FACULTY OF SCIENCE

School of Mathematics and Statistics

POTENTIAL MASTERS PROJECTS

SCHOOL OF MATHEMATICS AND STATISTICS The School of Mathematics and Statistics at UNSW is one of the largest and most successful in Australia. It is an active centre for research, both fundamental and applied. The School members have worked on a wide range of research projects in pure mathematics, applied mathematics, and statistics, and have participated in collaborative research activities with numerous distinguished researchers. Department members have received prestigious awards and international recognition for outstanding research in recent years. The School has three departments: Applied Mathematics, Pure Mathematics and Statistics. APPLIED MATHEMATICS Research areas of the Applied Department at UNSW cover 4 broad groups: Biomathematics; Computational Mathematics; Nonlinear Phenomena; and Optimisation. The indicated staff member will be most happy to discuss the possible topics with you further. Adelle Coster Email: a.coster@unsw.edu.au

• The Insulin Signalling Pathway in Adipocytes - A Mathematical Investigation:

The insulin signalling pathway in adipocytes (fat cells) is the main controller of the uptake of glucose in the cells. Understanding of this system is vital in the investigation of diabetes, which is a deficiency in this control system. One of the experimental techniques used to understand this system is total internal reflection fluorescence (TIRF) microscopy. In this technique, the glucose transporter proteins in the cell have been modified to contain a fluorescent segment. Under the TIRF microscope the first ~200nm of the cell is observed, so that the proteins, embedded in small spheres called vesicles, can be detected as they are transported to the surface. These vesicles can then fuse with the cell surface, releasing the glucose transporter proteins. This project will develop a mathematical description of the movement of these vesicles in the TIRF field of view, and also the diffusion of the proteins if the vesicle fuses with the cell surface. This involves the development and solution of differential equations defining the intensity of the resulting image as a function of both space and time, taking into account the TIRF system, the depth of the vesicle and also the relative opacity of the cell interior. This will involve analytical work and possibly some computational simulation of the system. Experimental data for comparison with the model will be available in collaboration with Prof David James at the Garvan Institute for Medical Research.

• Dynamics and Stability of Cardiac Pacemaker Cells: The ionic currents underlying the electrical behaviour of pacemaker cells in the heart may be described using coupled nonlinear differential equations. These have been derived in a number of different models from a wide range of electrophysiological data in the literature. In this project, the dynamics of this system, both in single cells, and coupled networks of cells will be explored, and the robustness of the models to changes in parameters ascertained. Different models for the sinoatrial node cells will be assessed and their different responses correlated to the ionic currents, with a view to further optimising the system. This project will be both analytical and computational. Josef Dick Email: Josef.dick@unsw.edu.au Josef Dick is a Queen Elizabeth 2 Fellow (ARC) and his research interests are Quasi-randomness. Further details of his research can be found at http://quasirandomideas.wordpress.com/ Gary Froyland Email: g.froyland@unsw.edu.au

• Topics in Dynamical Systems and Ergodic Theory: Ergodic theory is the study of the dynamics of distributions of points, in contrast to topological dynamics, which focusses on the dynamics of single points. A number of theoretical Honours projects are available in dynamical systems and ergodic theory, with the goals being to develop new tools to analyse the complex behaviour of nonlinear dynamical systems. Depending on your background, these projects may involve mathematics from the areas of Ergodic Theory, Measure Theory, Spectral Theory, Functional Analysis, Nonlinear Time Series Analysis, Mathematical Modelling, Nonlinear and Random Dynamical Systems, Markov chains, Graph Theory, and Coding and Information Theory. Successful completion of Math3201 is recommended for this topic.

• Transfer Operator Methods with Applications to Fluid Mixing:

A transfer operator is a linear operator that completely describes the evolution of probability densities of a nonlinear dynamical system. Transfer operators are therefore fundamental objects like maps and flows, but operate on probability distributions rather than single points. Spectral techniques using transfer operators have been shown to be particularly effective for identifying coherent dynamical structures in a variety of theoretical and physical systems. The identification and estimation of coherent structures in driven fluid flow is a current hot topic. This project will focus on developing and applying powerful spectral techniques to extract important geometric dynamical structures from fluid mixing models, including ocean models. Successful completion of Math3201 is recommended for this to complete this project.

• Integer Programming and Combinatorial Optimisation: •

Integer programming is a mathematical framework for solving large decision problems. Usually there is some underlying discrete structure for the problem such as a network or graph. Application areas may include scheduling airlines, rail, or mining processes; theoretical projects are also possible. Successful completion of Math2140 is required for this project.

