Coursework

# Master of Science (Mathematics and Statistics)

- CRICOS Code: 094599G

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## What will I study?

### Overview

#### Course structure

The Master of Science (Mathematics and Statistics) is a 200-point course, made up of:

- Discipline subjects (137.5 points), including compulsory subjects and electives
- A professional skills subject (12.5 points)
- A research project (50 points, or 25 points with approval and other subjects to compensate).

You’ll choose from one of five areas to **specialise** in:

- Applied Mathematics and Mathematical Biology
- Operations Research and Industrial Optimisation
- Pure Mathematics
- Statistics and Stochastic Processes
- Mathematical Physics and Physical Combinatorics

You'll select your subjects from the large range of advanced mathematics and statistics subjects on offer. The course will be made up of subjects from your chosen specialty and others. You can even choose masters level subjects in physics, computer science or bioinformatics. You can also take subjects through the Australian Mathematical Sciences Institute (AMSI) National Graduate Summer School.

You'll choose your **professional skills** subject from a selection that includes modelling, science communication, and scientific computing, giving you an extra skill that you can take with you through your career.

All students undertake a **research project**, over 12–18 months, working on a mathematics and statistics research question. To support you and provide direction, you’ll be matched with one of our expert researchers as a supervisor. During the first semester of study you’ll select your research topic and supervisor, with the research project usually beginning in the second semester.

### Sample course plan

View some sample course plans to help you select subjects that will meet the requirements for this degree.

### Pure Mathematics specialisation

Year 1

100 pts

**Semester 1**50 pts

**Semester 2**50 pts

Year 2

100 pts

**Semester 1**50 pts

**Semester 2**50 pts

### Explore this course

Explore the subjects you could choose as part of this degree.

#### Core

Complete both of the following subjects:

**Advanced Methods: Transforms**12.5 ptsThis subject develops the mathematical methods of applied mathematics and mathematical physics with an emphasis on integral transform and related techniques. An introduction is given to the calculus of variations and the Euler-Lagrange equation. Advanced complex contour integration techniques are used to evaluate and invert Fourier and Laplace transforms. The general theory includes convolutions, Green’s functions and generalized functions. The methods of Laplace, stationary phase, steepest descents and Watson’s lemma are used to asymptotically approximate integrals. Throughout, the theory is set in the context of examples from applied mathematics and mathematical physics such as the brachistochrone problem, Fraunhofer diffraction, Dirac delta function, heat equation and diffusion.

**Advanced Methods: Differential Equations**12.5 ptsThis subject develops the mathematical methods of applied mathematics and mathematical physics with an emphasis on ordinary differential equations. Both analytical and approximate techniques are used to determine solutions of ordinary differential equations. Exact solutions by localised series expansion techniques of second-order linear ordinary differential equations and Sturm-Liouville boundary value problems are explored. Special functions are introduced here. Regular and singular perturbation expansion techniques, asymptotic series solutions, dominant balance, and WKB theory are used to determine approximate solutions of linear and nonlinear differential equations. Throughout, the theory is set in the context of examples from applied mathematics and mathematical physics such as nonlinear oscillators, boundary layers and dispersive phenomena.

#### Elective

Complete three of the following subjects:

**Random Matrix Theory**12.5 ptsRandom matrix theory is a diverse topic in mathematics. It draws together ideas from linear algebra, multivariate calculus, analysis, probability theory and mathematical physics, amongst other topics. It also enjoys a wide number of applications, ranging from wireless communication in engineering, to quantum chaos in physics, to the Reimann zeta function zeros in pure mathematics. A self contained development of random matrix theory will be undertaken in this course from a mathematical physics viewpoint. Topics to be covered include Jacobians for matrix transformation, matrix ensembles and their eigenvalue probability density functions, equilibrium measures, global and local statistical quantities, determinantal point processes, products of random matrices and Dyson Brownian motion.

**Bayesian Statistical Learning**12.5 ptsBayesian inference treats all unknowns as random variables, and the core task is to update the probability distribution for each unknown as new data is observed. After introducing Bayes’ Theorem to transform prior probabilities into posterior probabilities, the first part of this subject introduces theory and methodological aspects underlying Bayesian statistical learning including credible regions, prior choice, comparisons of means and proportions, multi-model inference and model selection. The second part of the subject will cover practical implementations of Bayesian methods through Markov Chain Monte Carlo computing and real data applications, focusing on (generalised) linear models and concluding by exploring machine learning techniques such as Gaussian processes.

**Advanced Biological Modelling: Dynamics**12.5 ptsThis subject builds on your knowledge of how biological modelling provides insight into complex biological phenomena. With a focus on mechanistic modelling and viewing biological systems as dynamic in nature, you will learn how to develop and implement “real-world” models, applicable to current open problems in computational biology. Advanced approaches to model-based analysis of data will be introduced, including Bayesian hierarchical modelling. Software languages and packages for modelling and statistical analysis (e.g. SBML and STAN) will be introduced. Motivating problems will be drawn from across the spectrum of biology from genetics to ecology.

**Computational Differential Equations**12.5 ptsMany processes in the natural sciences, engineering and finance are described mathematically using ordinary or partial differential equations. Only the simplest or those with special structure can be solved exactly. This subject discusses common techniques for computing numerical solutions to differential equations and introduces the major themes of accuracy, stability and efficiency. Understanding these basic properties of scientific computing algorithms should prevent the unwary from using software packages inappropriately or uncritically, and provide a foundation for devising methods for nonstandard problems. We cover both time-independent problems, in one and higher space dimensions, and evolution equations of hyperbolic or parabolic type.

