Honours Workshop 2003 |

Maths & Stats honours students have each been working on a research project this whole year, and this is their chance to impress! Each student will give a short presentation of their project. This is a great opportunity to see what honours students do, especially if you are considering doing honours in the future. The talks will span two days, and will be held in a variety of venues (details are provided below). Each talk will be 20-25 minutes in duration (with 5 minutes left for questions), and refreshments will be served towards the end of the day. ## Stats Talks## Details
## Timetable*2:15 pm :*Diana Zannino - Statistical Process Control: Cumulative Sum Control Charts*2:45 pm :*Charles Tran - Two models for genotype by environment interaction*3:30 pm :*Anna Kozlowska - Modelling using Generalized Ornstein-Uhlenbeck Processes*4:00 pm :*Fan Yang - Time series analysis of hospital data*4:45 pm :*Catherine Lai - Phylogenetic Inference and Hypothesis Testing
## Applied Maths Talks## Details
## Timetable*9:00 am :*Alan Motyer - On an embedded spatial pure birth process and stochastically-evolving networks*9:30 am :*Sarah Hicks - Theory to Interpret Direct Force-Displacement Measurements taken by the Atomic Force Microscope of the Forced Collision of Two Elastic Spheres*10:15 am :*Matthew Wightwick - A Phenomenological Model for the Segregation of a Granular Mixture in a Rotating Drum*10:45 am :*Timothy Baxter - Analysis of walks in the Lorentz Lattice Gas*11:30 am :*Malek Ghantous - The behaviour of polyelectrolytes in poor solvent, and the creation of pearl necklaces*12:00 pm :*Anna Qianyao Cai - Modelling Cell Growth and Vascularisation inside a Polymer Scaffold*12:45 pm :*Jennifer Slater - Travelling waves: Diffusion and chemotaxis in cell migration
## Pure Maths Talks## Details
## Timetable*9:00 am :*Paul Incani - The Higman Embedding Theorem*9:30 am :*Loretta Bartolini - Heegaard Splittings and the Reducibility of 3-Manifolds*10:15 am :*Alegra Dajic - Compact and trace class operators*10:45 am :*Huy Nguyen - Idempotents in Hilbert Spaces*11:30 am :*Raymond Lubansky - Semigroups of operators*12:00 pm :*Luke Mawbey - Improving a Cut Locus Approximation Through an Alternative Triangulation*12:45 pm :*Jonathan Bowden - Minimal surfaces and Plateau's Problem*1:15 pm :*Thomas Poole - Random Unitary Matrices and The Wigner Semicircle Law
## Abstracts## Stats Talks
Statistical process control (SPC) is a widely used method of quality control. Control charts are key tools of SPC and one particular method is the Cumulative Sum (CUSUM) procedure. The CUSUM scheme is reviewed and applied to both real life and simulated examples.
I will talk about two models commonly used to analyse the genotype by environment interaction in plant breeding programs. These models characterise the interaction differently from the additive and interactive ANOVA models usually encountered in a standard statistical course. I will briefly describe the model, explain why they are fitted that way and give an interpretation so it can be related to the real world problem of improving varieties.
In my project I looked at finding a good model for financial data. The aim was to use the Ornstein-Uhlenbeck process, which incorporates many of the features exhibited by financial log returns, but modify it such that it has tails that are heavier than the Gaussian tails. The project shows that a compound Poisson driven O-U process, in both discrete and continuous time, exhibits tail behaviour that is of the right form for modelling financial data. Some exploratory data analysis and model fitting is also preformed.
Time series analysis is a widely used technique for analysing data, especially for forecasting. In this project forecasting methods are applied to four series of data on the daily numbers of patients entering a hospital. An important finding is that, contrary to what was expected, the data do not exhibit any seasonality.
Phylogenetics is a field of biology that seeks to unlock the evolutionary history of life on earth. The aim is to understand relationships between species and through this the process of evolution itself. These relationships can be represented with a graph structure - traditionally simplified to evolutionary trees. The current approach is to try to reconstruct these trees from the blueprint of life: DNA sequences. Reconstruction methods are difficult to design and evaluate because the biological evidence is often ambiguous. Parametric approaches exploit the elementary knowledge we have of evolution while non-parametric approaches have been developed to avoid the possibility of inaccurate preconceptions. I consider methods of evaluating confidence in results from such reconstruction methods when there may be conflict. This leads to an examination of current hypothesis testing methods and consideration of the validity of the generalised least squares approach. ## Applied Maths Talks
We derive a limit theorem for an embedded spatial pure birth process and investigate properties of stochastically-evolving networks that involve this process. In particular, we compare properties to those found in scale-free networks.
One elastic sphere is driven towards another with a known driving displacement. The collision of two elastic spheres is modelled using Reynolds lubrication theory and linear elastic theory for small deformations. The atomic force microscope (AFM) may be used to measure the displacement of one sphere as the other approaches. The AFM is a cantilever whose tip deflects when forces are applied to it. The deflection of this tip is measured with a laser. The model of the collision of two elastic spheres is extended to include a spring attached to one sphere representing the cantilever of the atomic force microscope. Numerical methods investigate the behaviour of the elastic spheres for different geometries and physical properties. The use of a known forcing displacement and the addition of a spring modelling the AFM cantilever is novel.
Among the many interesting phenomena concerning granular materials is their tendency to segregate when they might be expected to mix. This can be seen in many different cases, one of which is a binary mixture in a rotating drum. I will be using a phenomenological model developed in Puri and Hayakawa in 2000 to investigate the two types of segregation seen in such a system, namely radial segregation, where a core rich in one material develops that is surrounded by the other material, and axial segregation, where bands that are rich in the two constituents are observed along the length of the drum.