• Stochastic Integer Programming with Applications:

Almost all real world models have significant uncertainty in their measured data. A naive approach is to replace probability distributions of data with their mean value and create a single deterministic model. However, optimising this deterministic model typically results in decisions that are far from optimal. In order to make the optimal decisions, the underlying probability distributions must be factored into the optimisation process. This is the aim of stochastic programming. The goal of this project is to develop rigorous optimization methods that take into account uncertainties in the forecast data and evaluate all possible options in light of the latest information. Successful completion of Math2140 is required for this project. Familiarity with basic probability theory will be beneficial. Application areas may include scheduling airlines, rail, or mining processes.

• Industrial Modelling and Optimization: Depending on availability there may be a project where you would work directly on modelling and optimising problems from a company. Bruce Henry Email: B.Henry@unsw.edu.au

• Pattern formation with anomalous diffusion: How did the leopard get its spots? This is one of many examples of pattern formation that has been mathematically modelled using reaction-diffusion equations. In this mathematical model the diffusion is treated as standard Brownian motion. In recent years there have been numerous experiments in biological systems showing diffusion that cannot be treated as standard Brownian motion. How can we model this anomalous diffusion mathematically and how can we model pattern formation in these systems?

• Mathematical modelling of plaque build up in Alzheimer's disease: Currently about 150,000 Australian's suffer from cognitive impairment due to Alzheimer's disease and this number is expected to double over the next few decades. One of the features of this disease at the neural level is the accumulation of extracellular amyloid beta plaque but the mechanism for this accumulation and the role that it plays in Alzheimer's is not yet understood. In this project you will review the literature in this field and you will develop mathematical models for the growth of plaque deposits.

• Fractional calculus for fractals: A fractal function such as the Mandebrot-Weistrass function is everywhere continuous but nowhere differentiable. However this function is fractionally differentiable. In this topic you will learn about fractional calculus and to what extent fractional calculus provides a calculus for fractals.

• Random walks on discrete lattices: A grasshopper jumps from one brick to another on a brick wall until it reaches a brick at the edge and then it is promptly squashed. If it starts in the middle of a brick wall that is one hundred bricks high what is the probability that it is squashed at the edge brick that is eleventh from the top? This is an example of a random walk problem on the triangular lattice. In this project you will be solving the random walk problem on other lattices.

• Statistical Mechanics of small particle systems:

The fundamental postulate of equilibrium statistical mechanics is that in an isolated system all parts of the energy surface are equally probable states. But the phase space trajectories for most isolated systems with just a few particles do not visit all regions of their energy surface with equal frequency. They hang around in some regions for a long time and hardly ever visit other regions. In this project you will use perturbation methods to find the regions that are visited most frequently and you will then use this information to determine statistical mechanical properties of small systems. Vaithilingam Jeyakumar Email: v.jeyakumar@unsw.edu.au

• Mathematical Programming under Uncertainty. In many real world problems, the data associated with the underlying mathematical programming problem are often uncertain due to modelling errors. Various techniques, such as stochastic programming and scenario optimization, have been developed to address these optimization problems under uncertainty. In this project, we will examine deterministic approaches to identifying solutions of mathematical programming problems such as linear programming problems and convex optimization problems.

• Robust Optimization and Data Mining: Robust optimization, which is based on a description of uncertainty by sets, instead of probability distributions, is emerging as a powerful methodology to examine un- certain optimization problems. This project will examine various robust optimization approaches to solving uncertain optimization problems. Application areas may include data mining and machine learning. Frances Kuo Email: f.kuo@unsw.edu.au Frances Kuo is a Queen Elizabeth 2 Fellow ARC and her research interests are Quasi-Monte Carlo methods for high-dimensional numerical integration and approximation, information-based complexity, and applications in mathematical finance and statistics. Bill McLean Email: w.mclean@unsw.edu.au

• Hierarchical Matrices: In 1999, Wolfgang Hackbusch introduced the concept of a hierarchical matrix or H-matrix. The idea grew out of a computational algorithm for solving integral equations, known as panel clustering. If A is an N-by-N, dense matrix then the obvious representation of A stores the N2 matrix entries, and computing a matrix-vector product Ax in the obvious way costs N2 operations. However, for a wide class of matrices arising as discretizations of integral and other linear operators, an H-matrix representation uses only O(N) storage and allows the computation of Ax using only O(N) operations, accurate to within the order of the discretization error. It is even possible to develop a whole fast algebra of matrices, with many interesting applications for large-scale numerical simulations. A project in this area would involve a mix of theory and computation, depending on the background of the student.