**Mathematical Biology**12.5 ptsModern techniques have revolutionised biology and medicine, but interpretative and predictive tools are needed. Mathematical modelling is such a tool, providing explanations for counter-intuitive results and predictions leading to new experimental directions. The broad flavour of the area and the modelling process will be discussed. Applications will be drawn from many areas including population growth, epidemic modelling, biological invasion, pattern formation, tumour modelling, developmental biology and tissue engineering. A large range of mathematical techniques will be discussed, for example discrete time models, ordinary differential equations, partial differential equations, stochastic models and cellular automata.

**Mathematical Statistical Mechanics**12.5 ptsThe goal of statistical mechanics is to describe the behaviour of bulk matter starting from a physical description of the interactions between its microscopic constituents. This subject introduces the Gibbs probability distributions of classical statistical mechanics, the relations to thermodynamics and the modern theory of phase transitions and critical phenomena. The central concepts of critical exponents, universality and scaling are emphasized throughout. Applications include the ideal gases, magnets, fluids, one-dimensional Ising and Potts lattice spin models, random walks and percolation as well as approximate methods of solution.

**Continuum Mechanics**12.5 ptsThis subject develops mathematical methods for the study of the mechanics of fluids and solids and illustrates their use in several contexts. Topics covered include Newtonian fluids at low and at high Reynolds number and the linear theory of elasticity. Applications may be drawn from biological, earth sciences, engineering or physical contexts.

**Infectious Disease Dynamics**12.5 ptsThis subject introduces the fundamental mathematical models used to study infectious diseases at both the epidemiological and within-host scale. The emphasis is on: 1) how models are developed, from conceptualisation through to implementation in software; and 2) how to apply models to questions of epidemiological, public health and biological importance. Statistical techniques for the model-based analysis of relevant data resources will be introduced.

- Epidemiology: epidemic/endemic behaviour and intervention strategies to reduce transmission, the SIR model, including demography, threshold behaviour, phase-plane analysis;
- Viral dynamics: host-pathogen interactions, the mediating influences of immunomodulatory agents and antimicrobials, the TIV model, including the immune response, pharmacokinetic-pharmacodynamic models;
- Model sensitivity and uncertainty analysis, scenario analysis, parameter estimation, model comparison

#### Core

Complete two of the following subjects:

**Mathematical Statistical Mechanics**12.5 ptsThe goal of statistical mechanics is to describe the behaviour of bulk matter starting from a physical description of the interactions between its microscopic constituents. This subject introduces the Gibbs probability distributions of classical statistical mechanics, the relations to thermodynamics and the modern theory of phase transitions and critical phenomena. The central concepts of critical exponents, universality and scaling are emphasized throughout. Applications include the ideal gases, magnets, fluids, one-dimensional Ising and Potts lattice spin models, random walks and percolation as well as approximate methods of solution.

**Advanced Methods: Transforms**12.5 ptsThis subject develops the mathematical methods of applied mathematics and mathematical physics with an emphasis on integral transform and related techniques. An introduction is given to the calculus of variations and the Euler-Lagrange equation. Advanced complex contour integration techniques are used to evaluate and invert Fourier and Laplace transforms. The general theory includes convolutions, Green’s functions and generalized functions. The methods of Laplace, stationary phase, steepest descents and Watson’s lemma are used to asymptotically approximate integrals. Throughout, the theory is set in the context of examples from applied mathematics and mathematical physics such as the brachistochrone problem, Fraunhofer diffraction, Dirac delta function, heat equation and diffusion.

**Advanced Discrete Mathematics**12.5 ptsThe subject consists of three main topics. The bijective principle with applications to maps, permutations, lattice paths, trees and partitions. Algebraic combinatorics with applications rings, symmetric functions and tableaux. Ordered sets with applications to generating functions and the structure of combinatorial objects.

#### Elective

Complete three of the following subjects:

**Random Matrix Theory**12.5 ptsRandom matrix theory is a diverse topic in mathematics. It draws together ideas from linear algebra, multivariate calculus, analysis, probability theory and mathematical physics, amongst other topics. It also enjoys a wide number of applications, ranging from wireless communication in engineering, to quantum chaos in physics, to the Reimann zeta function zeros in pure mathematics. A self contained development of random matrix theory will be undertaken in this course from a mathematical physics viewpoint. Topics to be covered include Jacobians for matrix transformation, matrix ensembles and their eigenvalue probability density functions, equilibrium measures, global and local statistical quantities, determinantal point processes, products of random matrices and Dyson Brownian motion.

**Introduction to String Theory**12.5 ptsThe first half of this subject is an introduction to two-dimensional conformal field theory with emphasis on the operator formalism and explicit calculations. The second half is an introduction to string theory based on the first half. For concreteness, the representation theory of Virasoro algebra and bosonic strings will be emphasized.

**Lie Algebras**12.5 ptsThe theory of Lie algebras is fundamental to the study of groups of continuous symmetries acting on vector spaces, with applications to diverse areas including geometry, number theory and the theory of differential equations. Moreover, since quantum mechanical systems are described by Hilbert spaces acted on by continuous symmetries, Lie algebras and their representations are also fundamental to modern mathematical physics. This subject develops the basic theory in a way accessible to both pure mathematics and mathematical physics students, with an emphasis on examples. The main theorems are: the classification of complex semi-simple Lie algebras in terms of Cartan matrices and Dynkin diagrams, and the classification of finite-dimensional representations of these algebras in terms of highest weight theory.

