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This thesis investigates statistical approaches to interpreting the output of persistent homology, a multi-resolution algorithm for discovering topological structure in data. We provide a brief introduction to the theory of topology and homology. The output is a set of intervals, visualised either as a 'barcode' or as a set of points called a persistence diagram. We discuss suitable metrics for persistence diagrams. The following chapter demonstrates how to compute persistent homology using R. Following this foundational work, we find a confidence set for the true persistence diagram of the underlying space using a sample diagram. Such sets aid with the interpretation of persistence diagrams by identifying points that are likely representative of true topological features, and those points that are noise due to sampling. We present two methods of constructing confidence sets. The first assumes that the support of the sampling density is not too 'spiky'. The second method uses a stronger assumption that the data are a realisation of a homogeneous Poisson process, which leads to a less conservative confidence set. In the middle section of this thesis, we investigate further sampling properties of persistence diagrams. Sampling on the circle leads us to propose a barcode test of sampling uniformity. We look at the diagrams of samples from the unit square, which is topologically simple, and propose these as a model for the noise in diagrams from other spaces. We propose density corrected persistent homology that makes sample diagrams less sensitive to the geometry of the underlying space and the sampling density. In the last section of this thesis, we demonstrate how persistent homology can be used to identify topological structure in correlation and partial correlation matrices. This relates to the problem of structure learning in graphical models.

The pulling nature of crises on the 'frontiers' stimulates Western interventionism focused on state building. This interventionism is fundamentally dependent upon collaboration with indigenous politics and 'collaborative systems', the relationships linking interventionist actors with indigenous ones, determine its structure and dynamics. State building interventions rely on collaborative systems because they define the interface of the external forces of the intervening power with indigenous politics. Unless the energy and resources of the intervening power can be translated and internalised into terms of indigenous politics, the intervention will be unable to achieve its state building goals. Presently, Western states are both failing to build appropriate collaborative systems and to manage their collaborative partners. However, if Western states can improve their approach to and implementation of collaboration with indigenous politics, they can better manage insecurity on the frontiers through state-building interventionism.

Background: Age is the strongest risk factor for most chronic diseases, yet individuals age biologically at different rates. Summarising patterns of biomarkers that contribute strongly to overall and body system-specific ageing into biological and body system (bodily) ages may aid health risk communication and disease prevention. A systematic review (undertaken within this thesis) found that biological ages indicated or predicted poorer health, but few studies conducted validation or followed good practice for estimating and reporting biological ages. Methods: Among 480,019 UK Biobank participants aged 40-70 followed up for 6-12 years via linked death registry and hospital records, analyses focused on 141,254 (29.4%) participants healthy at baseline. Sex-specific biological ages were estimated from biomarker principal components (characterised from 72 physical and biochemical biomarkers) via two main approaches: (1) the age-based Klemera Doubal method, which emphasised biomarkers strongly related to chronological age, and (2) a novel disease risk-based approach of aggregating 8 body system ages (artery, musculoskeletal, gut, cardiac, metabolic disease, inflammatory, neurological and lung ages, each estimated using Cox lasso models) using a multi-state model. The proportions of the overall biological and chronological age effects on mortality from chronic disease, age-related frailty and the 8 groups of diseases explained by bodily ages were assessed using likelihood-based measures. Results: In healthy participants, reduced lung function, reduced kidney function, slower reaction time, lower insulin-like-growth factor 1 and lower hand grip strength strongly featured in the age-based biological age, while higher general adiposity was a shared risk factor across body system ages and therefore featured strongest in the disease risk-based biological age (together with lower central adiposity in men and lower hand grip strength in women). Although key biomarkers of body system ages were generally different from each other and from biomarkers strongly related to chronological age, biomarker patterns of body system ages apart from neurological age were moderately correlated (p=0.15-0.69). Gut, cardiac and neurological ages for men, and musculoskeletal and neurological ages for women contributed substantially to the prediction of mortality and frailty in the disease risk-based biological age. The age-based biological age overlapped with chronological age to explain two-thirds of the overall biological and chronological age effects on mortality and frailty. Biomarker constituents of the disease risk-based vs age-based biological ages alone explained 11-17% vs 2-8% of the overall effects on mortality and frailty, and the largest improvement was for frailty in healthy women. These proportions were higher when the analysis was repeated in the whole UK Biobank population (25-38% vs 15-35% for disease risk-based vs age-based biological ages). Biomarker constituents of body system ages explained 24-81% of the overall effects for their respective diseases in healthy participants. Conclusions: Bodily ages, particularly the novel disease risk-based ages, improved our understanding of the biomarker-disease relationships represented by chronological age and improved prognosis of later life health outcomes. Biomarkers across a range of body systems described a common ageing effect and substantial proportions of age effects on later life health, supporting a broader, multi-system risk-based approach to research and prevention of diseases of ageing.

Microtubules are critically important cytoskeletal elements. Together with microtubule associated proteins (MAPs), they form the latticework on which eukaryotic life exists. Simply put, microtubules are polymers of tubulin heterodimers, which are composed of the globular proteins alpha and beta tubulin. In vivo, these monomers associate with one another to form heterodimers, which then polymerize to form microtubules. In mammals, microtubule polymerization is a temperature-dependent process with an optimum of 37°C (Detrich et al., 2000). If temperatures exceed this optimal temperature by even a few degrees, the microtubule will begin to dissemble due to denaturation of the tubulin subunit and permanent loss of both shape and function will occur. This thermal barrier seems to be consistent in most eukaryotic organisms. Two exceptions are the thermophilic red algae, Cyanidium caldarium and Cyanidioschyzon merolae. These thermophilic acidophiles have been discovered in volcanic vents around the globe from Yellow Stone Park to Italy and grow at optimal temperatures of around 55°C. These organisms have been primarily studied in the context of evolutionary biology because of their primitive characteristics. Very little is known about the molecular biology of these organisms, and certainly nothing is known about how the biochemistry of these organisms brings about the ability to survive the harsh conditions of their environment. Currently, my hypothesis concerning the thermostable tubulin expressed within these organisms is that there may be key amino acid differences in the tubulin's primary structure that confer enhanced thermostability. I am testing this hypothesis by sequencing the alpha and beta tubulin genes of Cyanidium caldarium and Cyanidioschyzon merolae, generating homology models of the tubulin dimers, and comparing these models to a known mesophilic tubulin heterodimer structure in order to identify potential structural differences.