Senior Mathematics and Statistics from 2006
- Joint Pure and Applied Mathematics Units in Semester 1
- Joint Pure and Applied Mathematics Units in Semester 2
- Applied Mathematics Units in Semester 1
- Applied Mathematics Units in Semester 2
- Pure Mathematics Units in Semester 1
- Pure Mathematics Units in Semester 2
- Statistics Units in Semester 1
- Statistics Units in Semester 2
Joint Pure and Applied Mathematics Units in Semester 1
MATH3063 Differential Equations and Biomathematics
Prerequisite: 12 credit points of Intermediate Mathematics.
Assumed knowledge: MATH2061.
Prohibitions: MATH3003, MATH3923, MATH3020, MATH3920, MATH3963.
This unit of study is an introduction to the theory of systems of ordinary differential equations. Such systems model many types of phenomena in engineering, biology and the physical sciences. The emphasis will not be on finding explicit solutions, but instead on the qualitative features of these systems, such as stability, instability and oscillatory behaviour. The aim is to develop a good geometrical intuition into the behaviour of solutions to such systems. Some background in linear algebra, and familiarity with concepts such as limits and continuity, will be assumed. The applications in this unit will be drawn from predator-prey systems, transmission of diseases, chemical reactions, beating of the heart and other equations and systems from mathematical biology.
MATH3963 Differential Equations and Biomathematics (Advanced)
Prerequisite: 12 credit points of Intermediate Mathematics.
Assumed knowledge: MATH2961.
Prohibitions: MATH3003, MATH3923, MATH3020, MATH3920, MATH3063.
The theory of ordinary differential equations is a classical topic going back to Newton and Leibniz. It comprises a vast number of ideas and methods of different nature. The theory has many applications and stimulates new developments in almost all areas of mathematics. The applications in this unit will be drawn from predator-prey systems, transmission of diseases, chemical reactions, beating of the heart and other equations and systems from mathematical biology. The emphasis is on qualitative analysis including phase-plane methods, bifurcation theory and the study of limit cycles. The more theoretical part includes existence and uniqueness theorems, stability analysis, linearization, and hyperbolic critical points, and omega limit sets.
Joint Pure and Applied Mathematics Units in Semester 2
MATH3964 Complex Analysis with Applications (Advanced)
Prerequisite: 12 credit points of Intermediate Mathematics.
Assumed knowledge: MATH2962.
Prohibitions: MATH3904, MATH3915.
This unit continues the study of functions of a complex variable and their applications introduced in the second year unit Real and Complex Analysis (MATH2962). It is aimed at highlighting certain topics from analytic function theory and the analytic theory of differential equations that have intrinsic beauty and wide applications. This part of the analysis of functions of a complex variable will form a very important background for students in applied and pure mathematics, physics, chemistry and engineering.
The course will begin with a revision of properties of holomorphic functions and Cauchy's theorem with added topics not covered in the second year course. This will be followed by meromorphic functions, entire functions, harmonic functions, elliptic functions, elliptic integrals, analytic differential equations, hypergeometric functions. The rest of the course will consist of selected topics from Green's functions, complex differential forms and Riemann surfaces.
Applied Mathematics Units in Semester 1
MATH3076 Mathematical Computing
Prerequisite: 12 credit points of intermediate mathematics.
Prohibitions: MATH3976, MATH3016, MATH3916.
This unit of study provides an introduction to Fortran 95 programming and numerical methods. Topics covered include computer arithmetic and computational errors, systems of linear equations, interpolation and approximation, solution of nonlinear equations, quadrature, initial value problems for ordinary differential equations and boundary value problems.
MATH3974 Fluid Dynamics (Advanced)
Prerequisite: 12 credit points of intermediate mathematics with an average grade of credit or better.
Assumed knowledge: MATH2961 and MATH2965.
Prohibitions: MATH3914.
This unit of study provides an introduction to fluid dynamics, starting with a description of the governing equations and the simplifications gained by using stream functions or potentials. It develops elementary theorems and tools, including Bernoulli's equation, the role of vorticity, the vorticity equation, Kelvin's circulation theorem, Helmholtz's theorem, and an introduction to the use of tensors. Topics covered include viscous flows, lubrication theory, boundary layers, potential theory, and complex variable methods for 2-D airfoils. The unit concludes with an introduction to hydrodynamic stability theory and the transition to turbulent flow.
MATH3976 Mathematical Computing (Advanced)
Prerequisite: 12 credit points of Intermediate Mathematics and one of MATH1903 or MATH1907 or Credit in MATH1003
Prohibitions: MATH3076, MATH3016, MATH3916.