**Exactly Solvable Models**12.5 ptsIn mathematical physics, a wealth of information comes from the exact, non-perturbative, solution of quantum models in one-dimension and classical models in two-dimensions. This subject is an introduction to this beautiful and deep subject. Yang-Baxter equations, Bethe ansatz and matrix product techniques are developed in the context of the critical two-dimensional Ising model, dimers, free fermions, the 6-vertex model, percolation, quantum spin chains and the stochastic asymmetric simple exclusion model. The algebraic setting incorporates the quantum groups, and the Temperley-Lieb and braid-monoid algebras.

**Enumerative Combinatorics**12.5 ptsThe subject is about the use of generating functions for enumeration of combinatorial structures, including partitions of numbers, partitions of sets, permutations with restricted cycle structure, connected graphs, and other types of graph. The subject covers the solution of recurrence relations, methods of asymptotic enumeration, and some applications in statistical mechanics. The methods covered have widespread applicability, including in areas of pure and applied mathematics and computer science.

**Advanced Methods: Differential Equations**12.5 ptsThis subject develops the mathematical methods of applied mathematics and mathematical physics with an emphasis on ordinary differential equations. Both analytical and approximate techniques are used to determine solutions of ordinary differential equations. Exact solutions by localised series expansion techniques of second-order linear ordinary differential equations and Sturm-Liouville boundary value problems are explored. Special functions are introduced here. Regular and singular perturbation expansion techniques, asymptotic series solutions, dominant balance, and WKB theory are used to determine approximate solutions of linear and nonlinear differential equations. Throughout, the theory is set in the context of examples from applied mathematics and mathematical physics such as nonlinear oscillators, boundary layers and dispersive phenomena.

#### Core

Complete both of the following subjects:

**Optimisation for Industry**12.5 ptsThe use of mathematical optimisation is widespread in business, where it is a key analytical tool for managing and planning business operations. It is also required in many industrial processes and is useful to government and community organizations. This subject will expose students to operations research techniques as used in industry. A heavy emphasis will be placed on the modelling process that turns an industrial problem into a mathematical formulation. The focus will then be on how to solve the resulting mathematical problem with mixed-integer programming techniques.

**Approximation, Algorithms and Heuristics**12.5 ptsMany discrete optimisation problems encountered in practice are too difficult to solve exactly in a reasonable time frame. Approximation algorithms and heuristics are the most widely used approaches for obtaining reasonably accurate solutions to such hard problems. This subject introduces the basic concepts and techniques underlying these “inexact” approaches. We will address the following fundamental questions in the subject: How difficult is the problem under consideration? How closely can an optimal solution be approximated? And how can we go about finding near-optimal solutions in an efficient way? We will discuss methodologies for analysing the complexity and approximability of some important optimisation problems, including the travelling salesman problem, knapsack problem, bin packing, scheduling, network design, covering problems and facility location. We will also learn about various metaheuristics (simulated annealing, Tabu search, GRASP, genetic algorithms) and matheuristics (relax-and-fix, fix-and-optimise, local branching) that are widely used in solving real-world optimisation problems.

#### Elective

Complete three of the following subjects:

**Network Optimisation**12.5 ptsMany practical problems in management, operations research, telecommunication and computer networking can be modelled as optimisation problems on networks. Here the underlying structure is a graph. This subject is an introduction to optimisation problems on networks with a focus on theoretical results and efficient algorithms. It covers classical problems that can be solved in polynomial time, such as shortest paths, maximum matchings, maximum flows, and minimum cost flows. Other topics include complexity and NP-completeness, matroids and greedy algorithms, approximation algorithms, multicommodity flows, and network design. This course is beneficial for all students of discrete mathematics, operations research, and computer science.

**Scheduling and Optimisation**12.5 ptsScheduling is critical to manufacturing, mining, and logistics, and is of increasing importance in healthcare and service industries. Most automated systems, ranging from elevators to industrial robots, embed some kind of scheduling algorithms. Building on the Optimisation background provided in Optimisation for Industry, this subject teaches students how to solve more advanced problems. A particular focus will be scheduling problems, but other more general assignment problems will be discussed.

**Mathematical Game Theory**12.5 ptsGame theory is a branch of mathematics where the interactions between rational decision makers (players) are modelled and analysed. Game theory can broadly be divided into the study of noncooperative games and cooperative games. For noncooperative games we study two-player games, games in extensive form, games of perfect and imperfect information, games with complete and incomplete information, games with chance moves, repeated games, and Bayesian games. To analyse these games we introduce the concepts of Nash equilibria, evolutionary stable strategies, subgame perfect equilibria, and belief spaces. For cooperative games we study coalitional games with transferable utility, and introduce the concepts of coalitions, characteristic functions, the core, the Shapley value, the nucleolus, and dual games. We prove the well known Bonderava-Shapley theorem which gives conditions for the nonemptyness of the core. This subject provides a rigorous mathematical treatment of game theory, and will include applications selected from queueing theory, biology, population dynamics, resource allocation, auction theory, political science, and military applications.

**Advanced Nonlinear Optimisation**12.5 ptsMany optimisation problems in the real world are inherently nonlinear. A variety of industries, including telecommunications networks, underground mining, microchip design, computer vision, facility location and supply chain management, depend on the efficient solution of nonlinear programs. This subject introduces the foundational mathematical concepts behind nonlinear optimisation. Some of the concepts covered include convex analysis, optimality conditions, conic programming, and duality. Various methods to solve nonlinear programs are covered, including iterative methods such as conjugate gradient methods, barrier methods and subgradient methods. This subject also explores the application of geometric methods such as perturbation and variational approaches to problems in facility location and network design.

#### Core

Complete both of the following subjects:

**Measure Theory**12.5 ptsMeasure Theory introduces the modern conceptual framework of analysis that has led to a transformation and generalisation of such basic objects as functions, and such notions as continuity, differentiability and integrability.

