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

在数据驱动的科学研究中，一个中心问题是如何解释现代工具发现的大数据集中的结构。拓扑数据分析领域通过同源语言提供了一种潜在的解决方案，该语言将感兴趣的特征编码为循环。原则上，这些可以被定位和理解为生成器，它们揭示了原始数据中的明确结构。然而，基本的数学和计算挑战将大多数拓扑分析限制在持久性图的研究上，持久性图是省略生成器的数字总结，因此极大地限制了建模能力和可解释性。这个项目借鉴了代数拓扑学、数值线性代数、范畴和序格理论、计算和组合学等数学领域以及生物聚集、大脑和医学成像等科学和工程领域的不同想法。ExHACT项目将通过为持久同调代数创建统一的理论和计算工具集，提供实现生成循环的完整建模和解释能力所需的工具。由其中一个PI开发的拟阵理论和精确范畴(项目由此得名)领域的最新结果为使用众所周知的矩阵操作有效地执行必要的计算提供了基础。PI将利用这一新的机会，开发理论和计算工具，用于研究持久生成元、持久模的诱导同态、精确序列和谱序列以及相对持久同调等方法。他们将用数据可视化能力来增强这一计算核心，以便于没有广泛数学背景的科学家以直观的方式对同源数据进行图形探索，并为目前在材料科学、神经科学、生物化学和生物聚合中应用拓扑方法的现有研究小组提供新的工具。ExHACT还将使更有经验的用户能够构建自定义功能和工作流程，为开发拓扑数据分析的新方法提供一个稳定的社区平台。所有软件功能都将被广泛记录，包括技术规范和详细的用例，以使广大受众能够访问全套计算和可视化能力。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.