See the description of MATH3076 Mathematical Computing (Normal).
Applied Mathematics Units in Semester 2
MATH3075 Financial Mathematics
Prerequisite: 12 credit points of intermediate mathematics.
Prohibitions: MATH3975, MATH3015, MATH3933.
This unit is an introduction to the mathematical theory of modern finance. Topics include: notion of arbitrage, pricing riskless securities, risky securities, utility theory, fundamental theorems of asset pricing, complete markets, introduction to options, binomial option pricing model, discrete random walks, Brownian motion, derivation of the Black-Scholes option pricing model, extensions and introduction to pricing exotic options, credit derivatives. A strong background in mathematical statistics and partial differential equations is an advantage, but is not essential. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills.
Note that students enrolled in MATH3075 and those enrolled in the advanced level unit MATH3975 attend the same lectures, but the assessment tasks for MATH3975 are more challenging than those for MATH3075.
MATH3078 Partial Differential Equations and Waves
Prerequisite: 12 credit points of intermediate mathematics.
Assumed knowledge: MATH2061 (or MATH2961) and MATH2065 (or MATH2965).
Prohibitions: MATH3978, MATH3018, MATH3921.
This unit of study introduces Sturm-Liouville eigenvalue problems and their role in finding solutions to boundary value problems. Analytical solutions of linear PDEs are found using separation of variables and integral transform methods. Three of the most important equations of mathematical physics – the wave equation, the diffusion (heat) equation and Laplace's equation – are treated, together with a range of applications. There is particular emphasis on wave phenomena, with an introduction to the theory of sound waves and water waves.
MATH3975 Financial Mathematics (Advanced)
Prerequisite: 12 credit points of intermediate mathematics with an average grade of credit or better.
Prohibitions: MATH3075, MATH3015, MATH3933.
This unit is an introduction to the mathematical theory of modern finance. Topics include: notion of arbitrage, pricing riskless securities, risky securities, utility theory, fundamental theorems of asset pricing, complete markets, introduction to options, binomial option pricing model, discrete random walks, Brownian motion, derivation of the Black-Scholes option pricing model, extensions and introduction to pricing exotic options, credit derivatives. A strong background in mathematical statistics and partial differential equations is an advantage, but is not essential. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills.
Note that students enrolled in MATH3975 and those enrolled in the normal level unit MATH3075 attend the same lectures; students enrolled in MATH3975 will be required to undertake more challenging assessment tasks.
MATH3977 Lagrangian and Hamiltonian Dynamics (Advanced)
Prerequisite: 12 credit points of intermediate mathematics with an average grade of credit or better.
Prohibitions: MATH2904, MATH2004, MATH3917.
This unit provides a comprehensive treatment of dynamical systems using the mathematically sophisticated framework of Lagrange and Hamilton. This formulation of classical mechanics generalizes elegantly to modern theories of relativity and quantum mechanics.
The unit develops dynamical theory from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to make apparently complicated dynamical problems appear very simple. The unit will also explore connections between geometry and different physical theories beyond classical mechanics.
Students will be expected to solve fully dynamical systems of some complexity including planetary motion and to investigate stability using perturbation analysis. Hamilton-Jacobi theory will be used to elegantly solve problems ranging from geodesics (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes.
This unit is a useful preparation for units in dynamical systems and chaos, and complements units in differential equations, quantum theory and general relativity.
MATH3978 Partial Differential Equations and Waves (Advanced)
Prerequisite: 12 credit points of intermediate mathematics with an average grade of credit or better.
Assumed knowledge: MATH2961 or MATH2061 and MATH2965 or MATH2065.
Prohibitions: MATH3078, MATH3018, MATH3921.
This unit of study introduces Sturm-Liouville eigenvalue problems and their role in finding solutions to boundary value problems. Analytical solutions of linear PDEs are found using separation of variables and integral transform methods. Three of the most important equations of mathematical physics – the wave equation, the diffusion (heat) equation and Laplace's equation – are treated, together with a range of applications. There is particular emphasis on wave phenomena, with an introduction to the theory of sound waves and water waves.
Pure Mathematics Units in Semester 1
MATH3065 Logic and Foundations
Prerequisite: 6 credit points of Intermediate Mathematics.