It is fundamental to many areas of mathematics and probability and has applications in other fields such as physics and economics. Students will be introduced to the core topics of Lebesgue's theory of integration, and abstract measure theory, in particular signed measures, the Hahn-Jordan decomposition, the Radon-Nikodym derivative. Additional topics may include rudiments of probability theory (conditional expectation, Borel sets and measures) and geometric analysis (rectifiable curves, Hausdorff measure and dimension).

**Algebraic Topology**12.5 ptsThis subject studies topological spaces and continuous maps between them. It demonstrates the power of topological methods in dealing with problems involving shape and position of objects and continuous mappings, and shows how topology can be applied to many areas, including geometry, analysis, group theory and physics. The aim is to reduce questions in topology to problems in algebra by introducing algebraic invariants associated to spaces and continuous maps. Important classes of spaces studied are manifolds (locally Euclidean spaces) and CW complexes (built by gluing together cells of various dimensions). Topics include: homotopy of maps and homotopy equivalence of spaces, homotopy groups of spaces, the fundamental group, covering spaces; homology theory, including singular homology theory, the axiomatic approach of Eilenberg and Steenrod, and cellular homology.

#### Elective

Complete three of the following subjects:

**Algebraic Geometry**12.5 ptsThis course is an introduction to algebraic geometry. Algebraic geometry is the study of zero sets of polynomials. It exploits the interplay between rings of functions and the underlying geometric objects on which they are defined. It is a fundamental tool in may areas of mathematics, including number theory, physics and differential geometry. The syllabus includes affine and projective varieties, coordinate ring of functions, the Nullstellensatz, Zariski topology, regular morphisms, dimension, smoothness and singularities, sheaves, schemes.

**Commutative and Multilinear Algebra**12.5 ptsThe subject covers aspects of multilinear and commutative algebra as well as two substantial applications. Within multilinear algebra this includes bilinear forms and `multilinear products’ of vector spaces, such as tensor products. Commutative algebra concerns itself with properties of commutative rings, such as polynomial rings and their quotients and to modules over such rings. Both topics have wide application, both to other parts of mathematics and to physics. Much of this theory was developed for applications in geometry and in number theory, and the theorems can be used to cast substantial light on problems from geometry and number theory.

**Groups, Categories & Homological Algebra**12.5 ptsAs well as being beautiful in its own right, algebra is used in many areas of mathematics, computer science and physics. This subject provides a grounding in several fundamental areas of modern advanced algebra including Lie groups, combinatorial group theory, category theory and homological algebra.

The material complements that covered in the subject Commutative and Mutlilinear Algebra without assuming it as prerequisite.

**Functional Analysis**12.5 ptsFunctional analysis is a fundamental area of pure mathematics, with countless applications to the theory of differential equations, engineering, and physics.

The students will be exposed to the theory of Banach spaces, the concept of dual spaces, the weak-star topology, the Hahn-Banach theorem, the axiom of choice and Zorn's lemma, Krein-Milman, operators on Hilbert space, the Peter-Weyl theorem for compact topological groups, the spectral theorem for infinite dimensional normal operators, and connections with harmonic analysis.

**Representation Theory**12.5 ptsSymmetries arise in mathematics as groups and Representation Theory is the study of groups via their actions on vector spaces. It has important applications in many fields: physics, chemistry, economics, biology and others. This subject will provide the basic tools for studying actions on vector spaces. The course will focus on teaching the basics of representation theory via favourite examples: symmetric groups, diagram algebras, matrix groups, reflection groups. In each case the irreducible characters and irreducible modules for the group (or algebra) will be analysed, developing more and more powerful tools as the course proceeds. Examples that will form the core of the material for the course include SL2, cyclic and dihedral groups, diagram algebras: Temperley-Lieb, symmetric group and Hecke algebras, Brauer and BMW algebras, compact Lie groups. Among the tools and motivation that will play a role in the study are characters and character formulas, induction, restriction and tensor products, and connections to statistical mechanics, mathematical physics and geometry.

If time permits, there may be some treatment of loop groups, affine Lie algebras and Dynkin diagrams.

**Differential Topology and Geometry**12.5 ptsThis subject extends the methods of calculus and linear algebra to study the geometry and topology of higher dimensional spaces. The ideas introduced are of great importance throughout mathematics, physics and engineering. This subject will cover basic material on the differential topology of manifolds including integration on manifolds, and give an introduction to Riemannian geometry. Topics include: Differential Topology: smooth manifolds, tangent spaces, inverse and implicit function theorems, differential forms, bundles, transversality, integration on manifolds, de Rham cohomology; Riemanian Geometry: connections, geodesics, and curvature of Riemannian metrics; examples coming from Lie groups, hyperbolic geometry, and other homogeneous spaces.

**Riemann Surfaces and Complex Analysis**12.5 ptsRiemann surfaces arise from complex analysis. They are central in mathematics, appearing in seemingly diverse areas such as differential and algebraic geometry, number theory, integrable systems, statistical mechanics and string theory.

The first part of the subject studies complex analysis. It assumes students have completed a first course in complex analysis so begins with a quick review of analytic functions and Cauchy's theorem, emphasising topological aspects such as the argument principle and Rouche's theorem.

Topics also include: Schwarz's lemma; limits of analytic functions, normal families, Riemann mapping theorem; multiple-valued functions, differential equations and Riemann surfaces. The second part of the subject studies Riemann surfaces and natural objects on them such as holomorphic differentials and quadratic differentials.

