# Mathematical Sciences

##### Errata

item | Errata | Date |
---|---|---|

1. | MATH3976 Mathematical Computing (Advanced) Prerequisites should read: 12 credit points of MATH2XXX and [3 credit points from (MATH1923 or MATH1903 or MATH1933 or MATH1907), or a mark of 65 or above in (MATH1023 or MATH1003)] |
6/2/2018 |

## MATHEMATICAL SCIENCES

## Mathematical Sciences program

The Mathematical Sciences program is available only to students enrolled in the Dalyell stream.

A program in Mathematical Sciences requires 60 credit points including:

(i) A 48 credit point major in either Financial Mathematics and Statistics, Mathematics or Statistics, and:

(a) For students with a major in Mathematics, 6 credit points of 2000-level selective units and 6 credit points of 3000-level selective mathematics or statistics units in addition to those counted towards the major.

(b) For students with a major in Financial Mathematics and Statistics or Statistics, 12 credit points of 2000-level selective mathematics or statistics units in addition to those counted towards the major.

### Units of study

The units of study are listed below.

#### 2000-level units of study

###### Selective

**MATH2021 Vector Calculus and Differential Equations**

Credit points: 6 Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class per week Prerequisites: (MATH1X21 or MATH1931 or MATH1X01 or MATH1906) and (MATH1XX2) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) Prohibitions: MATH2921 or MATH2065 or MATH2965 or MATH2061 or MATH2961 or MATH2067 Assessment: assessment for this unit consists of quizzes, assignments, and a final exam Mode of delivery: Normal (lecture/lab/tutorial) day

This unit opens with topics from vector calculus, including vector-valued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; path-independent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals, polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, through cylinders, spheres and other parametrised surfaces), Gauss' and Stokes' theorems. The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a basic grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2921 Vector Calculus and Differential Eqs (Adv)**

Credit points: 6 Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class per week Prerequisites: [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] Prohibitions: MATH2021 or MATH2065 or MATH2965 or MATH2061 or MATH2961 or MATH2067 Assessment: assessment for this unit consists of quizzes, assignments, and a final exam. Mode of delivery: Normal (lecture/lab/tutorial) day

This is the advanced version of MATH2021, with more emphasis on the underlying concepts and mathematical rigour. The vector calculus component of the course will include: parametrised curves and surfaces, vector fields, div, grad and curl, gradient fields and potential functions, lagrange multipliers line integrals, arc length, work, path-independent integrals, and conservative fields, flux across a curve, double and triple integrals, change of variable formulas, polar, cylindrical and spherical coordinates, areas, volumes and mass, flux integrals, and Green's Gauss' and Stokes' theorems. The Differential Equations half of the course will focus on ordinary and partial differential equations (ODEs and PDEs) with applications with more complexity and depth. The main topics are: second order ODEs (including inhomogeneous equations), series solutions near a regular point, higher order ODEs and systems of first order equations, matrix equations and solutions, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, elementary Sturm-Liouville theory, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series). The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a more thorough grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2022 Linear and Abstract Algebra**

Credit points: 6 Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class per week Prerequisites: MATH1XX2 Prohibitions: MATH2922 or MATH2968 or MATH2061 or MATH2961 Assessment: quizzes, assignments and final exam Mode of delivery: Normal (lecture/lab/tutorial) day

Linear and abstract algebra is one of the cornerstones of mathematics and it is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit investigates and explores properties of linear functions, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract group theory.

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2922 Linear and Abstract Algebra (Advanced)**

Credit points: 6 Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class per week Prerequisites: MATH1902 or (a mark of 65 or above in MATH1002) Prohibitions: MATH2022 or MATH2968 or MATH2061 or MATH2961 Assessment: quizzes, assignments and final exam Mode of delivery: Normal (lecture/lab/tutorial) day

Linear and abstract algebra is one of the cornerstones of mathematics and it is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit is an advanced version of MATH2022, with more emphasis on the underlying concepts and on mathematical rigour. This unit investigates and explores properties of vector spaces, matrices and linear transformations, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract group theory. The unit culminates in studying inner spaces, quadratic forms and normal forms of matrices together with their applications to problems both in mathematics and in the sciences and engineering.

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2023 Analysis**

Credit points: 6 Session: Semester 2 Classes: lecture 3hrs/week; practice class 1hr/week; tutorial 1hr/week Prerequisites: (MATH1X21 or MATH1931 or MATH1X01 or MATH1906) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) and (MATH1XX2) Prohibitions: MATH2923 or MATH3068 or MATH2962 Assessment: assessment for this unit consists of quizzes, an assignment, and a final exam Mode of delivery: Normal (lecture/lab/tutorial) day

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. It is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. This unit introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. It shows the utility of abstract concepts and teaches an understanding and construction of proofs in mathematics. This unit will be useful to students of mathematics, science and engineering and in particular to future school mathematics teachers, because we shall explain why common practices in the use of calculus are correct, and understanding this is important for correct applications and explanations. The unit starts with the foundations of calculus and the real numbers 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: elementary functions of complex variable, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2923 Analysis (Advanced)**

