Braid groups, mapping class groups, and Torelli groups

This thesis discusses subgroups of mapping class groups of particular surfaces. First, we study the Torelli group, that is, the subgroup of the mapping class group that acts trivially on the first homology. We investigate generators of the Torelli group, and we give an algorithm that factorizes elements of the Torelli group into products of particular generators. Furthermore, we investigate normal closures of powers of standard generators of the mapping class group of a punctured sphere. By using the Jones representation, we prove that in most cases these normal closures have infinite index in the mapping class group. We prove a similar result for the hyperelliptic mapping class group, that is, the group that consists of mapping classes that commute with a fixed hyperelliptic involution. As a corollary, we recover an older theorem of Coxeter (with 2 exceptional cases), which states that the normal closure of the m-th power of standard generators of the braid group has infinite index in the braid group. Finally, we study finite index subgroups of braid groups, namely, congruence subgroups of braid groups. We discuss presentations of these groups and we provide a topological interpretation of their generating sets.

The Shrine of ʿAbd al-Qādir al-Jīlānī in Baghdad & the Shrine of ʿAbd al-ʿAzīz al-Jīlānī in ʿAqra: Mapping the multiple orientations of two Qādirī Sufi shrines in Iraq

This thesis charts the stakeholder communities, physical environment and daily life of two little studied Qādiriyya Sufi shrines associated with Shaikh ʿAbd al-Qādir al-Jīlānī (1077 – 1165 AD), a 12th century Ḥanbalī Muslim theologian and the posthumous founder of one of the oldest Sufi orders in Islam. The first shrine is based in Baghdad and houses his burial chamber; and the second shrine, on the outskirts of the city of ‘Aqra in the Kurdish region of northern Iraq, is that of his son Shaikh ʿAbd al-ʿAzīz (died 1206 AD). The latter was also known for lecturing in Ḥanbalī theology in the region, and venerated for this as well as his association with Shaikh ʿAbd al-Qādir. Driven by the research question “What shapes the identity orientations of these two Qādiriyya Sufi shrines in modern times?” the findings presented here are the result of field research carried out between November 2009 and February 2014. This field research revealed a complex context in which the two shrines existed and interacted, influenced by both Sufi and non-Sufi stakeholders who identified with and accessed these shrines to satisfy a variety of spiritual and practical needs, which in turn influenced the way each considered and viewed the two shrines from a number of orientations. These overlapping orientations include the Qādirī Sufi entity and the resting place of its patron saint; the orthodox Sunnī mosque with its muftī-imams, who are employed by the Iraqi government; the local Shīʿa community’s neighbourhood saint’s shrine and its destination for spiritual and practical aid; and the local provider of welfare to the poor of the city (soup kitchen, funeral parlour and electricity-generation amongst other services). The research findings also revealed a continuously changing and adapting Qādirī Sufi scene not immune from the national and regional socio-religio-political environments in which the two shrines exist: a non-Sufi national political class vying to influence and manipulate these shrines for their own purposes; and powerful national sectarian factions jostling to do the same. The mixture of stakeholders using and associating with the two shrines were found to be influential shapers of these entities, both physically and spiritually. Through encountering and interacting with each other, most stakeholders contributed to maintaining and rejuvenating the two shrines, but some also sought to adapt and change them driven by their particular orientation’s perspective.

Mapping and modulating spatial attention asymmetries in young and older adults

Healthy young adults demonstrate a group-level, systematic preference for stimuli presented in the left side of space relative to the right (‘pseudoneglect’) (Bowers & Heilman, 1980). This results in an overestimation of features such as size, brightness, numerosity and spatial frequency in the left hemispace, probably as a result of right cerebral hemisphere dominance for visuospatial attention. This spatial attention asymmetry is reduced in the healthy older population, and can be shifted entirely into right hemispace under certain conditions. Although this rightward shift has been consistently documented in behavioural experiments, there is very little neuroimaging evidence to explain this effect at a neuroanatomical level. In this thesis, I used behavioural methodology and electroencephalography (EEG) to map spatial attention asymmetries in young and older adults. I then use transcranial direct current stimulation (tDCS) to modulate these spatial biases, with the aim of assessing age-related differences in response to tDCS. In the first of three experiments presented in this thesis, I report in Chapter Two that five different spatial attention tasks provide consistent intra-task measures of spatial bias in young adults across two testing days. There were, however, no inter-task correlations between the five tasks, indicating that pseudoneglect is at least partially driven by task-dependent patterns of neural activity. In Chapter Three, anodal tDCS was applied separately to the left (P5) and right (P6) posterior parietal cortex (PPC) in young and older adults, with an aim to improve the detection of stimuli appearing in the contralateral visual field. There were no age differences in response to tDCS, but there were significant differences depending on baseline performance. Relative to a sham tDCS protocol, tDCS applied to the right PPC resulted in maintained visual detection across both visual fields in adults who were good at the task at baseline. In contrast, left PPC tDCS resulted in reduced detection sensitivity across both visual fields in poor performers. Finally, in Chapter Four, I report a right-hemisphere lateralisation of EEG activity in young adults that was present for long (but not short) landmark task lines. In contrast, older adults demonstrated no lateralised activity for either line length, thus providing novel evidence of an age-related reduction of hemispheric asymmetry in older adults. The results of this thesis provide evidence of a highly complex set of factors that underlie spatial attention asymmetries in healthy young and older adults.