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Browsing by Author "Miranda, Rick, committee member"

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    A simplicial homotopy group model for K2 of a ring
    (Colorado State University. Libraries, 2010) Whitfield, JaDon Saeed, author; Duflot, Jeanne, advisor; Miranda, Rick, committee member; Achter, Jeffrey D., committee member; Gelfand, Martin Paul, committee member
    We construct an isomorphism between the group K2(R) from classical, algebraic K-Theory for a ring R and a simplicial homotopy group constructed using simplicial homotopy theory based on that same ring R. First I describe the basic aspects of simplicial homotopy theory. Special attention is paid to the use of category theory, which will be applied to the construction of a simplicial set. K-Theory for K0(R), K1(R) and K2(R) is then described before we set to work describing explicitly the nature of isomorphisms for K0(R) and K1(R) based on previous work. After introducing some theory related to K-Theory, some considerations and corrections on previous work motivate more new theory that helps the isomorphism with K2(R). Such theory is developed, mainly with regards to finitely generated projective modules over R and then elementary matrices with entries from R, culminating in the description of the Steinberg Relations that are central to the understanding of K2(R) in terms of homotopy classes. We then use new considerations on the previous work to show that a map whose image is constructed through this article is an isomorphism since it is the composition of isomorphisms.
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    Investigating individually expressed motives and collectively generated goals for equity-oriented reform in undergraduate mathematics education
    (Colorado State University. Libraries, 2024) Tremaine, Rachel, author; Hagman, Jess Ellis, advisor; Arnold, Elizabeth, committee member; Miranda, Rick, committee member; Basile, Vincent, committee member; DiGregorio, Gaye, committee member
    Supporting diversity, equity, and inclusion (DEI) is an explicitly stated goal of many mathematics departments across the country, and addressing ongoing disparities in outcomes and experiences within undergraduate mathematics is a shared responsibility among undergraduate mathematics community members. Despite the prevalence of ideological, political, and contextual barriers to equity-oriented action within undergraduate mathematics spaces, many community members can and do take a responsive stance toward enhancing DEI within their department and at their institution. Understanding how mathematics faculty members, administrators, and students are personally motivated to take up work toward these aims within their own mathematics departments is paramount in ensuring that such work continues. In this dissertation I present two investigations which draw on cultural historical activity theory (CHAT) as a conceptual and theoretical lens. In the first investigation, I analyze the motives of 30 undergraduate mathematics community members (five administrators, 17 faculty members, and eight students) across three institutions to understand their reasoning for participation in an intradepartmental community focused on creating transformative, equity-oriented change within introductory mathematics courses. A reflexive thematic analysis of journal entries and individual interviews with participants resulted in five themes which motivated participation in collaborative equity reform within their mathematics department: a relational motive, a self-improvement motive, a student experience motive, an influence motive, and a values to action motive. With these themes in mind, I then consider how a Networked Improvement Community (NIC) at one institution developed a shared object for their work through a CHAT lens, highlighting what rules, communities, subjects, artifacts, and divisions of labor proved salient to this development. The prevalence and pervasiveness of self-interests, identity neutrality, and paternalism are critically discussed within the context of these investigations, and I build on existing literature to produce recommendations for disrupting such ideologies to produce transformative change in undergraduate mathematics environments. Among these recommendations are the need for critical engagement to see beyond self-interest in the context of one's own reform work, and the need for collaborative reform groups to not only position students as experts on their own experiences, but to also conceptualize instructors as novices on student experiences. I conclude with a discussion of future work supporting continued theorizing of the link between individually expressed motives and collectively generated goals in undergraduate mathematics reform efforts.
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    Multiplicities and equivariant cohomology
    (Colorado State University. Libraries, 2010) Lynn, Rebecca E., author; Duflot, Jeanne, advisor; Miranda, Rick, committee member; Hulpke, Alexander, committee member; Iyer, Hariharan K., committee member
    The aim of this paper is to address the following problem: how to relate the algebraic definitions and computations of multiplicity from commutative algebra to computations done in the cohomology theory of group actions on manifolds. Specifically, this paper is concerned with applications of commutative algebra to the study of cohomology rings arising from group actions on manifolds, in the way that Quillen initiated. This paper synthesizes two distinct areas of pure mathematics (commutative algebra and cohomology theory) and two ways of computing multiplicities in order to link them. In order to accomplish this task, a discussion of commutative algebra will be followed by a discussion of cohomology theory. A link between commutative algebra and cohomology theory will be presented, followed by its application to a significant example. In commutative algebra, we discuss graded rings, Pioncaré Series, dimension, and multiplicities. Whereas the theory for multiplicities has been developed for local rings, we give an exposition of the theory for graded rings. Several definitions for dimension will be presented, and it will be proven that all of these distinct definitions are equal. The basic properties of multiplicities will be introduced, and a brief discussion of a classical multiplicity in commutative algebra, the Samuel multiplicity, will be presented. Then, Maiorana's C-multiplicity will be defined, and a relationship between all of these multiplicities will be observed. In cohomology theory, we address smooth actions of finite groups on manifolds. As a part of this study in cohomology theory, we will consider group actions on topological spaces and the Borel construction (equivariant cohomology), completing this part of the paper with a discussion of smooth (or differentiable) actions, setting some notation necessary for our discussion of Maiorana's results, which inspire some of our main theorems, but on which we do not rely in this dissertation. Following the treatments of commutative algebra and cohomology theory, we present one of Quillen's main results without proof, linking these two distinct areas of pure mathematics. Quillen's work results in a formula for finding the multiplicity of the equivariant cohomology of a compact G-manifold with G a compact Lie group. We apply these results to the compact G-manifold U/S, where G (a compact Lie group) is embedded in a unitary group U=U(n) and S=S(n) is the diagonal p-torus of rank n in U(n), resulting in a nice topological formula for computing multiplicities. Finally, we end the paper with a proposal for future research.