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

该奖项是2021财年数学和物理科学提升博士后研究奖学金(MPS-Ascend Program)的一部分。Ranthony Edmonds被授予这一奖学金，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研究和教育项目，包括对其他学科的应用。这个项目致力于拓扑数据分析(TDA)的一些具有挑战性的问题，TDA是一个利用代数拓扑和许多其他领域的工具来研究大数据形状的新的、快速增长的领域。除了这项研究，Edmonds将与现有的地方和国家网络合作，开展与数据科学有关的教育活动，重点是增加未被充分代表的少数群体在数学科学中的代表性。第一个是研究度量空间，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距离小扰动下的稳定性。第三个目标是适用的，涉及选区重新划分，其基本思想是使用GW指标来比较重新划分选区的计划。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。

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.