There has been a considerable amount of research performed on deterministic motion in random environments. The classical example of such a system is the `Lorentz gas' in which a moving particle undergoes elastic collisions with fixed, rigid obstacles (scatterers). In a Lorentz gas model, the scatterers may be placed evenly on a periodic lattice or distributed randomly on the lattice. There is also the opportunity to place scatterers with different characteristics that result in the particle rebounding at different angles after a collision. The interest is in knowing, for an arbitrary starting position and initial velocity direction, such things as the probability that the motion will be periodic, aperiodicbut bounded, or unbounded. The aim for this project is to simulate movements in these environments so as to answer such questions as: Where will the particle be after a given number of steps? Where has the particle traveled during the walk? And, is the particle trapped in a loop?
Polyelectrolytes are found in a wide range of systems and have gained great im-portance in industry in everything from paint to diapers (and my mum never knew...). Because of the many competing forces present their behaviour can be quite complex and difficult to model. Experiments can tell us some things but due to the incredibly small time and distance scales involved they are often incabable of furnishing conclusive answers. An important method for overcoming these problems is by simulation. In this study we examine the behaviour of a polyelectrolyte in poor solvent, comparing two different models for it (a square well attraction between the monomers and a Lenard-Jones type attraction, which has already been studied in some detail for these systems). Specifically we are looking at the region in between where the polymer is fully extended and fully collapsed. According to theory the polymer chain should form large clusters separated by stringy bits, usually referred to in the literature as ``pearl necklaces". These structures are as yet not detectable in experiment and we hope to suggest some way in which they might be, at least indirectly.
A major cause of experimental failure during soft tissue engineering is inadequate oxygen delivery to growing cells. Cells can die if lacking oxygen for extended periods of time and dead cells affect health of neighbouring cells , this could lead to experimental failure. High cell density with large blood vessel network is characteristic of soft tissue, which suggests high oxygen demand. Hence successful generation of healthy soft tissue involves not only growing cells but also developing a blood vessel network to sustain the cells. This thesis aims to provide to provide an understanding of various mechanisms involved in soft tissue generation by modelling cell growth and development of blood vessel network (vascularisation), based on the arterio-venous experimental model from Bernard O'Brien Institute.
The process of chemotactic cellular migration can be modelled by a convection/diffusion equation, coupled with another pde. The convection term represents chemotaxis, a biological phenomenon whereby cells migrate towards higher concentrations of an attractive chemical (chemoattractant), and also consume this chemoattractant. The diffusion terms allows linear diffusion of cells to be included in the model. If only diffusion is present, the model reduces to Fisher's equation, which supports smooth travelling wave solutions, for some parameter values. If chemotaxis is the only cell migration mechanism present, the model supports both smooth and discontinuous travelling wave solutions, for some parameter values. The existence of travelling wave solutions is determined by phase plane analysis, while wavespeed is calculated from numerical solutions of the migration equations. I have investigated the existence and speed of travelling wave solutions to the combined problem by considering it as a perturbation away from the convection problem and again as a perturbation away from the diffusion problem. ## Pure Maths Talks
The Higman Embedding Theorem is a remarkable result which blends together algebra and mathematical logic. The theorem states that a finitely generated group G can be embedded in a finitely presented group if and only if G is recursively presented. We give a proof of this theorem based on the approach of J. R. Shoenfield, and have actually corrected a mistake in Shoenfield's proof. We then give some applications of the Higman Embedding Theorem, and in particular show how it can be used to prove that 1
The project covers the general background of 3-manifolds and how Heegaard splittings can be used to study them. It concentrates on two of the landmark re-sults in Heegaard splitting theory: Haken's Lemma (1968), and the Casson-Gordon result for weakly reducible splittings (1987). While the former has changed the way the area is approached, having a widespread influence over subsequent work, the latter introduced the concept of strong irreducibility, which is the topic of significant current research.
The project gives an introduction to compact operators and the main results regarding compact operators on Hilbert spaces, and provides a novel proof of the existence of a nontrivial eigenvalue of a normal compact operator. Singular values and the Schmidt representation for a compact operator are introduced and the definition of the trace, as known from linear algebra, is extended to a class of compact operators, called the trace class. Hilbert-Schmidt operators on Hilbert spaces are also introduced.
This is a project concerned with extending an unpublished result known for matrices to orthogonal projections in a Hilbert space. It investigates the invertibility of the sum of two orthogonal projections P and Q, first under the assumption that the difference P - Q is invertible, and then in the general case. Necessary and sufficient conditions for the invertibility of P + Q are given, and explicit formulae for the inverse are derived. This is a new result applicable to the theory of differential equations in Hilbert spaces.
A strongly continuous semigroup of operators is a generalisation of the operator exponential on bounded linear operators in a Banach space to a class of densely defined closed linear operators (which includes bounded linear operators). In this talk I will present the theory leading up to the Laplace transform representation of the resolvent and an application of the theory to partial differential equations.
The cut locus on a convex two dimensional surface embedded in R
The Belgian physicist J.Plateau investigated the propeties of soap-films bounded by certain contours in space. This led him to conjecture that any curve in space bounds a soap film like disc, which of necessity has the least possible area. I will give an overview of the basic theory of minimal surfaces, including some beautiful connections with complex analysis and give an outline of the solution Plateau's Problem.
In this talk I will give a brief introduction to random matrices. I will then talk about the various versions of the Wigner semicircle law and briefly describe the proofs for each version. Finally I will describe two ergodic theorems and show that they are equivalent to the two versions of the Wigner semicircle law. |