Prohibitions: MATH3005
This unit is in two halves. The first half provides a working knowledge of the propositional and predicate calculi, discussing techniques of proof, consistency, models and completeness. The second half discusses notions of computability by means of Turing machines (simple abstract computers). (No knowledge of computer programming is assumed.) It is shown that there are some mathematical tasks (such as the halting problem) that cannot be carried out by any Turing machine. Results are applied to first-order Peano arithmetic, culminating in Gödel’s Incompleteness Theorem: any statement that includes first-order Peano arithmetic contains true statements that cannot be proved in the system. A brief discussion is given of Zermelo-Fraenkel set theory (a candidate for the foundations of mathematics), which still succumbs to Gödel’s Theorem.
MATH3068 Analysis
Prerequisite: 12 credit points of Intermediate Mathematics.
Prohibitions: MATH3008, MATH2007, MATH2907, MATH2962
Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. The aim of the unit is to present enduring beautiful and practical results that continue to justify and inspire the study of analysis. The unit starts with the foundations of calculus and the real number system. It goes on to study the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform convergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: analytic functions, Taylor expansions and the Cauchy Integral Theorem.
Power series are not adequate to solve the problem of representing periodic phenomena such as wave motion. This requires Fourier theory, the expansion of functions as sums of sines and cosines. This unit deals with this theory, Parseval’s identity, pointwise convergence theorems and applications.
The unit goes on to introduce Bernoulli numbers, Bernoulli polynomials, the Euler MacLaurin formula and applications, the gamma function and the Riemann zeta function. Lastly we return to the foundations of analysis, and study limits from the point of view of topology.
MATH3961 Metric Spaces (Advanced)
Prerequisite: 12 credit points of Intermediate Mathematics.
Assumed knowledge: MATH2961 or MATH2962.
Prohibitions: MATH3901, MATH3001
Topology, developed at the end of the 19th Century to investigate the subtle interaction of analysis and geometry, is now one of the basic disciplines of mathematics. A working knowledge of the language and concepts of topology is essential in fields as diverse as algebraic number theory and non-linear analysis. This unit develops the basic ideas of topology using the example of metric spaces to illustrate and motivate the general theory. Topics covered include: Metric spaces, convergence, completeness and the contraction mapping theorem; Metric topology, open and closed subsets; Topological spaces, subspaces, product spaces; Continuous mappings and homeomorphisms; Compact spaces; Connected spaces; Hausdorff spaces and normal spaces, Applications include the implicit function theorem, chaotic dynamical systems and an introduction to Hilbert spaces and abstract Fourier series.
MATH3962 Rings, Fields and Galois Theory (Advanced)
Prerequisite: 12 credit points of Intermediate Mathematics.
Assumed knowledge: MATH2961.
Prohibitions: MATH3902, MATH3002, MATH3062
This unit of study investigates the modern mathematical theory that was originally developed for the purpose of studying polynomial equations. The philosophy is that it should be possible to factorize any polynomial into a product of linear factors by working over a "large enough" field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory.
The basic theoretical tool needed for this program is the concept of a ring, which generalizes the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions.
Pure Mathematics Units in Semester 2
MATH3061 Geometry and Topology
Prerequisite: 12 credit points of Intermediate Mathematics.
Prohibitions: MATH3006, MATH3001.
The aim of the unit is to expand visual/geometric ways of thinking. The geometry section is concerned mainly with transformations of the Euclidean plane (that is, bijections from the plane to itself), with a focus on the study of isometries (proving the classification theorem for transformations which preserve distances between points), symmetries (including the classification of frieze groups) and affine transformations (transformations which map lines to lines). The basic approach is via vectors and matrices, emphasizing the interplay between geometry and linear algebra. The study of affine transformations is then extended to the study of collineations in the real projective plane, including collineations which map conics to conics. The topology section considers graphs, surfaces and knots from a combinatorial point of view. Key ideas such as homeomorphism, subdivision, cutting and pasting and the Euler invariant are introduced first for graphs (1-dimensional objects) and then for triangulated surfaces (2-dimensional objects). The classification of surfaces is given in several equivalent forms. The problem of colouring maps on surfaces is interpreted via graphs. The main geometric fact about knots is that every knot bounds a surface in 3-space. This is proved by a simple direct construction, and is then used to show that every knot is a sum of prime knots.
MATH3062 Algebra and Number Theory
Prerequisite: 12 credit points of Intermediate Mathematics.
Assumed knowledge: MATH2068 (or MATH2968) recommended but not essential.
