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The NSF has awarded a three-year grant to a multi-university team led by Professor Chad Giusti for the development of theoretical and computational tools for application of homological algebra to data analysis. Professor Giusti, in collaboration with Doctor Gregory Henselman-Petreusek of Princeton University and Professor Lori Ziegelmeier of Macalester College, will utilize the $560,000 grant from the NSF Division of Mathematical Sciences' Computational and Data-Enabled Science and Engineering in the Mathematical Sciences to support a postdoctoral researcher, provide stipends for undergraduate researchers, and support travel and summer salary.
Topology provides a rigorous mathematical tool kit for the study of coarse, qualitative features of geometric objects. These types of features can be tuned to be robust in the presence of noise or uncertainty, driving increasing interest in their use as tools for analysis of real data. Such applications have been made possible by recent algorithmic developments which allow computers to handle simple computations of topological measures on thousands or millions of data points. However, the current state of the art software can only enumerate topological features, not allowing examination or direct comparison of specific features across data sets. Such fine control requires the introduction of advanced techniques from a field called homological algebra. In this project, Professor Giusti and his collaborators will develop the appropriate theoretical and algorithmic foundations needed to implement these techniques in software, and implement them in a modular, open, and extensible software platform. In addition, in collaboration with the broader mathematical and scientific communities, they will develop visualization tools, documentation, and use cases which will make these new tools accessible to a broad audience of scientists and data analysts.
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