AF: Small: Expanding the Reach of Topological Data Analysis

数据科学和分析在现代占据了中心舞台，为日常生活提供了一系列应用程序。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，一种基于经典数学领域拓扑学的技术已经发展起来，它有望补充人工智能技术，在某些情况下还可以成为人工智能技术的替代。随着这个被称为拓扑数据分析(TDA)的新兴领域的扩展，从多样化、复杂的数据中提取智能摘要的任务变得越来越具有挑战性。为了迎接这一挑战，TDA还需要通过在数学和算法概念之间建立更深层次的联系来扩大其目前的技术储备。这个项目旨在开发这些技术，同时将数据科学的研究生和其他工作人员作为序列进行培训。该项目尤其将结合计算来研究几个新的数学概念，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Morse理论与进一步的应用相联系，(Iii)发展组合动力系统的持续理论，以及(Iv)从计算的角度发展多参数持续。对于上述四个主题，一个由可靠的数学支持的成功的算法理论可以为工程和科学领域的数据探索和分析提供一个强有力的工具。拟议的外联活动还将进一步巩固正在进行的将科学和工程领域的不同群体聚集在一起进行现代数据分析的努力。该项目将帮助培训研究生，他们将发展数学和理论计算机科学方面的技能，尤其是算法和拓扑学方面的技能，编写高效和可用的软件，以及应用于分析真实世界的数据集。将努力从代表性不足的群体中招募和指导学生，就像调查人员已经做的那样。这一奖项反映了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。

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.