Russel Morison & Michael Banner Email: R.Morison@unsw.edu.au

• Analysis of Wave Time Series. Time series data is common in both the physical and financial worlds. A Problem occurs where there are fluctuations on many different time scales. However small amplitude high frequency data can be masked by larger amplitude lower frequency data. This project will examine the counting of waves at different wavelengths and frequencies. A knowledge of matlab, or some scientific programming would be an advantage. Russel Morison Email: R.Morison@unsw.edu.au

• Examination of Renewable Energy in the GCC. The global cost of energy is increasing rapidly as communities look for alternatives to fossil fuels. This project will examine the available resources for wind and solar power generation in selected areas (preferably the Eastern province of Saudi Arabia) utilising either observed of modelled data. A knowledge of fortran / scientific programming and/or some meteorology would be an advantage. John Murray Email: J.Murray@unsw.edu.au

• Within host dynamics of emerging drug resistant hepatitis B virus: Approximately 400 million people world-wide are chronically infected with hepatitis B virus (HBV). Over the long-term it can lead to cirrhosis of the liver and hepatocellular carcinoma. Viral replication can be limited through the use of antiviral drugs but these can fail with the development of drug resistance. Combination therapy is now being used to combat this type of failure. This project will include mathematical modelling of the development of drug resistant HBV, and analyse the advantages of different schedules of drug combinations.

• The effects of contact network structure on the spread of diseases: The transmission of disease within a community is not uniform among individuals. In many instances a small number of "high-transmitters" can greatly influence the resulting epidemic. If the interaction between individuals is modelled by a network, then the high-transmitters can be thought of as nodes with many connections. This project will model transmission over a network with differing degrees of connectivity.

• Scheduling of cancer chemotherapy with feedback of information during treatment:

Most cancer chemotherapy modelling assumes information is either available completely during treatment, or is known to within some error before treatment but that no new information is obtained after that point. Neither of these assumptions is accurate for clinical treatment of cancer, where there is a large uncertainty in a patient's response to therapy but that often some information is obtained during treatment that can lead to modification of therapy. This project will use optimization with uncertainty to model clinical practice of cancer chemotherapy scheduling.

John Roberts Email: jag.roberts@unsw.edu.au

• Arithmetic dynamics This topic is broadly taken to be the intersection of algebra, number theory, and dynamical systems. This interdisciplinary area of research is cutting edge and exciting, and has important applications to e.g. cryptography, random matrix theory, materials science and engineering.

• Discrete integrable systems The study of integrable (partial) difference equations and integrable maps is presently a very active field of research. In the first instance, this is due to the increasingly numerous areas of physics in which such systems feature. The study of discrete integrable systems also has intrinsic mathematical appeal, broadly speaking to do with finding analogues of concepts or properties (e.g. the Painleve property, Lax pairs, Hamiltonian structure) that exist in integrable systems with continuous time. Chris Tisdell Email: cct@unsw.edu.au

• Dynamic Equations on Time Scales Dynamic equations on time scales (measure chains) is a new and exciting theory that models phenomena whose behaviour can change smoothly at one level, but irregularly at another. The field was first created by Stefan Hilger in 1990 in an attempt to unify the ideas of differential calculus and difference calculus. In fact, the field of dynamic equations on time scales contains the theory of differential equations and the theory of difference equations both as special cases. Due to the short history of time scales, there are many open questions and significant problems to work on. Moreover, because of the versatility of time scales, it has potential to model a wide range of continuous and discrete phenomena under the one framework. The theory is expected to apply to such "stop-start" behaviour as: insect population models, stock markets, crop harvests and combustion. In addition, numerical techniques involving time scales are becoming increasingly more popular and versatile. This project will lead to advancements in basic theory and applications of dynamic equations on time scales. Students who undertake this project will be well-equipped to contribute to this research area.

• Difference Equations: Theory, Method and Application The field of difference equations provides a natural framework for modeling discrete processes and is also used in the numerical approximation of solutions to ordinary differential equations. Difference equations are particularly useful in economics and ecology, where many situations are (sometimes crudely) described by simple difference equations. A striking point of interest is that even simple difference equations can possess a very rich spectrum of dynamical behaviour. This project will lead to advancements in theory and applications of difference equations. Students who undertake this project will be well-equipped to contribute to this research area.

• Elliptic Partial Differential Equations Partial differential equations are essential mathematical tools for a better understanding a range of rich, complex phenomena. This project will involve a qualitative and quantitative investigation of the boundary value problems associated with elliptic partial differential equations. The basic ideas involve maximum principles, estimates and fixed-point techniques. Thanh Tran Email: Thanh.Tran@unsw.edu.au

• A posteriori error estimates for the finite-element method of lines for nonlinear parabolic equations

Many time-dependent phenomena can be formulated as a boundary value problem with a nonlinear parabolic equation. When approximation techniques, e.g. the finite-element method of lines, are used to find the approximate solution, there is a need 'measure' the error. A posteriori error estimates are designed to judge where the computed solution better approximates the exact solution, and where it fails to do so. These error estimates are a fundamental component in the design of reliable and efficient adaptive algorithms for solving the equations. In this project you will learn some of these techniques and do experiments with some practical problems.

