Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

A Feminist Affective Turn for Public Relations: Mothers, Passionate Publics, and the Childhood Vaccine Debate

This project proposes a feminist intervention in how affect and publics are theorized in public relations research. Drawing from extant literature, I argue that public relations theories of affect and publics have been apolitical and lack depth and context (Leitch & Motion, 2010a). Using the context of the online childhood vaccine debate, I illustrate several theories and concepts of the new feminist affective turn, as well as postmodern theories of affect, relevant to public relations research: (a) Public Feelings, “ugly” feelings, agency, and community (Cvetkovich, 2012; Ngai, 2007); (b) passionate politics (Mouffe, 2014); (c) postmodern assemblages, biopower, and body politics (Deleuze & Guattari, 1988; Foucault, 1984); (d) affective facts and logics of future threats (Massumi, 2010); and (e) affective ethics (Bertleson & Murphie, 2010). Scholarship in the areas of public relations, risk, feminist and postmodern affect theory, and the vaccine debate provided theoretical grounding for this project. My research questions asked: How is feminist affect theory embodied by mothers in the vaccine debate? How do mothers understand risks as affective facts in the vaccine debate (if at all)? What affective logics are used by mothers in the vaccine debate (if any)? And, What are sources of knowledge for mothers in the vaccine debate? Multi-sited online ethnographic methods were used to explore how feminist affect theory contributes to public relations research, including 29 one-on-one in-depth interviews with mothers of young children and participant observation of 15 online discussions about vaccines on parenting websites BabyCenter.com, TheBump.com, and WhatToExpect.com. I used snowball sampling to recruit interview participants and grounded theory (Glaser & Strauss, 1967) to analyze interview and online data. Results show that feminist affect theory contributes to theoretical and practical knowledge in public relations by politicizing and contextualizing understandings of publics and elucidating how affective facts and logics inform publics’ knowledge and choices, specifically in the context of risk. I also found evidence of suppression of dissent (Martin, 2015) and academic bias in vaccine debate research, which resulted in cultures of silence. Further areas of study included how specific contexts such as motherhood and issues of privilege and access affect publics’ experiences, knowledges, and choices.