Topics may also include: divisors, Riemann-Roch theorem; the moduli space of Riemann surfaces, Teichmueller space; integrable systems.

**Advanced Topics in Geometry and Topology**12.5 ptsThis subject will present an introduction to an advanced topic in geometry or topology serving to prepare students for a PhD in mathematics. The specific content will vary depending on the subject coordinator.

**Lie Algebras**12.5 ptsThe theory of Lie algebras is fundamental to the study of groups of continuous symmetries acting on vector spaces, with applications to diverse areas including geometry, number theory and the theory of differential equations. Moreover, since quantum mechanical systems are described by Hilbert spaces acted on by continuous symmetries, Lie algebras and their representations are also fundamental to modern mathematical physics. This subject develops the basic theory in a way accessible to both pure mathematics and mathematical physics students, with an emphasis on examples. The main theorems are: the classification of complex semi-simple Lie algebras in terms of Cartan matrices and Dynkin diagrams, and the classification of finite-dimensional representations of these algebras in terms of highest weight theory.

**Partial Differential Equations**12.5 ptsThis subject offers a wide ranging introduction to the modern theory of partial differential equations (PDEs) in pure mathematics. Thus we will study questions of existence, uniqueness, regularity, and long time behaviour (e.g.\ energy dispersion) for solutions to PDEs. We will discuss these questions first for the classical equations (Laplace's equation, the heat equation, and the wave equation) which will lead us to the broader theory of elliptic, parabolic, and hyperbolic equations. The course covers mostly linear equations, but exposes the student also to some of the most interesting non-linear equations arising in physics and geometry.

Further topics may include: Calculus of variations, Hamilton-Jacobi equations, Systems of Conservation laws; Non-linear elliptic equations, Schauder theory; Quasi-linear hyperbolic equations, propagation of singularities, blow up phenomena.

**Algebraic Number Theory**12.5 ptsThis course is an introduction to algebraic number theory. Algebraic number theory studies the structure of the integers and algebraic numbers, combining methods from commutative algebra, complex analysis, and Galois theory. This subject covers the basic theory of number fields, rings of integers and Dedekind domains, zeta functions, decomposition of primes in number fields and ramification, the ideal class group, and local fields. Additional topics may include Dirichlet L-functions and Dirichlet’s theorem; quadratic forms and the theorem of Hasse-Minkowski; local and global class field theory; adeles; and other topics of interest.

#### Core

Complete both of the following subjects:

**Mathematical Statistics**12.5 ptsThe theory of statistical inference is important for applied statistics and as a discipline in its own right. After reviewing random samples and related probability techniques including inequalities and convergence concepts the theory of statistical inference is developed. The principles of data reduction are discussed and related to model development. Methods of finding estimators are given, with an emphasis on multi-parameter models, along with the theory of hypothesis testing and interval estimation. Both finite and large sample properties of estimators are considered. Applications may include robust and distribution free methods, quasi-likelihood and generalized estimating equations. It is expected that students completing this course will have the tools to be able to develop inference procedures in novel settings.

**Advanced Probability**12.5 ptsThis subject explores a range of key concepts in modern Probability Theory that are fundamental for Mathematical Statistics and are widely used in other applications. We study measurable space, product measure, Fubini's theorem, conditional expectation and conditional probability, construction of i.i.d. and beyond, discrete-time martingales.

#### Elective

Complete three of the following subjects:

**Stochastic Calculus with Applications**12.5 ptsThis subject provides an introduction to stochastic calculus and mathematics of financial derivatives. Stochastic calculus is essentially a theory of integration of a stochastic process with respect to another stochastic process, created for situations where conventional integration will not be possible. Apart from being an interesting and deep mathematical theory, stochastic calculus has been used with great success in numerous application areas, from engineering and control theory to mathematical biology, theory of cognition and financial mathematics.

**Multivariate Statistical Techniques**12.5 ptsMultivariate statistics concerns the analysis of collections of random variables that has general applications across the sciences and more recently in bioinformatics. It overlaps machine learning and data mining, and leads into functional data analysis. Here random vectors and matrices are introduced along with common multivariate distributions. Multivariate techniques for clustering, classification and data reduction are given. These include discriminant analysis and principal components. Classical multi-variate regression and analysis of variance methods are considered. These approaches are then extended to high dimensional data, such as that commonly encountered in bioinformatics, motivating the development of multiple hypothesis testing techniques. Finally, functional data is introduced.

**Computational Statistics & Data Science**12.5 ptsComputing techniques and data mining methods are indispensable in modern statistical research and data science applications, where “Big Data” problems are often involved. This subject will introduce a number of recently developed methods and applications in computational statistics and data science that are scalable to large datasets and high-performance computing. The data mining methods to be introduced include general model diagnostic and assessment techniques, kernel and local polynomial nonparametric regression, basis expansion and nonparametric spline regression, generalised additive models, classification and regression trees, forward stagewise and gradient boosting models. Important statistical computing algorithms and techniques used in data science will be explained in detail. These include the bootstrap resampling and inference, cross-validation, the EM algorithm and Louis method, and Markov chain Monte Carlo methods including adaptive rejection and squeeze sampling, sequential importance sampling, slice sampling, Gibbs sampler and Metropolis-Hastings algorithm.

**Random Processes**12.5 ptsThe subject covers some key aspects of the theory of stochastic processes that plays a central role in modern probability and has numerous applications in natural sciences and industry. We discuss the following topics: ways to construct and specify random processes, functional central limit theorem, Levy processes, renewal processes and Markov processes (discrete and continuous state space). Applications to modelling random phenomena evolving in time are discussed throughout the course.