• Interpolation on the sphere by spherical splines Data interpolation and fitting problems where the underlying domain is the sphere arise in many areas including, e.g., geodesy and earth science in which the sphere is taken as a model for the earth. In this project you will learn how to use piecewise polynomials on the sphere (which are called spherical splines) to solve these problems. Experiments with scattered data obtained from a NASA satellite will be carried out.

• Boundary element methods Boundary element methods have long been used in engineering to solve boundary value problems. These problems are formulated from many physical phenomena, ranging from mechanical engineering (e.g. in car design) to petroleum engineering (e.g. for simulation of fractured reservoirs). In this project, you will first learn basic concepts of boundary element methods, how to implement and analyse efficiency and accuracy of the methods. Then, depending on your needs and interests, you will use the methods to solve practical problems in engineering or geodesy. Problems in geodesy will involve programming with data collected by a NASA satellite, which may contain up to almost 30 million points. Rob Womersley Email: R.Womersley@unsw.edu.au Rob Womersleyʼs research interest are optimization methods and applications and computational mathematics and high performance computing Mathematical finance

PURE MATHEMATICS The Pure Mathematics Department is one of the strongest in Australia. Its interests span a wide spectrum of harmonic, and functional analysis with depth also in algebra and discrete mathematics through to aspects of computer science and physics. There are strong links with Applied Mathematics: indeed the Nonlinear Phenomena Group spans both departments. Thomas Britz

• On the Union-Closed Sets Conjecture Description: The Union-Closed Sets Conjecture states that for each (non-trivial) union-closed family of sets, there is an element that is contained in at least half of the sets. In this project, you would survey the presently-known results related to this conjecture, and you would attempt to find new such results. Peter Brown Email: peter@unsw.edu.au

• Gauss Sums

Suppose ζ is an n-th root of unity. Then clearly,

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ζk

k=0

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∑ = 0 if gcd(k,n) = 1. Gauss looked at the sum

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ζk2

k=0

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∑ which leads to the classical gauss sum

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χ(k)ζkk=0

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∑ , where χ is a Dirichlet character. This

thesis will examine the theory, generalisations and applications of (quadratic) Gauss sums. It is assumed that the student will have completed a first course in Number Theory - other- wise, the student will need to undertake some preliminary reading. Daniel Chan Email. danielc@unsw.edu.au Daniel Chanʼs research interests are in algebraic geometry and non- commutative algebra. Specific interests include non-commutative surfaces, maximal orders, non-commutative duality, moduli spaces in non-commutative algebra, derived categories, Brauer groups and Brauer-Severi varietie

Diana Combe Email: d.combe@unsw.edu.au Graph labellings are labellings of graphs (incidene structures). They can be labellings of vertices (or of edges, or of both edges and vertices) over the natural numbers, (or over the integers, or over finite or infinite groups) satisfying particular constraints. Graph labellings can be used to solve particular practical or theoretical problems or to explore the structure and geometry of the graph or the labelling set (or group). Some suggested projects in this area would be to: i) Determine the edge-magic (or vertex-magic, or both edge-magic and vertex-magic) labellings for a family of graphs over (a family of) finite abelian groups. ii) Explore extending a type of graph labelling to a labelling of directed graphs (and, possibly investigate the practical applications.) . Tony Dooley Email: a.dooley@unsw.edu.au Tony Dooleyʼs research interests are harmonic analysis on Lie groups and ergodic theory. Ian Doust Email: i.doust@unsw.edu.au * Functional analysis, in particular Banach space theory. A possible topic is recent generalizations of the Banach Stone Theorem which relates the topological structure of a compact set $K$ to the structure of the function space $C(K)$. * Operator theory. A possible topic is to look at recent work on how the spectra of the truncations of an infinite matrix approximate the spectrum of that matrix. Jie Du Email: j.du@unsw.edu.au Jie Du research interests lie in the representation theories on algebraic and quantum groups, finite groups of Lie type, finite dimensional algebras, and related topics. His recent work has concentrated mainly on the Ringel-Hall approach to quantum groups and q-Schur and generalised q-Schur algebras and their associated monomial and canonical basis theory. He is also interested in combinatorics arising from generalised symmetric groups, Kazhdan-Lusztig cells and representations of finite algebras. Jim Franklin Email: j.franklin@unsw.edu.au In bank operational risk, or estimating rare events such as biosecurity breaches, it is necessary to combine small amounts of data with expert opinion. The project looks at mathematical methods such as extreme value theory and how to use them to assist experts in forming their judgment.

