Theses and Dissertations at Montana State University (MSU)
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Item Directed graph descriptors and distances for analyzing multivariate time series data(Montana State University - Bozeman, College of Letters & Science, 2022) Belton, Robin Lynne; Chairperson, Graduate Committee: Tomas GedeonLocal maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty in the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise L1 distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison.Item Additivity of factorization algebras & the cohomology of real Grassmannians(Montana State University - Bozeman, College of Letters & Science, 2021) Berry, Eric Daniel; Chairperson, Graduate Committee: David Ayala; Ryan Grady (co-chair)This dissertation is composed of two separate projects. The first chapter proves two additivity results for factorization algebras. These provide a way to understand factorization algebras on the product of two spaces. Our results can be thought of as a generalization of Dunn's additivity for En-algebras. In particular, our methods provide a new proof of Dunn's additivity. The second chapter is an examination of the Schubert stratification of real Grassmann manifolds. We use this extra structure to identify the quasi-isomorphism type of the Schubert CW chain complex for real Grassmannians. We provide explicit computations using our methods.Item Adjunction spaces and k-spaces(Montana State University - Bozeman, College of Letters & Science, 1973) Behrens, Dale W.