**Statistical Modelling**12.5 ptsStatistical models are central to applications of statistics and their development motivates new statistical theories and methodologies. Commencing with a review of linear and generalized linear models, analysis of variance and experimental design, the theory of linear mixed models is developed and model selection techniques are introduced. Approaches to non and semiparametric inference, including generalized additive models, are considered. Specific applications may include longitudinal data, survival analysis and time series modelling.

**Practice of Statistics & Data Science**12.5 ptsThis subject builds on methods and techniques learned in theoretical subjects by studying the application of statistics in real contexts. Emphasis is on the skills needed for a practising statistician, including the development of mature statistical thinking, organizing the structure of a statistical problem, the contribution to the design of research from a statistical point of view, measurement issues and data processing. The subject deals with thinking about data in a broad context, and skills required in statistical consulting.

**Mathematics of Risk**12.5 ptsMathematical modelling of various types of risk has become an important component of the modern financial industry. The subject discusses the key aspects of the mathematics of market risk. Main concepts include loss distributions, risk and dependence measures, copulas, risk aggregation and allocation principles, elements of extreme value theory. The main theme is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers.

**Analysis of High-Dimensional Data**12.5 ptsModern data sets are growing in size and complexity due to the astonishing development of data acquisition and storage capabilities. This subject focuses on developing rigorous statistical learning methods that are needed to extract relevant features from large data sets, assess the reliability of the selected features, and obtain accurate inferences and predictions. This subject covers recent methodological developments in this area such as inference for high-dimensional inference regression, empirical Bayes methods, model selection and model combining methods, and post-selection inference methods.

**Advanced Statistical Modelling**12.5 ptsComplex data consisting of dependent measurements collected at different times and locations are increasingly important in a wide range of disciplines, including environmental sciences, biomedical sciences, engineering and economics. This subject will introduce you to advanced statistical methods and probability models that have been developed to address complex data structures, such as functional data, geo-statistical data, lattice data, and point process data. A unifying theme of this subject will be the development of inference, classification and prediction methods able to cope with the dependencies that often arise in these data.

**Advanced Topics in Stochastic Models**12.5 ptsThis subject develops the advanced topics and methods of stochastic processes and discusses possible applications of the models covered in the course. It serves to prepare students for research in Probability Theory. The specific content will vary depending on the subject coordinator.

**Inference for Spatio-Temporal Processes**12.5 ptsModern data collection technologies are creating unprecedented challenges in statistics and data science related to the analysis and interpretation of massive data sets where observations exhibit patterns through time and space. This subject introduces probability models and advanced statistical inference methods for the analysis of temporal and spatio-temporal data. The subject balances rigorous theoretical development of the methods and their properties with real-data applications. Topics include inference methods for univariate and multivariate time series models, spatial models, lattice models, and inference methods for spatio-temporal processes. The subject will also address aspects related to computational and statistical trade-offs, and the use of statistical software.

**Bayesian Statistical Learning**12.5 ptsBayesian inference treats all unknowns as random variables, and the core task is to update the probability distribution for each unknown as new data is observed. After introducing Bayes’ Theorem to transform prior probabilities into posterior probabilities, the first part of this subject introduces theory and methodological aspects underlying Bayesian statistical learning including credible regions, prior choice, comparisons of means and proportions, multi-model inference and model selection. The second part of the subject will cover practical implementations of Bayesian methods through Markov Chain Monte Carlo computing and real data applications, focusing on (generalised) linear models and concluding by exploring machine learning techniques such as Gaussian processes.

**Multivariate Statistics for Data Science**12.5 ptsModern statistics and data science deals with data having multiple dimensions. Multivariate methods are used to handle these types of data. Approaches to supervised and unsupervised learning with multivariate data are discussed. In particular, methods for classification, clustering, and dimension reduction are introduced, which are particularly suited to high-dimensional data. Both parametric and nonparametric approaches are discussed.

#### Physics

**Quantum Mechanics**12.5 ptsQuantum Mechanics introduces a dramatically new and rich understanding of the universe. In addition to providing a much deeper insight into the world of atoms and subatomic particles than afforded by classical Newtonian physics, Quantum Mechanics underpins advances in science across all disciplines, from molecular biology to astrophysics. This subject provides a rigorous mathematical formalism for advanced quantum mechanics, laying the foundation for further fundamental theoretical physics and research-level experimental physics in frontier areas such as quantum communication and quantum computation.

The subject describes the Hilbert-space formulation of quantum wave mechanics, including density matrix descriptions for single and joint Hilbert space systems; symmetries and conservation laws including rotations and angular momentum; many-body systems of identical particles; time-dependent perturbation theory, and scattering theory.

**Quantum Field Theory**12.5 ptsThis subject introduces quantum field theory, the combination of quantum mechanics and relativity that explains the fundamental structure of matter and the physics of the early universe. The course has an emphasis on quantum electrodynamics. Specific topics will include an introduction to classical field theory, the Euler-Lagrange equations and Noether’s theorem; the Dirac and Klein-Gordon equations; the quantisation of free scalar, Dirac and vector fields; covariant perturbation theory, the Smatrix and Feynman diagrams; the computation of elementary processes in quantum electrodynamics.

**Physical Cosmology**12.5 ptsThis subject provides an advanced introduction to physical cosmology. Specific topics may include the isotropic homogeneous Universe, the Robertson Walker metric, the Friedmann equations, baryogenesis, inflation, big-bang nucleosynthesis, the recombination era, density fluctuations as the origin of galaxies, the cosmic microwave background, linear and non-linear growth of structure, the Press-Schechter mass function, reionization of the IGM and gravitational lensing. Examples are drawn from past and current cosmological observations.