Catherine Greenhill Email: c.greenhill@unsw.edu.au

• The web graph and small world networks In recent years there has been a lot of interest in graphs, which can be used to model various "real world" networks, such as the internet and social networks. Features of real world networks include the "small world" or "six degrees of separation" phenomena, where nodes in large networks are almost always connected by relatively short paths. These features are not seen in the standard random graph models introduced by Erdos, and so new random graph models (called "preferential attachment models") have been constructed to try to model real-world networks. The aim of the project is to survey the topic, to gain some understanding of the new models and their applications.

• The differential equations method for analysing combinatorial algorithms

The differential equations method is a way of analysing a discrete-time combinatorial process. Equations describing the expected change per unit time of certain discrete variables are treated as differential equations, turning a discrete problem into a continuous one. Wormald (1995, 1999) gave conditions under which the discrete system is well-approximated by the solution to the system of differential equations. The method has been applied to algorithms for random graphs and random SAT instances. The aim of the project is to survey this area, learning about the method and how it has been applied. The focus is not on calculus, but on the interplay between discrete and continuous mathematics. Hendrik Grundling Email: h.grundling@unsw.edu.au Hendrik Grundling works on mathematical physics problems in the two areas of quantum field theory and quantization theory. In quantum field theory he is concerned with issues arising from gauge theory and in particular the problem of quantum constraints. The analysis is done in the context of operator algebras, so he is also concerned with problems in that area as well as with the representations and actions of groups which are not locally compact. In quantization theory, he studies the existence and uniqueness of quantizations of Poisson algebras associated with simple manifolds other than the plane. John Steele Email: j.steele@unsw.edu.au John Steele's research interests are in the area of General Relativity, particularly in exact solutions of the Einstein Field Equations, their symmetries and interpretation. He is also interested in geometric aspects of mathematical physics and the history of mathematical physics.

Norman Wildberger Email: n.wildberger@unsw.edu.au Norman Wildberger is interested in Lie group representation theory and the connections with quantization procedures, specifically geometric quantization, star products and Kirillov theory. Another interest is that of commutative hypergroups, of which the fusion rule algebras from conformal field theories provide rich examples, and connections with duality. Lately he has been very interested in combinatorial constructions of Lie algebra representations, trying to obtain explicit models of for example sl(3) modules (the symmetry algebra of the strong force) which are easier to implement than Gelfand Tsetlin. He is especially interested in extending these and related ideas coming from games on graphs to obtain modules for general Kac Moody algebras. Norman Wildberger has worked on the orbit method and moment map on compact and nilpotent Lie groups. He has recently been developing the theory of hypergroups, which impacts significantly in this area and also has applications to diophantine equations. Fedor Sukochev Email: f.sukochev@unsw.edu.au Fedor Sukochev has wide interests in non-commutative functional analysis and its applications to non-commutative geometry, in non-commutative integration theory and in Banach space geometry and its applications.

STATISTICS The Department of Statistics at UNSW is an innovative and research intensive department, with ambitions of becoming the leading department for statistical research in Australia. We collaborate locally and internationally with researchers and industry groups in areas as diverse as medicine, finance, ecology and engineering. The department has particular research strengths in the following areas: Stochastic Processes and Financial Mathematics, Bayesian Statistics and Monte Carlo Methods, Biostatistics, Smoothing methods and Non-parametric Statistics Bruce Brown Email: Bruce.Brown@unsw.edu.au Bruce Brownʼs research interests are nonparametric statistics, random matrices in statistics, categorical data analysis and statistics of financial diffusions. Feng Chen Email: Feng.Chen@unsw.edu.au Feng Chen is interested in the inference and applications of stochastic processes, especially the point processes.

• NONPARAMETRIC ESTIMATION OF THE INTENSITY OF COUNTING PROCESSES

Counting process is a basic type of point processes, which has applications in a wide variety of applied fields. The evolution of a counting process over time is determined by its intensity process. Therefore estimating intensity process has been an important topic. When there is not enough prior information for us to assume a parametric form for the intensity, we will have to rely on various nonparametric procedures to estimate the intensity, such as kernel smoothing, sieve MLE, penalised spline, local polynomial, wavelet shrinkage methods, etc. While the asymptotic behaviours of these estimation procedures are largely known, their relative merits in the small sample context still need further investigation. In this project, we will compare the nonparametric intensity estimation procedures from a computational perspective.

• ULTRA HIGH FREQUENCY FINANCIAL DATA MODELLING Ultra high frequency (UHF) financial data is the limit of the high frequency financial data where each trade is recorded. Typically there are hundreds to thousands of trades within a single trading day. One of the stylized facts of UHF data is that the trades happen in irregularly spaced time points. Such type of data is naturally modelled by a marked point process (MPP), with the trade arrival process modelled by a simple counting process and the other associated trade information (type of trade, price, volume, etc) by the marks associated with each event point. In this project we will consider tractable models for the intensity process and the mark distribution of the MPP and their relationship.