Prohibitions: MATH3002, MATH3902, MATH3962, MATH3009
The first half of the unit continues the study of elementary number theory, with an emphasis on the solution of Diophantine equations (for example, finding all integer squares which are one more than twice a square). Topics include the Law of Quadratic Reciprocity, representing an integer as the sum of two squares, and continued fractions. The second half of the unit introduces the abstract algebraic concepts which arise naturally in this context: rings, fields, irreducibles, and unique factorization. Polynomial rings, algebraic numbers, and constructible numbers are also discussed.
MATH3067 Information and Coding Theory
Prerequisite: 12 credit points of Intermediate Mathematics.
Prohibitions: MATH3007, MATH3010
The related theories of information and coding provide the basis for reliable and efficient storage and transmission of digital data, including techniques for data compression, digital broadcasting and broadband internet connectivity. The first part of this unit is a general introduction to the ideas and applications of information theory, where the basic concept is that of entropy. This gives a theoretical measure of how much data can be compressed for storage or transmission. Information theory also addresses the important practical problem of making data immune to partial loss caused by transmission noise or physical damage to storage media. This leads to the second part of the unit, which deals with the theory of error-correcting codes. We develop the algebra behind the theory of linear and cyclic codes used in modern digital communication systems such as compact disk players and digital television.
MATH3966 Modules and Group Representations (Advanced)
Prerequisite: 12 credit points of Intermediate Mathematics.
Assumed knowledge: MATH3962.
Prohibitions: MATH3907, MATH3906
This unit deals first with generalized linear algebra, in which the field of scalars is replaced by an integral domain. In particular we investigate the structure of modules, which are the analogues of vector spaces in this setting, and which are of fundamental importance in modern pure mathematics. Applications of the theory include the solution over the integers of simultaneous equations with integer coefficients and analysis of the structure of finite abelian groups.
In the second half of this unit we focus on linear representations of groups. A group occurs naturally in many contexts as a symmetry group of a set or space. Representation theory provides techniques for analysing these symmetries. The component will deals with the decomposition of representation into simple constituents, the remarkable theory of characters, and orthogonality relations which these characters satisfy.
MATH3968 Differential Geometry (Advanced)
Prerequisite: 12 credit points of Intermediate Mathematics, including MATH2961
Assumed knowledge: at least 6 credit points of advanced level senior or intermediate mathematics.
Prohibitions: MATH3903
This unit is an introduction to Differential Geometry, using ideas from calculus of several variables to develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. Differential geometry also plays an important part in both classical and modern theoretical physics. The initial aim is to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface. A second aim is to present the calculus of differential forms as the natural setting for the key ideas of vector calculus, along with some applications.
MATH3969 Measure Theory and Fourier Analysis (Advanced)
Prerequisite: 12 credit points of Intermediate Mathematics.
Assumed knowledge: at least 6 credit points of advanced level senior or intermediate mathematics.
Prohibitions: MATH3909
Measure theory is the study of such fundamental ideas as length, area, volume, arc length and surface area. It is the basis for the integration theory used in advanced mathematics since it was developed by Henri Lebesgue in about 1900. Moreover, it is the basis for modern probability theory. The course starts by setting up measure theory and integration, establishing important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. This is then applied to Fourier Analysis, and results such as the Inversion Formula and Plancherel's Theorem are derived. Probability Theory is then discussed, with topics including independence, conditional probabilities, and the Law of Large Numbers.
Statistics Units in Semester 1
STAT3011 Stochastic Processes and Time Series
Prerequisite: STAT2011 or STAT2911 or STAT2001 or STAT2901, and MATH1003 or MATH1903 or MATH1907.
Prohibitions: STAT3911, STAT3003, STAT3903, STAT3005, STAT3905.
Section I of this course will introduce the fundamental concepts of applied stochastic processes and Markov chains used in financial mathematics, mathematical statistics, applied mathematics and physics. Section II of the course establishes some methods of modeling and analysing situations which depend on time. Fitting ARMA models for certain time series are considered from both theoretical and practical points of view. Throughout the course we will use the S-PLUS (or R) statistical packages to give analyses and graphical displays.
There will be 3 lectures and 1 tutorial per week, and a total of 10 computer lab sessions in the semester.
STAT3911 Stochastic Processes and Time Series (Advanced)
Prerequisite: STAT2911 or credit in STAT2901, and MATH1003 or MATH1903 or MATH1907.
Prohibitions: STAT3011, STAT3003, STAT3903, STAT3005, STAT3905.
This is an Advanced version of STAT3011. There will be 3 lectures in common with STAT3011. In addition to STAT3011 material, theory on branching processes and birth and death processes will be covered. There will be more advanced tutorial and assessment work associated with this unit.