**Statistical Mechanics**12.5 ptsThis subject provides an advanced introduction to non-equilibrium statistical mechanics. The subject focuses on collective phenomena in complex many-body systems with an emphasis on diffusive processes, stability and the emergence of long-range order, with examples drawn from physics, chemistry, biology and economics. Specific topics include diffusive stochastic processes (Fokker-Planck equations), birth-death processes (master equations), kinetic transport, and spatio-temporal pattern formation in unstable nonlinear systems (bifurcations, chaos, reaction-diffusion equations).

**Particle Physics**12.5 ptsParticle Physics is the study of the elementary constituents of matter, and the fundamental forces of nature. The subject introduces modern elementary particle physics, with an emphasis on the theoretical description of the Standard Model of Particle Physics and its experimental basis. Specific topics may include basic group theory; parity and CP violation; global and local symmetries; non-abelian gauge theory; QCD and the quark model; running coupling constants and asymptotic freedom; spontaneous symmetry breaking and the Higgs mechanism; the complete Standard Model Lagrangian; interactions of particles with matter; accelerators and detectors; deep inelastic scattering and structure functions; flavour mixing and neutrino oscillations.

**General Relativity**12.5 ptsThis subject provides an advanced introduction to Einstein's theory of general relativity. Specific topics may inlcude special relativity, manifolds and curvature, experimental tests, Einstein's equations, the Schwarzschild solution and black holes, weak fields and gravitational radiation. Examples will be drawn from particle physics, astrophysics and cosmology.

#### Computer Science

**Algorithms and Complexity**12.5 pts**AIMS**The aim of this subject is for students to develop familiarity and competence in assessing and designing computer programs for computational efficiency. Although computers manipulate data very quickly, to solve large-scale problems, we must design strategies so that the calculations combine effectively. Over the latter half of the 20th century, an elegant theory of computational efficiency developed. This subject introduces students to the fundamentals of this theory and to many of the classical algorithms and data structures that solve key computational questions. These questions include distance computations in networks, searching items in large collections, and sorting them in order.

**INDICATIVE CONTENT**

Topics covered include complexity classes and asymptotic notation; empirical analysis of algorithms; abstract data types including queues, trees, priority queues and graphs; algorithmic techniques including brute force, divide-and-conquer, dynamic programming and greedy approaches; space and time trade-offs; and the theoretical limits of algorithm power.

**Declarative Programming**12.5 pts**AIMS**Declarative programming languages provide elegant and powerful programming paradigms which every programmer should know. This subject presents declarative programming languages and techniques.

**INDICATIVE CONTENT**

- The dangers of destructive update
- Functional programming
- Recursion
- Strong type systems
- Parametric polymorphism
- Algebraic types
- Type classes
- Defensive programming practice
- Higher order programming
- Currying and partial application
- Lazy evaluation
- Monads
- Logic programming
- Unification and resolution
- Nondeterminism, search, and backtracking

**Cryptography and Security**12.5 pts**AIMS**The subject will explore foundational knowledge in the area of cryptography and information security. The overall aim is to gain an understanding of fundamental cryptographic concepts like encryption and signatures and use it to build and analyse security in computers, communications and networks. This subject covers fundamental concepts in information security on the basis of methods of modern cryptography, including encryption, signatures and hash functions.

This subject is an elective subject in the Master of Engineering (Software). It can also be taken as an advanced elective in Master of Information Technology.

**INDICATIVE CONTENT**

The subject will be made up of three parts:

- Cryptography: the essentials of public and private key cryptography, stream ciphers, digital signatures and cryptographic hash functions
- Access Control: the essential elements of authentication and authorization; and
- Secure Protocols; which are obtained through cryptographic techniques.

A particular emphasis will be placed on real-life protocols such as Secure Socket Layer (SSL) and Kerberos.

Topics drawn from:

- Symmetric key crypto systems
- Public key cryptosystems
- Hash functions
- Authentication
- Secret sharing
- Protocols
- Key Management.

**Statistical Machine Learning**12.5 pts**AIMS**With exponential increases in the amount of data becoming available in fields such as finance and biology, and on the web, there is an ever-greater need for methods to detect interesting patterns in that data, and classify novel data points based on curated data sets. Learning techniques provide the means to perform this analysis automatically, and in doing so to enhance understanding of general processes or to predict future events.

Topics covered will include: supervised learning, semi-supervised and active learning, unsupervised learning, kernel methods, probabilistic graphical models, classifier combination, neural networks.

This subject is intended to introduce graduate students to machine learning though a mixture of theoretical methods and hands-on practical experience in applying those methods to real-world problems.

**INDICATIVE CONTENT**Topics covered will include: linear models, support vector machines, random forests, AdaBoost, stacking, query-by-committee, multiview learning, deep neural networks, un/directed probabilistic graphical models (Bayes nets and Markov random fields), hidden Markov models, principal components analysis, kernel methods.

**Constraint Programming**12.5 pts**AIMS**The aims for this subject is for students to develop an understanding of approaches to solving combinatorial optimization problems with computers, and to be able to demonstrate proficiency in modelling and solving programs using a high-level modelling language, and understanding of different solving technologies. The modelling language used is MiniZinc.

**INDICATIVE CONTENT**

Topics covered will include:

- Modelling with Constraints
- Global constraints
- Multiple Modelling
- Model Debugging
- Scheduling and Packing
- Finite domain constraint solving
- Mixed Integer Programming

#### Bioinformatics

**Elements of Bioinformatics**12.5 ptsBioinformatics is a key research tool in modern agriculture, medicine, and the life sciences in general. It forms a bridge between complex experimental and clinical data and the elucidation of biological knowledge. This subject presents bioinformatics in the context of its role in science, using examples from a variety of fields to illustrate the history, current status, and future directions of bioinformatics research and practice.