William Dunsmuir Email: W.Dunsmuir@unsw.edu.au William Dunsmuir will supervise projects on time series analysis and modelling. His current interests are mainly concerned with discrete valued time series, which arise in financial applications (high frequency intraday trading data) and in public health (assessing the impact of policy interventions on violent deaths and injury and diseases counts). He has also over recent years supervised projects on modelling stochastic volatility for financial time series." Yanan Fan Email: Y.Fan@unsw.edu.au Yanan Fan is a Bayesian statistician whose is primarily interested in the development of Monte Carlo simulation techniques, which is vital for Bayesian inference Bayesian inference is about the incorporation of prior information into a likelihood-based inference. In such situations, inference on the posterior distribution (prior and likelihood) can be numerically challenging, as they often involve high dimensional integrals. One of her primary interest is in the development of Monte Carlo (MCMC) methods for numerical integration. Among these methods, I am particularly interested in Markov chain Monte Carlo, sequential importance sampling, sequential Monte Carlo, the slice sampler and nested sampling. A potential project may be to study the use of MCMC in the particular case of approximate Bayesian computation, where efficiency of the method has been shown to be poor, and compare its performance with other alternatives, such as sequential importance sampling or nested sampling. Another may be to study nested sampling, and understand its connections to other sampling methods, such as rejection sampling. Another area of research of Yanan Fan is the capture-recapture modelling for the estimation of population size. The method is often applied in Ecology, where we may be interested in estimating population size, or survival probabilities of wild animals. These methods have also been applied to heroin users, whose true population is unknown, but users are sighted every now and then. Sally Galbraith Email: sally.galbraith@unsw.edu.au Sally Galbraithʼs research interests include biostatistical methods for the design and analysis of clinical and other biomedical studies, in particular methods for missing data, longitudinal data, surrogate outcomes, survival analysis, and gene expression studies. These interest involves researchers in the Faculty of Medicine and other medical research institutes at UNSW. Gery Geenens Email: GGeenens@unsw.edu.au Functional Data Analysis (FDA) is clearly an ideal area to carry on advanced research in the future. This domain has lately been attracting a good amount of interest, since recent technological advances in measuring devices allow us to observe spatio-temporal phenomena on arbitrarily fine grids, while the limits of these devices previously imposed to discretize any such phenomenon. By considering random functions in their whole allows us to analyse their underlying structure (through their derivatives, for example) rather than to view them just as infinite dimensional vectors, which opens up new prospects in data analysis. However, by accepting that any graphical representation (scatter plots, histograms, etc) is inconceivable when infinite dimensional data are involved, and that in the classical vector case, this kind of plot is often the primary tool to define a suitable parametric pattern

for the data, it appears that the risk of model misspecification is still much more important in the functional case than in the vector case. The flexibility ensured by nonparametric methods in that framework date back to only a couple of years, much is left to do in the nonparametric functional data analysis theory, which represents a thrilling challenge. In addition, the possible applications of this kind of theory are obviously numerous and various, as any domain where spatio-temporal phenomena, that is random functions, are observed is potentially concerned: quantum mechanics, communications, musicology, climatology, hydrology, risk analysis, mathematical finance or econometrics, among others. These ideas would also be useful in the much more general setting of time series analysis, as forecasting problem can be efficiently tackled by a functional regression model. Among other projects, a powerful system of signature digital recognition is under consideration. Actually, the current digital signature recognition systems are almost exclusively based on the drawing only: the ʻsignature curveʼ is finely discretized and analyzed through some key positions and parameters, that is features discriminating at most genuine signatures from fakes. Yet, it seems easy for features discriminating at most genuine signatures from fakes. Yet, it seems easy for a dexterous person to reproduce or trace somebody elseʼs signature when very few error, which makes those procedures hardly reliable. However, we claim that a signature can be modeled by a random function S from R+ (time) to R2 (position of the pen at the time). Then, many underlying features, inherent to the functional nature of the so defined random element, can be analysed. For example, the tangential derivatives of S provide the temporal evolution of the speed and the acceleration of the pen. Precisely, it is not difficult to understand that the acceleration of the pen is directly dictated by the movement of the wrist of the person who is signing, and that the ʻgenuineʼ wrist movement would be very hard, not to say impossible to reproduce. This movement and the acceleration that is induced are consequently much more peculiar to each and every one than the drawing itself, and are therefore very efficient discriminating characteristics. Of course, analyzing this acceleration is possible only via an approach able to consider the whole curve itself as argument, since any intrinsically functional feature (derivative in our case) is lost during the discretization step. The above mentioned ideas of nonparametrics functional data analysis therefore appear essential to set up this problem. Ben Goldys Email: B.Goldys@unsw.edu.au Stochastic, ordinary and partial, differential equations are indispensable tools for the modeling and analysis of complex systems evolving randomly in space and time. They are applied in the Climate Science, Mathematical Finance, Population Biology, Statistical Physics and Quantum Field Theory. Stochastic differential equations have also become a tool to tackle certain difficult problems in Pure Mathematics, in particular in Differential Geometry, Functional Analysis, Analysis on loop groups and the theory of deterministic differential equations. Ben Goldys has provided four projects. Please consult the following website for these projects: http://www.maths.unsw.edu.au/~yanan/honours/ben.pdf Jake Olivier Email: j.olivier@unsw.edu.au Jake Olivierʼs research interests are biostatistics and binomial confidence intervals.

