## Degree structure

Students undertake the Diploma in Mathematical Sciences together with an undergraduate degree, to make a total program of between 350 and 400 points.

Up to 50 points of appropriate Mathematics and Statistics subjects in the undergraduate degree program may also be attributed to the Diploma.

Students are required to select subjects in the undergraduate degree and the Diploma, so that all requirements of the undergraduate degree are met and, in addition through the Diploma, the student meets the specific requirements for the award of any one specialisation of the major in Mathematics and Statistics, as defined for the Bachelor of Science.

Specialisations available in the Diploma of Mathematical Sciences:

- Pure Mathematics
- Applied Mathematics
- Discrete Mathematics and Operations Research
- Statistics / Stochastic Processes

## Pure Mathematics specialisation

Year 1, Semester 1 | |
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Core Subjects | Points |

Calculus 2Core | 12.5 |

## Calculus 2This subject will extend knowledge of calculus from school. Students are introduced to hyperbolic functions and their inverses, the complex exponential and functions of two variables. Techniques of differentiation and integration will be extended to these cases. Students will be exposed to a wider class of differential equation models, both first and second order, to describe systems such as population models, electrical circuits and mechanical oscillators. The subject also introduces sequences and series including the concepts of convergence and divergence. Calculus topics include: intuitive idea of limits and continuity of functions of one variable, sequences, series, hyperbolic functio... Detailed Information MAST10006Type Core |

Year 1, Semester 2 | |
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Core Subjects | Points |

Linear AlgebraCore | 12.5 |

## Linear AlgebraThis subject gives a solid grounding in key areas of modern mathematics needed in science and technology. It develops the concepts of vectors, matrices and the methods of linear algebra. Students should develop the ability to use the methods of linear algebra and gain an appreciation of mathematical proof. Little of the material here has been seen at school and the level of understanding required represents an advance on previous studies. Systems of linear equations, matrices and determinants; vectors in real n-space, cross product, scalar triple product, lines and planes; vector spaces, linear independence, basis, dimension; linear transformations, eigenvalues, eigenvectors; inner produc... Detailed Information MAST10007Type Core |

Year 2, Semester 1 | |
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Core Subjects | Points |

Real AnalysisCore | 12.5 |

## Real AnalysisThis subject introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. Many of the important results are proved rigorously and students are introduced to methods of proof such as mathematical induction and proof by contradiction.The important distinction between the real numbers and the rational numbers is emphasized and used to motivate rigorous notions of convergence and divergence of sequences, including the Cauchy criterion. These ideas are extended to cover the theory of infinite series, including common tests for convergence and divergence. A similar treatment of continuity and differentiability of functions of a single... Detailed Information MAST20026Type Core |

Year 2, Semester 2 | |
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Core Subjects | Points |

Group Theory and Linear AlgebraCore | 12.5 |

## Group Theory and Linear AlgebraThis subject introduces the theory of groups, which is at the core of modern algebra, and which has applications in many parts of mathematics, chemistry, computer science and theoretical physics. It also develops the theory of linear algebra, building on material in earlier subjects and providing both a basis for later mathematics studies and an introduction to topics that have important applications in science and technology. Topics include: modular arithmetic and RSA cryptography; abstract groups, homomorphisms, normal subgroups, quotient groups, group actions, symmetry groups, permutation groups and matrix groups; theory of general vector spaces, inner products, linear transformations,... Detailed Information MAST20022Type Core | |

Elective Subjects | Points |

Vector CalculusCore | 12.5 |

## Vector CalculusThis subject studies the fundamental concepts of functions of several variables and vector calculus. It develops the manipulation of partial derivatives and vector differential operators. The gradient vector is used to obtain constrained extrema of functions of several variables. Line, surface and volume integrals are evaluated and related by various integral theorems. Vector differential operators are also studied using curvilinear coordinates. Functions of several variables topics include limits, continuity, differentiability, the chain rule, Jacobian, Taylor polynomials and Lagrange multipliers. Vector calculus topics include vector fields, flow lines, curvature, torsion, gradient, div... Detailed Information MAST20009Type Core |

Year 3, Semester 1 | |
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Core Subjects | Points |

Complex AnalysisCore | 12.5 |

## Complex AnalysisComplex analysis is a core subject in pure and applied mathematics, as well as the physical and engineering sciences. While it is true that physical phenomena are given in terms of real numbers and real variables, it is often too difficult and sometimes not possible, to solve the algebraic and differential equations used to model these phenomena without introducing complex numbers and complex variables and applying the powerful techniques of complex analysis. Topics include:the topology of the complex plane; convergence of complex sequences and series; analytic functions, the Cauchy-Riemann equations, harmonic functions and applications; contour integrals and the Cauchy Integral Theorem; ... Detailed Information MAST30021Type Core | |

AlgebraCore | 12.5 |

## AlgebraAlgebra has a long history of important applications throughout mathematics, science and engineering, and is also studied for its intrinsic beauty. In this subject we study the algebraic laws satisfied by familiar objects such as integers, polynomials and matrices. This abstraction simplifies and unifies our understanding of these structures and enables us to apply our results to interesting new cases. Students will gain further experience with abstract algebraic concepts and methods. General structural results are proved and algorithms developed to determine the invariants they describe. Detailed Information MAST30005Type Core |

Year 3, Semester 2 | |
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Core Subjects | Points |

Metric and Hilbert SpacesCore | 12.5 |

## Metric and Hilbert SpacesThis subject provides a basis for further studies in modern analysis, geometry, topology, differential equations and quantum mechanics.It introduces the idea of a metric space with a general distance function, and the resulting concepts of convergence, continuity, completeness, compactness and connectedness. The subject also introduces Hilbert spaces: infinite dimensional vector spaces (typically function spaces) equipped with an inner product that allows geometric ideas to be used to study these spaces and linear maps between them. Topics include: metric and normed spaces, limits of sequences, open and closed sets, continuity, topological properties, compactness, connectedness; Cauchy se... Detailed Information MAST30026Type Core | |

Elective Subjects | Points |

Discrete MathematicsCore | 12.5 |

## Discrete MathematicsThis subject is concerned with the study of objects, which are finite in number and typically computable. At a computational level one seeks efficient algorithms and methods for construction and counting of the objects. The main topics to be covered are: enumeration, permutations, designs, finite geometry, words, Ramsey theory and physical combinatorics. Designs are relevant to statistics, Ramsey theory to computer science, and physical combinatorics to mathematical physics. Words are useful for representing and constructing objects and relating combinatorial objects to algebraic structures. Detailed Information MAST30012Type Core |