AF: Small: Expanding the Reach of Topological Data Analysis

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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.