Spiro Penev Email: S.Penev@unsw.edu.au

• Wavelet methods in non-parametric inference. These include applications in density estimation, non-parametric regression and in signal analysis. Wavelets are orthonormal bases used for series expansions of curves that exhibit local irregularities. When estimating spatially inhomogeneous curves wavelets outperform traditional nonparametric methods. Typical wavelet estimators are of non-linear threshold-type. My research is focused on improving their flexibility by making a finer balance between stochastic and approximation terms in the mean integrated squared error. I also make data-based recommendations for choosing tuning parameters in the estimation. I am also implementing modifications as to make the estimators satisfy additional shape constraints like non-negativity, integral equal to one, convexity etc. My recent interest in wavelet methods is in applying them for shape-constrained estimation, for analyzing non-regularly sampled images and surfaces, and in applying Bayesian inference techniques in wavelet methodology.

• Saddlepoint approximations Saddlepoint approximation for densities and tail-area probabilities of certain statistics turn out to be surprisingly accurate down to small sample sizes. Although being asymptotic in spirit, they sometimes give accurate approximations even down to a sample size of one. They are also accurate in the tails where other methods fail. I have derived the saddlepoint approximation of the joint density of slope and intercept in the Linear Structural Relationship model. Similar in spirit to Saddlepoint is the Wiener germ approximation which I have applied to approximating the non-central chi-square distribution and its quantiles. It performs better than any other approximations of the non-central chi-square in the literature. (Higher order) Edgeworth expansions deliver better approximations to the limiting distribution of a statistic in comparison to the approximations delivered by the Central Limit Theorem. They may be non-trivial to derive for complicated estimators such as the kernel estimator of the p-th quantile. We study the Edgeworth expansion of the latter and demonstrate the improvement in comparison to the normal approximation.

• Latent Variable Models (LVM) Latent Variable Models (LVM) is an area of significant interest to me. I have worked on aspects of model choice and model equivalence in LVM. These are practically relevant issues- model equivalence appears more often in LVM than in other statistical models and represents a threat to the validity of inferences. I am also interested in inference for latent correlations and for scale reliability. I also have studied the relationship between the concepts of maximal reliability and maximal validity and have established useful inequalities between them. Within the setting of the LVM, I am interested in a subset of models called GLLAMM (generalized linear latent and mixed models). They are general enough yet with a sufficient structure to allow flexible modeling of responses of mixed type (continuous and categorical).

• Investigation of dependencies and associations Studying dependence of random variables when their joint distribution is not multivariate normal is important. A single correlation coefficient is not enough to describe the dependence in such cases and copula functions are a useful tool. A particularly simple copula is the Archimedean copula. We use spline methods in estimating Archimedean copulas.

Gareth Peters Email: GarethPeters@unsw.edu.au Students are invited to become a part of the newly formed Quantitative Risk Solutions Laboratory) QRSLab in the Department of Mathematics and Statistics, UNSW. The agenda of this laboratory is to produce innovative and cutting edge statistical solutions to modelling in applied financial risk; hedge fund models and risk; time series model; traditional - market, credit and operational risk; and insurance statistical modelling. This laboratory is specifically interested in modelling applied numerical and methodological financial risk problems of significance in the financial industry - as such it aims to develop strategic partnership of a research nature with industry. Partnerships are currently being developed with ETH RiskLab, a large Sydney based hedge fund as well as a large global financial services company with major offices in Sydney. If you are interested in the QRSLab - contact me or visit http://web.maths.unsw.edu.au/~peterga/

Some of the QRSLab areas of interest:

1. Model risk and insurance in Basel II 2. Computations with copula models in finance and model risk (eg. Operational Risk,

Insurance settings). 3. Computations with meta distributions for risk modelling and associated model risk

analysis 4. Risk sensitivity and stress testing frameworks under Basel II and Solvency II –

properties, algorithms, methodology, impact on capital calculation and allocation. 5. Dynamic latent factor models and copula for Operational Risk 6. Cointegration models, estimation, prediction and model risk. 7. Risk metrics and analysis for the hedge fund and private equity sectors. 8. Alpha stable heavy tailed models for financial risk 9. Claims reserving models for insurance - models and methodology 10. Project risk and pricing for mining / agriculture – via multifactor sde commodity

models – online calibration and estimation via Sequential Monte Carlo and Adaptive MCMC.

