项目将提供所需的工具，以实现通过创建一个统一的理论和计算工具集来实现持久的同源代数来实现产生周期的完整建模和解释能力。其中一个PI开发的Matroid理论和确切类别（从该项目绘制其名称的）领域的最新结果为有效地使用良好理解的矩阵操作有效执行必要的计算为基础提供了基础。 PI将通过开发理论和计算工具来研究持久发电机，诱导持久模块的同态，精确和频谱序列以及相对持久同源性等同质性以及其他方法以及其他方法以及相对持久的同源性，从而利用了这一新机会。他们将通过数据可视化功能增强这种计算核心，以直观地探索同源数据的图形探索，而没有广泛的数学背景，并为目前应用材料科学，神经科学，生物化学和生物学聚集的现有研究小组提供了新的工具。 Ounchact还将使经验丰富的用户能够构建自定义功能和工作流程，从而为在拓扑数据分析中开发新方法提供稳定的社区平台。所有软件功能都将被广泛记录，包括技术规格和详细用例，以使广泛受众均可访问一整套计算和可视化功能。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来通过评估来提供支持的。