**Statistics for Bioinformatics**12.5 ptsBioinformatics involves the analysis of biological data and randomness is inherent in both the biological processes themselves and the sampling mechanisms by which they are observed. This subject first introduces stochastic processes and their applications in Bioinformatics, including evolutionary models. It then considers the application of classical statistical methods including estimation, hypothesis testing, model selection, multiple comparisons, and multivariate statistical techniques in Bioinformatics.

#### Biomedical Engineering

**Systems and Synthetic Biology**12.5 pts**AIMS:**This subject introduces mathematical and computational modelling, simulation and analysis of biological systems. The emphasis is on developing models, with examples, using MATLAB.

**INDICATIVE CONTENT:**Topics include:

Modelling biochemical reactions. Law of mass action. Enzymes and regulation of enzyme reactions. Thermodynamics of reversible biochemical reactions. Cellular homeostasis. Application of ordinary differential equations to these problems.

Modelling large reaction networks. Flux balance analysis and constraint-based methods. Genome-scale models. Regulation of gene expression. Gene regulatory networks in systems and synthetic biology. Network inference and statistical modelling of –omic data. Knowledge-based modelling in systems biology.

#### Professional skills

Choose one of the following subjects:

**Systems Modelling and Simulation**12.5 ptsModern science and business makes extensive use of computers for simulation, because complex real-world systems often cannot be analysed exactly, but can be simulated. Using simulation we can perform virtual experiments with the system, to see how it responds when we change parameters, which thus allows us to optimise its performance. We use the language R, which is one of the most popular modern languages for data analysis.

**The Art of Scientific Computation**12.5 ptsThe physical, social and engineering sciences make widespread use of numerical simulations and graphical representations that link underlying their theoretical foundations with experimental or empirical data. These approaches are routinely designed and conducted by researchers with little or no formal training in computation, assembling instead the necessary skills from a variety of sources. There is an art to assembling computational tools that both achieve their goals and make good effective use of the available computational resources.

This subject introduces students to a wide range of skills that are commonly encountered in the design and construction of computational tools in research applications:

- Formulation of the task as a sequence of operations or procedures that express the context of the assigned problem in a form accessible to digital computing (Mathematica).
- Implementation of this formulation using computer languages appropriate for numerically intensive computation (C, C++, Fortran)
- Modularization of computationally intensive tasks, either as user-written procedures or existing libraries (for example BLAS, lapack)
- Documentation of the code to explain both its design, operation and limitations (LaTeX)
- Instrumentation of the code to verify its correct operation and monitor its performance (gprof)
- Optimization of the code, including the use of parallelization (OpenMPI)
- Visualization of data using graphical packages or rendering engines (Geomview, OpenGL)
- Interaction with the code through a graphical user interface (Python, Matlab)

These skills are introduced to the student by undertaking a short project that is selected in consultation with the Subject Coordinator.

#### Additional professional skills

If you have already received credit for MAST30028, COMP10001, COMP10002, COMP20005, or INFO10001, select one of the above subjects, or one of the following subjects:

**Science Communication**12.5 ptsWhy is it essential that scientists learn to communicate effectively to a variety of audiences? What makes for engaging communication when it comes to science? How does the style of communication need to change for different audiences? What are the nuts and bolts of good science writing? What are the characteristics of effective public speaking?

Weekly seminars and tutorials will consider the important role science and technology plays in twenty-first century society and explore why it is vital that scientists learn to articulate their ideas to a variety of audiences in an effective and engaging manner. These audiences may include school students, agencies that fund research, the media, government, industry, and the broader public. Other topics include the philosophy of science communication, talking about science on the radio, effective public speaking, writing press releases and science feature articles, science performance, communicating science on the web and how science is reported in the media.

Students will develop skills in evaluating examples of science and technology communication to identify those that are most effective and engaging. Students will also be given multiple opportunities to receive feedback and improve their own written and oral communication skills.

Students will work in small teams on team projects to further the communication skills developed during the seminar programme. These projects will focus on communicating a given scientific topic to a particular audience using spoken, visual, written or web-based communication.

**Communication for Research Scientists**12.5 ptsAs a scientist, it is not only important to be able to experiment, research and discover, it is also vital that you can communicate your research effectively in a variety of ways. Even the most brilliant research is wasted if no one knows it has been done or if your target audience is unable to understand it.

In this subject you will develop your written and oral communication skills to ensure that you communicate your science as effectively as possible. We will cover effective science writing and oral presentations across a number of formats: writing a thesis; preparing, submitting and publishing journal papers; searching for, evaluating and citing appropriate references; peer review, making the most of conferences; applying for grants and jobs; and using social media to publicise your research.

You will have multiple opportunities to practice, receive feedback and improve both your oral and written communication skills.

Please note: students must be undertaking their own research in order to enrol in this subject.

#### 50-Point project

Complete parts 1, 2 and 3.

**Research Project Part 1**12.5 ptsIn this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the School's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

**Research Project Part 1**25 ptsIn this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the School's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

**Research Project Part 2**12.5 ptsIn this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the School's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

**Research Project Part 2**25 pts

**Research Project Part 3**25 pts

**Research Project Part 3**12.5 pts

#### 25-Point project

Complete parts 1 and 2, and select a further 25 points of coursework subjects from your chosen specialisation.

**Minor Research Project Part 1**12.5 ptsIn this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the Department's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

**Minor Research Project Part 2**12.5 ptsIn this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the Department's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.