11. Tracking VaR and other risk measures dynamically through non-linear filtering techniques.

Rare-Event modelling and simulation methodology Donna Mary Salopek Email: dm.salopek@unsw.edu.au Donna Mary Salopekʼs research interests are financial modeling in general. In particular she is interested in non-semimartingales (eg. fractional Brownian motion) modeling, Levy modeling, and pricing of a variety of financial products. She is also interested in general problems in stochastic processes and stochastic analysis. For instance, she is currently working on stochastic evolution equations that are driven by H-cylindrical fractional Brownian motion and its possible application to financial mathematics.

Scott Sisson Email: Scott.Sisson@unsw.edu.au Scott Sisson is a Queen Elizabeth II research fellow at UNSW, with interests in Bayesian inference and computational techniques, in modeling the extremes of climate processes, and in genetic epidemiology. Projects are available in all of the below areas, and will be developed based on the interests and the strength of the student. Good computational and statistical skills are essential pre-requisites. For further information on these areas, including review articles and preprints, please visit: www.maths.unsw.edu.au/~scott Bayesian inference and computational techniques

Bayesian methods formulate both parameter and model uncertainty in distributional form, and thereby naturally incorporate this uncertainty in the statistical analysis. In practice, inference is performed by drawing samples from this posterior distribution. This can become extremely challenging, as the posterior distribution may exhibit numerous complex features. Stochastic simulation algorithms commonly used for this task include rejection sampling, importance sampling, Markov chain Monte Carlo (MCMC), adaptive MCMC, trans-dimensional MCMC and sequential Monte Carlo (SMC) techniques.

Approximate Bayesian computation Within Bayesian inference, the ability to numerically evaluate the likelihood function is a critical requirement in order to implement standard Bayesian procedures and algorithms. However, suppose that you wish to consider a set of models, which are sufficiently realistic, that the likelihood function is computationally intractable (that is, it may not be evaluated numerically). This obviously presents a number of modelling and practical computational challenges. Approximate Bayesian computation methods focus on the development and application of Bayesian inferential procedures in this setting. Environmental and climate extremes Statistical inference is typically interested in the mean levels of a process. But what happens if you're really interested in extreme levels? This is important in the study of e.g. extreme rainfall, temperature, wind speeds and droughts. The central limit theorem states that the limit distribution of a standardised sample mean is a Gaussian distribution. In analogy, the limit distribution of a standardised sample maxima is a Generalised extreme value distribution. This distribution may be used to model, for example, the maximum daily temperature in one year, and then provide predictions on, say, 1 in 100-year temperature events. Other asymptotic extreme value theory results provide a suite of potential frameworks for modelling extreme process levels. These include exceedances-above-threshold modelling and max-stable processes. Projects in this area may be in collaboration with UNSW's Climate Change Research Centre.

Genetic epidemiology Statistical techniques are essential tools in the analysis of molecular data from infectious diseases such as tuberculosis and HIV. Estimation of pathogen mutation and transmission rates, and possible changes with the onset of multiple drug-resistance are of critical importance for policy and intervention. Projects in this area will be in collaboration with Dr. Mark Tanaka (BABS, UNSW). David Warton Email: David.Warton@unsw.edu.au David Warton leads the Eco-Stats Research Group, a team of statistics researchers and students working to improve the methods for making use of data to answer research questions commonly asked in ecology. "Eco-Stats" is short for "Ecological Statistics". Our research ranges from theory and methods research (particularly regarding the analysis of high dimensional data) to applied research (introducing modern methods of analysis to ecology). Current research focuses on two distinct topics: (1) Developing model-based approaches to studying ecological communities and their responses to environmental impacts such as climate change. (2) Novel methods for modelling the distribution of species as a function of environmental variables. In both cases there is considerable demand in ecology for modern statistical methods, and considerable potential in terms of developing and adapting new methods for ecology. Examples of current Eco-Stats research: - Developing new methods for the analysis of high-dimensional count data (with applications in ecology!). - The use of the LASSO as a model selection tool in species distribution modellling, and its relation to maximum entropy methods. - Incorporating information on species function into models for community structure and its potential response to climate change. For more information, see http://web.maths.unsw.edu.au/~dwarton/EcoStats.pdf And for some project ideas: http://web.maths.unsw.edu.au/~dwarton/DWarton_proj.pdf