Collaborative Research: Multiparameter Topological Data Analysis

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Complex datasets arise in many disciplines of science and engineering and their interpretation requires Multiparameter Data Analysis, which broadly speaking, studies the dependency of a phenomenon or a space on multiple parameters. For instance, in climate simulations, scientists are interested in identifying, verifying, and evaluating trends in detecting, tracking, and characterizing weather patterns associated with high impact weather events such as thunderstorms and hurricanes. In recent years, topological data analysis (TDA) has evolved as an emerging area in data science. So far, most of its applications have been limited to the single parameter case, that is, to data expressing the behavior of a single variable. As its reach to applications expands, the task of extracting intelligent summaries out of diverse, complex data demands the study of multiparameter dependencies. This project will help address this demand by developing a sound mathematical theory supported by efficient algorithmic tools thus providing a powerful platform for data exploration and analysis in scientific and engineering applications. The educational impact will be accelerated by the synergy between mathematics and computer science and integrated applications. Graduate students supported by the project will be trained to develop skills in mathematics and theoretical computer science, most notably in algorithms and topology, and analyze some real-world data sets. The investigators will follow best practice to recruit and mentor students from underrepresented groups who will participate in the project. The investigators also plan to broaden research engagement via workshops or tutorials at computational topology and TDA venues.Although TDA involving a single parameter has been well researched and developed, the same is not yet true for the multiparameter case. At its current nascent stage, multiparameter TDA is yet to develop tools to practically handle complex, diverse, and high-dimensional data. To meet this challenge, this project will make both mathematical and algorithmic advances for multiparameter TDA. To scope effectively, focus will be mainly on three research thrusts to: (I) explore multiparameter persistence for generalized features and develop algorithms to compute them; (II) exploit the connections of zigzag persistence to multiparameter settings to support dynamic data analysis, and (III) generalize graphical topological descriptors. From a methodological point of view, the geometric and topological ideas behind the proposed work inject novel perspectives and directions to the important field of computational data analysis. In particular, the project team will investigate several novel mathematical concepts in conjunction with algorithms to address various challenges appearing in the aforementioned thrusts. The resulting TDA methodologies have the potential to complement and augment traditional data analysis approaches in fields such as machine learning and statistical data analysis. The investigators bring together expertise in theoretical computer science, algorithms design, mathematics, and in particular topological data analysis to conduct this research.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.