School of Mathematics and Statistics - Theses
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ItemMathematical approaches to pattern formation in dermatology(University of Melbourne, 2005)
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ItemPointwise axiomatic spectral theory in Banach algebras(University of Melbourne, 2008)
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ItemIntersection theory on moduli spaces of curves via hyperbolic geometry( 2008)This work draws together Kontsevich’s combinatorial approach and Mirzakhani’s hyperbolic approach to Witten’s conjecture into a coherent narrative.
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ItemBoundary crossing probabilities for diffusion processes and related problems( 2008)This thesis is concerned with boundary crossing probabilities and first crossing time densities for stochastic processes. This is a classical problem in probability that goes back to the famous ballot problem (first studied by W. A. Whitworth (1878) and J. Bertrand (1887)) and has numerous applications in diverse areas including mathematical statistics and financial mathematics. Our main objective is the study of approximation methods and control of the resulting approximation error for boundary crossing probabilities where a closed-form solution is unavailable. This leads to the study of bounds for the density of the first crossing time of the boundary, which in turn leads to the derivation of some analytic properties of the densities. This thesis presents a whole suite of closely related new results obtained when working on the outlined research program. (For complete abstract open document).
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ItemImproving intellectual and affective quality in mathematics lessons: how autonomy and spontaneity enable creative and insightful thinking( 2005-09)The nature of creative mathematical thinking undertaken by students in classroom settings was studied through analysis of the autonomy and spontaneity associated with these processes. The theoretical lens developed enabled simultaneous analysis of cognitive, social, and affective elements of the creative process, and student responses to successes and failures during their exploratory activity (resilience or optimism). Collective case study was employed, with each case progressively informing the analysis of subsequent cases. The classrooms of teachers who were seen by their school communities to display 'good teaching practice' were selected for study. It was anticipated that such classrooms would provide more opportunity to study creative thinking than classrooms chosen at random. During the research period, each student participated individually in post-lesson interviews that were stimulated by lesson video material. To generate data to study student thinking, and the social and personal influences upon it, students were asked to identify parts of the lesson that were important to them, and discuss what was happening, and what they were thinking and feeling. Through this process, students who explored mathematical complexities to generate new mathematical knowledge were identified. (For complete abstract open document)
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ItemModelling hydrodynamic interactions between deformable droplets( 2007-06)Understanding deformations during interaction of colloidal or nano droplets. has important implications in a wide range of applications such as flotation collection and emulsion stability. The present work is important and necessary because current models are unable to properly describe recent experimental results obtained using the Atomic Force Microscope (AFM) and Surface Force Apparatus (SFA). We revisit and improve on theories involving interacting deformable droplets. A detailed theoretical model of the experiments, which accounts for surface forces, hydrodynamic interactions, droplet deformation and AFM cantilever detection, has been developed. The new feature of the model is the use of matched asymptotic expansions to derive a new boundary condition at constant volume to obtain results that are independent of the size of the computational domain. The AFM experiments provide measurements of dynamical forces between two interacting droplets as well as between a drop and a colloidal particle of micrometre radii in solution when they are driven together and then are retracted from each other. The SFA experiments measure (time dependent) surface profiles of a millimetre size mercury drop interacting with a mica surface. Different configurations have been studied including mechanical and electrical perturbations to a thin film stabilized by double layer repulsion. Interesting experimental features were observed when approaching or retracting the mica from the mercury drop.
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ItemEvaluating the constructivist potential of the multimedia software "Stage struck" in drama education( 2005-01)This project compares the educational aims of the designers of the educational title “Stage Struck” with those of two separate groups in the process of learning and teaching; namely the aims of teachers and students. This method of comparing the observations of two key stakeholder groups with the objectives of the designer is part of an instrument proposed by Squires and McDougall (1994) called the Perspectives Interaction Paradigm. This instrument diverges from traditional models of software evaluation in that it has encompassed multiple points of view in the determination of whether a piece of software is useful in educational environments. This project hypothesizes that the Squires/McDougall approach shows how software can be utilized in ways that can be complimentary or contrary to the intentions of designers. It is for this reason and the multi-pronged method of data collection that the instrument is considered a more appropriate tool for the project than other evaluative checklists. “Stage Struck” has been chosen because a number of items in the education literature indicate clearly articulated aims that the software be used in a manner befitting constructivist approach. Using interviews with two teachers and observations of two students within the scope of the controlled setting, this project will identify where and why aims are transformed/modified and how the expectations of the educational merit of “Stage Struck” correspond or fail to correspond.
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ItemWeierstrass points and canonical cell decompositions of the moduli and Teichmuller Spaces of Riemann surfaces of genus two( 2007)A genus-two Riemann surface admits a canonical decomposition into Dirichlet polygons determined by its six Weierstrass points. All possible associated graphs are determined explicitly from circle packing problems, solved by systems of linear inequalities whose solutions determine a finite 6-dimensional polyhedral complex in 12-dimensional space. The 6-dimensional Moduli Space of genus-two Riemann surfaces inherits a canonical explicit decomposition into Euclidean polyhedra, giving new natural coordinates for the Teichmuller Space of all possible constant curvature geometries on a marked genus-two surface.
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ItemRandom graph processes and optimisation( 2006-01)Random graph processes are most often used to investigate theoretical questions about random graphs. A randomised algorithm can be defined specifically for the purpose of finding some structure in a graph, such as a matching, a colouring or a particular kind of sub graph. Properties of the related random graph process then suggest properties, or bounds on properties, of the structure. In this thesis, we use a random graph process to analyse a particular load balancing algorithm from theoretical computer science. By doing so, we demonstrate that random graph processes may also be used to analyse other algorithms and systems of a random nature, from areas such as computer science, telecommunications and other areas of engineering and mathematics. Moreover, this approach can lead to theoretical results on the performance of algorithms that are difficult to obtain by other methods. In the course of our analysis we are also led to some results of the first kind, relating to the structure of the random graph. The particular algorithm that we analyse is a randomised algorithm for an off-line load balancing problem with two choices. The load balancing algorithm, in an initial stage, mirrors an algorithm which finds the k-core of a graph. This latter algorithm and the related random graph process have been previously analysed by Pittel, Spencer and Wormald, using a differential equation method, to determine the thresholds for the existence of a k-core in a random graph. We modify their approach by using a random pseudograph model due to Bollobas and Frieze, and Chvatal, in place of the uniform random graph. This makes the analysis somewhat simpler, and leads to a shortened derivation of the thresholds and other properties of k-cores.(For complete abstract open document)