Mapping Class Groups of Rational Maps

Given a rational map f degree at least 2, it follows from work done by McMullen and Sullivan that in certain circumstances, the pure mapping class group PMCG(f) can be identified with a subgroup of the pure mapping class group of a Riemann surface. We investigate this identification and explore what types of subgroups of mapping class groups of surfaces arise in this way. We focus primarily on the case in which PMCG(f) can be viewed as a subgroup of a product of pure mapping class groups of punctured tori. A specific case of this setting --- namely, when f is a generic quadratic rational map --- was explored by Goldberg and Keen. The authors proved that for such a choice of f, PMCG(f) is an infinitely generated subgroup of the pure mapping class group of the twice-punctured torus. We prove the analogous statement in the setting of cubic polynomials, and explicitly write down a collection of generators of PMCG(f) in terms of point-pushes and a Dehn twist. We then prove a general result that is independent of the degree of the map. Specifically, we prove that for f in an open subset of rational maps of degree d, PMCG(f) is an infinitely generated subgroup of a product of pure mapping class groups of punctured tori.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163074/1/jtpowell_1.pd

The emergence of 4-cycles in polynomial maps over the extended integers

Let $f(x) \in \mathbb{Z}[x]$; for each integer $\alpha$ it is interesting to consider the number of iterates $n_{\alpha}$, if possible, needed to satisfy $f^{n_{\alpha}}(\alpha) = \alpha$. The sets $\{\alpha, f(\alpha), \ldots, f^{n_{\alpha} - 1}(\alpha), \alpha\}$ generated by the iterates of $f$ are called cycles. For $\mathbb{Z}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $\mathbb{Z}$ by adjoining reciprocals of primes. Let $\mathbb{Z}[1/p_1, \ldots, 1/p_n]$ denote $\mathbb{Z}$ extended by adding in the reciprocals of the $n$ primes $p_1, \ldots, p_n$ and all their products and powers with each other and the elements of $\mathbb{Z}$. Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $\mathbb{Z}\left[1/p_1, \ldots, 1/p_n\right][x]$ under the appropriate conditions. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $\pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.Comment: 14 pages, 1 figur

Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions

Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence $x, f_n(x), f_n(f_n(x))$, where $f_n(x) = x + n$, $n$ a nonzero integer. Thus avoiding three-term arithmetic progressions is equivalent to containing no three elements of the form $x, f_n(x), f_n(f_n(x))$ with $f_n \in\mathcal{F}_{\rm t}$, the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions ($f_n(x) = nx$ with $n > 1$ a natural number) and exponential progressions ($f_n(x) = x^n$). Progression-free sets are often constructed "greedily," including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1, and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density, however we consider several different notions of density, including the uniform and exponential densities.Comment: Version 1.0, 13 page

A blank screen? : digital divide discourses and practices

The existence of a “digital divide,” or inequalities of access to digital technologies among different American subpopulations, has been hotly debated and contested since the National Telecommunications and Information Administration first popularized the phrase in 1995. The purpose of this thesis is to critically examine the dominant discourses around the digital divide and articulate the existence of counterdiscourses to highlight the complexities of access to digital technologies and the implications of their various uses by African American college students at Oregon State University. After implementing a multidisciplinary approach incorporating interpretive analysis and ideas from cyberculture studies, using qualitative methods gleaned from cultural anthropology, a more multidimensional picture of the “digital divide” emerged. Early individual experiences with digital technologies, age, and location appeared to have a deeper impact on the students’ relationships with digital technologies, challenging the dominant discourse’s focus on race/ethnicity and socioeconomic status as primary determinants of an individual’s lack of access due to their membership in a “disadvantaged population.

What is the pathogenic CAG expansion length in Huntington’s disease?

Huntington’s disease (HD) (OMIM 143100) is caused by an expanded CAG repeat tract in the HTT gene. The inherited CAG length is known to expand further in somatic and germline cells in HD subjects. Age at onset of the disease is inversely correlated with the inherited CAG length, but is further modulated by a series of genetic modifiers which are most likely to act on the CAG repeat in HTT that permit it to further expand. Longer repeats are more prone to expansions, and this expansion is age dependent and tissue-specific. Given that the inherited tract expands through life and most subjects develop disease in mid-life, this implies that in cells that degenerate, the CAG length is likely to be longer than the inherited length. These findings suggest two thresholds – the inherited CAG length which permits further expansion, and the intracellular pathogenic threshold, above which cells become dysfunctional and die. This two-step mechanism has been previously proposed and modelled mathematically to give an intracellular pathogenic threshold at a tract length of 115 CAG (95% confidence intervals 70-165 CAG). Empirically, the intracellular pathogenic threshold is difficult to determine. Clues from studies of people and models of HD, and from other diseases caused by expanded repeat tracts, place this threshold between 60-100 CAG, most likely towards the upper part of that range. We assess this evidence and discuss how the intracellular pathogenic threshold in manifest disease might be better determined. Knowing the cellular pathogenic threshold would be informative for both understanding the mechanism in HD and deploying treatments

