Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

Fine mapping of qGW1, a major QTL for grain weight in sorghum

Key message We detected seven QTLs for 100-grain weight in sorghum using an F 2 population, and delimited qGW1 to a 101-kb region on the short arm of chromosome 1, which contained 13 putative genes. Abstract Sorghum is one of the most important cereal crops. Breeding high-yielding sorghum varieties will have a profound impact on global food security. Grain weight is an important component of grain yield. It is a quantitative trait controlled by multiple quantitative trait loci (QTLs); however, the genetic basis of grain weight in sorghum is not well understood. In the present study, using an F2 population derived from a cross between the grain sorghum variety SA2313 (Sorghum bicolor) and the Sudan-grass variety Hiro-1 (S. bicolor), we detected seven QTLs for 100-grain weight. One of them, qGW1, was detected consistently over 2 years and contributed between 20 and 40 % of the phenotypic variation across multiple genetic backgrounds. Using extreme recombinants from a fine-mapping F3 population, we delimited qGW1 to a 101-kb region on the short arm of chromosome 1, containing 13 predicted gene models, one of which was found to be under purifying selection during domestication. However, none of the grain size candidate genes shared sequence similarity with previously cloned grain weight-related genes from rice. This study will facilitate isolation of the gene underlying qGW1 and advance our understanding of the regulatory mechanisms of grain weight. SSR markers linked to the qGW1 locus can be used for improving sorghum grain yield through marker-assisted selection.

Mapping the carriage of flaA-restriction fragment length polymorphism Campylobacter genotypes on poultry carcasses through the processing chain and comparison to clinical isolates

Poultry are considered a major source for campylobacteriosis in humans. A total of 1866 Campylobacter spp. isolates collected through the poultry processing chain were typed using flaA-restriction fragment length polymorphism to measure the impact of processing on the genotypes present. Temporally related human clinical isolates (n = 497) were also typed. Isolates were obtained from whole chicken carcass rinses of chickens collected before scalding, after scalding, before immersion chilling, after immersion chilling and after packaging as well as from individual caecal samples. A total of 32 genotypes comprising at least four isolates each were recognised. Simpson's Index of Diversity (D) was calculated for each sampling site within each flock, for each flock as a whole and for the clinical isolates. From caecal collection to after packaging samples the D value did not change in two flocks, decreased in one flock and increased in the fourth flock. Dominant genotypes occurred in each flock but their constitutive percentages changed through processing. There were 23 overlapping genotypes between clinical and chicken isolates. The diversity of Campylobacter is flock dependant and may alter through processing. This study confirms that poultry are a source of campylobacteriosis in the Australian population although other sources may contribute.

Numerical computation of the module of a quadrilateral

The module of a quadrilateral is a positive real number which divides quadrilaterals into conformal equivalence classes. This is an introductory text to the module of a quadrilateral with some historical background and some numerical aspects. This work discusses the following topics: 1. Preliminaries 2. The module of a quadrilateral 3. The Schwarz-Christoffel Mapping 4. Symmetry properties of the module 5. Computational results 6. Other numerical methods Appendices include: Numerical evaluation of the elliptic integrals of the first kind. Matlab programs and scripts and possible topics for future research. Numerical results section covers additive quadrilaterals and the module of a quadrilateral under the movement of one of its vertex.

Making the most of spatial information in health: A tutorial in Bayesian disease mapping for areal data

Disease maps are effective tools for explaining and predicting patterns of disease outcomes across geographical space, identifying areas of potentially elevated risk, and formulating and validating aetiological hypotheses for a disease. Bayesian models have become a standard approach to disease mapping in recent decades. This article aims to provide a basic understanding of the key concepts involved in Bayesian disease mapping methods for areal data. It is anticipated that this will help in interpretation of published maps, and provide a useful starting point for anyone interested in running disease mapping methods for areal data. The article provides detailed motivation and descriptions on disease mapping methods by explaining the concepts, defining the technical terms, and illustrating the utility of disease mapping for epidemiological research by demonstrating various ways of visualising model outputs using a case study. The target audience includes spatial scientists in health and other fields, policy or decision makers, health geographers, spatial analysts, public health professionals, and epidemiologists.

Generalised quaternion methods in conformal geometry

A new approach to Penrose's twistor algebra is given. It is based on the use of a generalised quaternion algebra for the translation of statements in projective five-space into equivalent statements in twistor (conformal spinor) space. The formalism leads toSO(4, 2)-covariant formulations of the Pauli-Kofink and Fierz relations among Dirac bilinears, and generalisations of these relations.

Using relationship mapping to understand sustainable housing stakeholders’ actions

The production of sustainable housing requires the cooperation of a variety of participants with different goals, needs, levels of commitment and cultures. To achieve mainstream net zero energy housing objectives, there is arguably a need for a non-linear network of collaboration between all the stakeholders. In order to create and improve such collaborative networks between stakeholders, we first need to map stakeholders’ relationships, processes, and practices. This paper discusses compares and contrasts maps of the sustainable housing production life-cycle in Australia, developed from different perspectives. The paper highlights the strengths and weaknesses of each visualization, clarifying where gaps in connectivity exist within existing industry networks. Understanding these gaps will help researchers and practitioners identify how to improve the collaboration between participants in the housing industry. This in turn may improve decision making across all stakeholder groups, leading to mainstream implementation of sustainability into the housing industry.

Fine Mapping of qDor7, a Major QTL Affecting Seed Dormancy in Sorghum (Sorghum bicolor (L.) Moench)

Seed dormancy is a key domestication trait for major crops, which is acquired in long-term systems development processes and enables the survival of plants in adverse natural conditions. It is a complex trait under polygenic control and is affected by endogenous and environmental factors. In the present study, a major seed dormancy QTL in sorghum (Sorghum bicolor (L.) Moench), qDor7, detected previously, was fine mapped using a large, multi-generational population. The qDor7 locus was delimited to a 96-kb region which contains 16 predicted gene models. These results lay a solid foundation for cloning qDor7. In addition, the functional markers tightly linked to the seed dormancy QTL may be used in marker-assisted selection for seed dormancy in sorghum.

Conformal invariance and scalar-tensor theories

A new generalisation of Einstein’s theory is proposed which is invariant under conformal mappings. Two scalar fields are introduced in addition to the metric tensor field, so that two special choices of gauge are available for physical interpretation, the ‘Einstein gauge’ and the ‘atomic gauge’. The theory is not unique but contains two adjustable parameters ζ anda. Witha=1 the theory viewed from the atomic gauge is Brans-Dicke theory (ω=−3/2+ζ/4). Any other choice ofa leads to a creation-field theory. In particular the theory given by the choicea=−3 possesses a cosmological solution satisfying Dirac’s ‘large numbers’ hypothesis.