There will be 3 lectures and 1 tutorial per week, plus an extra lecture on advanced material in the first half of the semester. There will be 7 computer lab sessions (on time series) in the second half of the semester.
STAT3012 Applied Linear Models
Prerequisite: STAT2012 or STAT2912 or STAT2004, and MATH1002 or MATH1902.
Prohibitions: STAT3002, STAT3004, STAT3902, STAT3904, STAT3912.
This course will introduce the fundamental concepts of analysis of data from both observational studies and experimental designs using classical linear methods, together with concepts of collection of data and design of experiments. First we will consider linear models and regression methods with diagnostics for checking appropriateness of models. We will look briefly at robust regression methods here. Then we will consider the design and analysis of experiments considering notions of replication, randomization and ideas of factorial designs. Throughout the course we will use the S-PLUS (or R) statistical packages to give analyses and graphical displays.
STAT3912 Applied Linear Models (Advanced)
Prerequisite: STAT2912 or credit in STAT2004, and MATH1902 or MATH2061 or MATH2961.
Prohibitions: STAT3002, STAT3004, STAT3902, STAT3904, STAT3012.
This course will introduce the fundamental concepts of analysis of data from both observational studies and experimental designs using classical linear methods, together with concepts of collection of data and design of experiments. First we will consider linear models and regression methods with diagnostics for checking appropriateness of models. We will look briefly at robust regression methods here. Then we will consider the design and analysis of experiments considering notions of replication, randomization and ideas of factorial designs. Throughout the course we will use the S-PLUS (or R) statistical packages to give analyses and graphical displays.
There will be 3 lectures, 1 tutorial and 1 computer laboratory session per week.
Statistics Units in Semester 2
STAT3013 Statistical Inference
Prerequisite: STAT2012 or STAT2912 or STAT2003 or STAT2903.
Prohibitions: STAT3001, STAT3901, STAT3913.
In this course we will study basic topics in modern statistical inference. This will include traditional concepts of mathematical statistics: likelihood estimation, method of moments, properties of estimators, exponential families, decision-theory approach to hypothesis testing, likelihood ratio test, as well as more recent approaches such as Bayes estimation, Empirical Bayes and nonparametric estimation. During the weekly computer classes (using S-PLUS or R software packages) we will illustrate the various estimation techniques and give an introduction to computationally intensive methods like Monte Carlo, Gibbs sampling and EM-algorithm.
There will be 3 lectures, 1 tutorial and 1 computer laboratory session per week.
STAT3913 Statistical Inference (Advanced)
Prerequisite: STAT2912 or STAT2903.
Prohibitions: STAT3001, STAT3901, STAT3013.
This unit is essentially an Advanced version of STAT3013, with emphasis on the mathematical techniques underlying statistical inference together with proofs based on distribution theory. There will be 3 lectures per week in common with some material required only in this advanced course and some advanced material given in a separate advanced tutorial together with more advanced assessment work.
There will be 3 lectures, 1 tutorial and 1 computer laboratory session per week.
STAT3014 Applied Statistics
Prerequisite: STAT2012 or STAT2912 or STAT2004.
Assumed Knowledge: STAT3012 or STAT3912.
Prohibitions: STAT3914, STAT3006, STAT3002, STAT3902.
This unit has three distinct but related components: multivariate analysis, sampling and surveys, and generalized linear models. The first component deals with multivariate data covering simple data reduction techniques like principal components analysis and core multivariate tests including Hotelling’s T2, Mahalanobis’ distance, and Multivariate Analysis of Variance (MANOVA). The sampling section includes sampling without replacement, stratified sampling, ratio estimation, and cluster sampling. The final section looks at the analysis of categorical data via generalized linear models. Logistic regression and log-linear models will be looked at in some detail along with special techniques for analyzing discrete data with special structure.
There will be 3 lectures, 1 tutorial and 1 computer laboratory session per week.
STAT3914 Applied Statistics (Advanced)
Prerequisite: STAT2912 or Credit or better in STAT2004.
Assumed Knowledge: STAT3912.
Prohibitions: STAT3014, STAT3006, STAT3002, STAT3902, STAT3907.
This unit is an Advanced version of STAT3014. There will be 3 lectures per week in common with STAT3014. The unit will have extra lectures focusing on multivariate distribution theory developing results for the multivariate normal, partial correlation, the Wishart distribution and Hotellling’s T2. There will also be more advanced tutorial and assessment work associated with this unit.
There will be 3 lectures, 1 tutorial and 1 computer laboratory session per week.