Persistent Homology, Metrics, and Applications on the Collection of Enriched Metric Measure Spaces

该奖项是作为2021财年数学和物理科学提升博士后研究奖学金的一部分，MPS-Ascend计划。Ranthony Edmonds获此奖学金，在赞助科学家Facundo Memoli的指导下，在俄亥俄州立大学开展数学科学的研究和教育项目，包括对其他学科的应用。该项目致力于研究拓扑数据分析（TDA）的一些具有挑战性的问题，这是一个新兴且快速发展的领域，它利用代数拓扑和许多其他领域的工具来研究大数据的形状。随着这项研究的开展，Edmonds将与现有的地方和国家网络合作，开展与数据科学相关的教育活动，重点是增加数学科学中未被充分代表的少数民族的代表性。该项目有几个目标。首先是对度量度量空间的研究，并为Gromov-Wasserstein （GW）距离找到一个合适的概念。第二个目标是探索数据在GW距离的小扰动下的稳定性。第三个目标是应用和关注选举重划，其基本思想是使用GW度量来比较重划计划。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。

This award is made as part of the FY 2021 Mathematical and Physical Sciences Ascending Postdoctoral Research Fellowships, MPS-Ascend Program. Ranthony Edmonds is awarded this fellowship to conduct a program of research and education at the Ohio State University in the mathematical sciences, including applications to other disciplines, under the mentorship of the sponsoring scientist Facundo Memoli. This project is devoted to some challenging issues of topological data analysis (TDA), a new and rapidly growing field that leverages tools from algebraic topology and many other areas to study the shape of big data. Along with this research, Edmonds will develop educational activities related to data science in collaboration with existing partnerships with local and national networks focused on increasing representation of underrepresented minorities in the mathematical sciences.The project has several objectives. The first is the study of metric measure spaces, and finding an appropriate notion for the Gromov-Wasserstein (GW) distance. The second goal is to explore stability of data under small perturbations in the GW distance. The third goal is applied and concerns electoral redistricting where the basic idea is to use the GW metric to compare redistricting plans.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.