AF: Small: Expanding the Reach of Topological Data Analysis

在现代，数据科学和分析已经占据了日常生活中大量应用的中心舞台。总的来说，人工智能和机器学习一直是这一发展的支柱。最近，一种基于拓扑学经典数学领域的技术已经发展起来，它有望补充并在某些情况下替代人工智能技术。随着被称为拓扑数据分析（TDA）的新兴领域的扩展，从各种复杂数据中提取智能摘要的任务变得越来越具有挑战性。为了应对这一挑战，TDA还需要通过在数学和算法概念之间建立更深入的联系来扩展其现有的技术库。该项目旨在开发这些技术，同时培训研究生和数据科学方面的其他工作人员。特别是，该项目将研究几个新的数学概念与计算相结合，以解决以下主题中出现的各种挑战：(i)用几何增强拓扑总结，（ii）将离散莫尔斯理论与进一步应用联系起来，（iii）为组合动力系统开发持久性理论，以及（iv）从计算的角度开发多参数持久性。一个成功的算法理论，由上述四个主题的健全数学支持，可以为工程和科学领域的数据探索和分析提供一个强大的工具。拟议的外展活动还将进一步巩固正在进行的努力，将从事现代数据分析的科学和工程领域的不同社区聚集在一起。该项目将帮助培养培养数学和理论计算机科学技能的研究生，特别是在算法和拓扑方面，编写高效可用的软件，并将其应用于分析现实世界的数据集。将努力从代表性不足的群体中招募和指导学生，正如调查员已经做的那样。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Data science and analysis have taken a center stage in modern years for a swath of applicationsembracing daily life. In general, AI and machine learning have been the backbone of thisdevelopment. Recently, a technique based on the classical mathematical area of topology has evolvedthat is showing promise to complement and in some cases be an alternative to AI techniques. As thisemergent area, known as topological data analysis (TDA), expands its reach, the task of extractingintelligent summaries out of diverse, complex data becomes increasingly challenging. To meet thischallenge, TDA also needs to expand its current repertoire of techniques by making further anddeeper connections between mathematical and algorithmic concepts. This project aims to developthese techniques while training graduate students and other workforce in data science as aconsequence.The project, in particular, will investigate several novel mathematical concepts in conjunction withcomputations to address various challenges appearing in the following topics: (i) augmentingtopological summarization with geometry, (ii) connecting discrete Morse theory to furtherapplications, (iii) developing persistence theory for combinatorial dynamical systems, and (iv)developing multi-parameter persistence from a computational view-point. A successful algorithmictheory supported by sound mathematics for the four aforementioned themes can provide a powerfultool for data exploration and analysis in a spectrum of engineering and scientific fields. Theproposed outreach activities will also further consolidate the ongoing effort of bringing togetherdifferent communities in science and engineering engaged into modern data analysis. The projectwill help train graduate students who will develop skills in mathematics and theoreticalcomputer science, most notably in algorithms and topology, in writing efficient and usable software,and in its application to analyzing real-world data sets. Efforts will be made to recruit and mentorstudents from under-represented groups, as has already been done by the investigator.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.