Project B.L.A.C.K. Barriers Lifted After Cultivating Knowledge: Assessing Individualized Barriers To Obesity Prevention In Black Women Using The Teach-Back Method

Background: Black women are diagnosed, disabled, and die from obesity and associated chronic diseases at higher rates than any other race or sex. Further exploration is warranted on how advanced practice registered nurses (APRN) can improve culturally relevant health education and counseling delivery. Objective: Explore individualized barriers and the healthcare provider’s roles in providing care affecting obesity prevention among Black women. While simultaneously assessing the effectiveness of educating Black women using the Teach-back method to understand health habits and attitudes. Method: A mixed-method design was utilized in group sessions and surveys. Participants identifying with obesity and associated diseases were recruited from a predominantly Black church in Atlanta. After completing a demographic survey and pre-Readiness to Change (RCQ) questionnaire, they engaged in weekly, one-hour educational sessions via Zoom addressing the four common barriers identified in the literature. They ended with a 5-10 minute teach-back session. Participants completed a post-RCQ questionnaire after the 4-weeks. Results: Twenty women completed the intervention. Descriptive statistics and qualitative data from surveys, audio, and emails were used for analysis. Paired sample t-test revealed no statistical significance and showed no correlation between pre and post-test RCQ scores after tailored health education was provided using teach-back. However, correlational analysis between BMI, education, and income level was significant with a p-value of 0.05. Discussion: Black women depend on healthcare providers for counseling and solutions. Furthermore, they require different approaches in screening, health promotion, and interventions that consistently assess individual risks, tailored education, and the use of Teach-back. Results emphasized that Black women experience rates of obesity differently from other races despite income or education level that was predominantly cited to be secondary to stress. Stress was voiced as a considerable contributor to disordered eating, decreased engagement in physical activity, and lack of motivation

The autophagy protein ATG16L1 is required for Sindbis virus-induced eIF2α phosphorylation and stress granule formation

Sindbis virus (SINV) infection induces eIF2α phosphorylation, which leads to stress granule (SG) assembly. SINV infection also stimulates autophagy, which has an important role in controlling the innate immune response. The importance of autophagy to virus-induced translation arrest is not well understood. In this study, we show that the autophagy protein ATG16L1 not only regulates eIF2α phosphorylation and the translation of viral and antiviral proteins, but also controls SG assembly. Early in infection (2hpi), capsids were recruited by host factors Cytotoxic Granule-Associated RNA Binding Protein (TIA1), Y-box binding protein 1 (YBX1), and vasolin-containing protein 1 (VCP), to a single perinuclear body, which co-localized with the viral pattern recognition sensors, double stranded RNA-activated protein-kinase R (PKR) and RIG-I. By 6hpi, there was increased eIF2α phosphorylation and viral protein synthesis. However, in cells lacking the autophagy protein ATG16L1, SG assembly was inhibited and capsid remained in numerous small foci in the cytoplasm containing YBX1, TIA1 with RIG-I, and these persisted for over 8hpi. In the absence of ATG16L1, there was little phosphorylation of eIF2α and low levels of viral protein synthesis. Compared to wild type cells, there was potentiated interferon protein and interferon-stimulated gene (ISG) mRNA expression. These results show that ATG16L1 is required for maximum eIF2α phosphorylation, proper SG assembly into a single perinuclear focus, and for attenuating the innate immune response. Therefore, this study shows that, in the case of SINV, ATG16L1 is pro-viral, required for SG assembly and virus replication

CD200 ectodomain shedding into the tumor microenvironment leads to NK cell dysfunction and apoptosis

The basis of immune evasion, a hallmark of cancer, can differ even when cancers arise from one cell type such as in the human skin keratinocyte carcinomas: basal and squamous cell carcinoma. Here we showed that the basal cell carcinoma tumor initiating cell surface protein CD200, through ectodomain shedding, was responsible for the near absence of NK cells within the basal cell carcinoma tumor microenvironment. In situ, CD200 underwent ectodomain shedding by metalloproteinases MMP3 and MMP11, which released biologically active soluble CD200 into the basal cell carcinoma microenvironment. CD200 bound its cognate receptor on NK cells, to suppress MAPK pathway signaling that in turn blocked indirect (gamma interferon release) and direct cell killing. In addition, reduced ERK phosphorylation relinquished negative regulation of PPARγ regulated gene transcription and lead to membrane accumulation of the Fas/FADD death receptor and its ligand, FasL that resulted in activation-induced apoptosis. Blocking CD200 inhibition of MAPK or PPARγ signaling restored NK cell survival and tumor cell killing, with relevance to many cancer types. Our results thus uncover a paradigm for CD200 as a potentially novel and targetable NK cell specific immune checkpoint, which is responsible for NK cell associated poor outcomes in many cancers