CAREER: The Open Neighborhood of Applied Topology

Large sets of high-dimensional data are common in most branches of science, and their shapes reflect important patterns within. A proper treatment of the mathematical foundations behind a data analysis technique is necessary to develop algorithms which are interpretable, so that scientists can trust the conclusions they draw from the data. The investigator will study the geometric theory underlying persistent homology, a frequently used tool to measure the shape of data, with applications to machine learning and artificial intelligence. A key goal of the project is to contribute to the broader applied topology community, including online seminars and content targeting undergraduate researchers and their mentors. The growth of complex datasets motivates the need for techniques that summarize the shape of data. The main algorithm in applied topology, persistent homology, uses Vietoris-Rips and Cech complexes to provide insight into the shape of nonlinear, high-dimensional, time-varying, and noisy datasets at multiple resolutions or scales. As a dataset converges to (say) a manifold as more points are sampled, the Vietoris-Rips persistent homology of the data converges to the Vietoris-Rips persistent homology of the manifold. The investigator will advance the theory of Vietoris-Rips and Cech complexes of finite datasets and manifolds and, furthermore, apply this knowledge to bound Gromov-Hausdorff distances between metric spaces. 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. NSF Award ID: 2540172 | Program: 01003031DB NSF RESEARCH & RELATED ACTIVIT,01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Henry Adams | Institution: University of Florida, GAINESVILLE, FL | Award Amount: $353,016 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2540172 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2540172.html