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Browsing by Author "Patel, Amit, committee member"

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    Functional programming applied to computational algebra
    (Colorado State University. Libraries, 2018) Kessler, Ian Holm, author; Wilson, James B., advisor; Patel, Amit, committee member; Chitsaz, Hamidreza, committee member
    Underlying many, if not all, areas of mathematics is category theory, an alternative to set theory as a foundation that formalizes mathematical structures and relations between them. These relations abstract the idea of a function, an abstraction used throughout mathematics as well as throughout programming. However, there is a disparity between the definition of a function used in mathematics from that used in mainstream programming. For mathematicians to utilize the power of programming to advance their mathematics, there is a demand for a paradigm of programming that uses mathematical functions, as well as the mathematical categories that support them, as the basic building blocks, enabling programs to be built by clever mathematics. This paradigm is functional programming. We wish to use functional programming to represent our mathematical structures, especially those used in computational algebra.
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    Persistence and simplicial metric thickenings
    (Colorado State University. Libraries, 2024) Moy, Michael, author; Adams, Henry, advisor; Patel, Amit, committee member; Peterson, Christopher, committee member; Ben-Hur, Asa, committee member
    This dissertation examines the theory of one-dimensional persistence with an emphasis on simplicial metric thickenings and studies two particular filtrations of simplicial metric thickenings in detail. It gives self-contained proofs of foundational results on one-parameter persistence modules of vector spaces, including interval decomposability, existence of persistence diagrams and barcodes, and the isometry theorem. These results are applied to prove the stability of persistent homology for sublevel set filtrations, simplicial complexes, and simplicial metric thickenings. The filtrations of simplicial metric thickenings studied in detail are the Vietoris–Rips and anti-Vietoris–Rips metric thickenings of the circle. The study of the Vietoris–Rips metric thickenings is motivated by persistent homology and its use in applied topology, and it builds on previous work on their simplicial complex counterparts. On the other hand, the study of the anti-Vietoris–Rips metric thickenings is motivated by their connections to graph colorings. In both cases, the homotopy types of these spaces are shown to be odd-dimensional spheres, with dimensions depending on the scale parameters.
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    The group extensions problem and its resolution in cohomology for the case of an elementary abelian normal sub-group
    (Colorado State University. Libraries, 2018) Adams, Zachary W., author; Hulpke, Alexander, advisor; Patel, Amit, committee member; Bohm, Wim, committee member
    The Jordan-Hölder theorem gives a way to deconstruct a group into smaller groups, The converse problem is the construction of group extensions, that is to construct a group G from two groups Q and K where K ≤ G and G/K ≅ Q. Extension theory allows us to construct groups from smaller order groups. The extension problem then is to construct all extensions G, up to suitable equivalence, for given groups K and Q. This talk will explore the extension problem by first constructing extensions as cartesian products and examining the connections to group cohomology.
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    Topological, geometric, and combinatorial aspects of metric thickenings
    (Colorado State University. Libraries, 2021) Bush, Johnathan E., author; Adams, Henry, advisor; Patel, Amit, committee member; Peterson, Chris, committee member; Luong, Gloria, committee member
    The geometric realization of a simplicial complex equipped with the 1-Wasserstein metric of optimal transport is called a simplicial metric thickening. We describe relationships between these metric thickenings and topics in applied topology, convex geometry, and combinatorial topology. We give a geometric proof of the homotopy types of certain metric thickenings of the circle by constructing deformation retractions to the boundaries of orbitopes. We use combinatorial arguments to establish a sharp lower bound on the diameter of Carathéodory subsets of the centrally-symmetric version of the trigonometric moment curve. Topological information about metric thickenings allows us to give new generalizations of the Borsuk–Ulam theorem and a selection of its corollaries. Finally, we prove a centrally-symmetric analog of a result of Gilbert and Smyth about gaps between zeros of homogeneous trigonometric polynomials.
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    Vietoris–Rips metric thickenings and Wasserstein spaces
    (Colorado State University. Libraries, 2020) Mirth, Joshua, author; Adams, Henry, advisor; Peterson, Christopher, committee member; Patel, Amit, committee member; Eykholt, Richard, committee member
    If the vertex set, X, of a simplicial complex, K, is a metric space, then K can be interpreted as a subset of the Wasserstein space of probability measures on X. Such spaces are called simplicial metric thickenings, and a prominent example is the Vietoris–Rips metric thickening. In this work we study these spaces from three perspectives: metric geometry, optimal transport, and category theory. Using the geodesic structure of Wasserstein space we give a novel proof of Hausmann's theorem for Vietoris–Rips metric thickenings. We also prove the first Morse lemma in Wasserstein space and relate it to the geodesic perspective. Finally we study the category of simplicial metric thickenings and determine effects of certain limits and colimits on homotopy type.