Credit points: 6 Session: Semester 2 Classes: lecture 3hrs/week; practice class 1hr/week; tutorial 1hr/week Prerequisites: [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] Prohibitions: MATH2023 or MATH2962 or MATH3068 Assessment: assessment for this unit consists of quizzes, an assignment, and a final exam Mode of delivery: Normal (lecture/lab/tutorial) day

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. It is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. This advanced unit introduces the field of mathematical analysis both with a careful theoretical frame- work as well as selected applications. It shows the utility of abstract concepts and teaches an understanding and construction of proofs in mathematics. This unit will be useful to students with more mathematical maturity who study mathematics, science, or engineering. The unit starts with the foundations of calculus and the real numbers system, with more emphasis on the topology. 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 con- vergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: elementary functions of complex variable, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

Textbooks

As set out in the Intermediate Mathematics Handbook

**MATH2068 Number Theory and Cryptography**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour computer laboratory per week. Prerequisites: 6 credit points of Junior Mathematics units Prohibitions: MATH2988 or MATH3009 or MATH3024 Assumed knowledge: MATH1014 or MATH1002 or MATH1902 Assessment: 2 hour exam, assignments, quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

Cryptography is the branch of mathematics that provides the techniques for confidential exchange of information sent via possibly insecure channels. This unit introduces the tools from elementary number theory that are needed to understand the mathematics underlying the most commonly used modern public key cryptosystems. Topics include the Euclidean Algorithm, Fermat's Little Theorem, the Chinese Remainder Theorem, Möbius Inversion, the RSA Cryptosystem, the Elgamal Cryptosystem and the Diffie-Hellman Protocol. Issues of computational complexity are also discussed.

**MATH2988 Number Theory and Cryptography Advanced**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour computer laboratory per week. Prerequisites: [MATH19X1 or MATH1906 or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH19X3 or MATH1907 or (a mark of 65 or above in MATH1023 or MATH1003)] and [MATH1902 or (a mark of 65 or above in MATH1002)] Prohibitions: MATH2068 Assessment: One 2 hr exam, homework assignments (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study is an advanced version of MATH2068, sharing the same lectures but with more advanced topics introduced in the tutorials and computer laboratory sessions.

**MATH2069 Discrete Mathematics and Graph Theory**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour practice class per week. Prerequisites: 6 credit points of Junior Mathematics units Prohibitions: MATH2011 or MATH2009 or MATH2969 Assessment: One 2 hour exam, assignments, quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit introduces students to several related areas of discrete mathematics, which serve their interests for further study in pure and applied mathematics, computer science and engineering. Topics to be covered in the first part of the unit include recursion and induction, generating functions and recurrences, combinatorics. Topics covered in the second part of the unit include Eulerian and Hamiltonian graphs, the theory of trees (used in the study of data structures), planar graphs, the study of chromatic polynomials (important in scheduling problems).

**MATH2969 Discrete Mathematics and Graph Theory Adv**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour practice class per week. Prerequisites: 9 credit points of Junior Mathematics (advanced level or Credit at the normal level) Prohibitions: MATH2011 or MATH2009 or MATH2069 Assessment: One 2-hour exam, assignments, quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit will cover the same material as MATH2069 with some extensions and additional topics.

**MATH2070 Optimisation and Financial Mathematics**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour computer laboratory per week. Prerequisites: (MATH1X21 or MATH1011 or MATH1931 or MATH1X01 or MATH1906) and (MATH1014 or MATH1X02) Prohibitions: MATH2010 or MATH2033 or MATH2933 or MATH2970 or ECMT3510 Assumed knowledge: MATH1X23 or MATH1933 or MATH1X03 or MATH1907 Assessment: One 2 hour exam, assignments, quiz, project (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Students may enrol in both MATH2070 and MATH3075 in the same semester

Problems in industry and commerce often involve maximising profits or minimising costs subject to constraints arising from resource limitations. The first part of this unit looks at programming problems and their solution using the simplex algorithm; nonlinear optimisation and the Kuhn Tucker conditions.

The second part of the unit deals with utility theory and modern portfolio theory. Topics covered include: pricing under the principles of expected return and expected utility; mean-variance Markowitz portfolio theory, the Capital Asset Pricing Model, log-optimal portfolios and the Kelly criterion; dynamical programming. Some understanding of probability theory including distributions and expectations is required in this part.

Theory developed in lectures will be complemented by computer laboratory sessions using MATLAB. Minimal computing experience will be required.

The second part of the unit deals with utility theory and modern portfolio theory. Topics covered include: pricing under the principles of expected return and expected utility; mean-variance Markowitz portfolio theory, the Capital Asset Pricing Model, log-optimal portfolios and the Kelly criterion; dynamical programming. Some understanding of probability theory including distributions and expectations is required in this part.

Theory developed in lectures will be complemented by computer laboratory sessions using MATLAB. Minimal computing experience will be required.

**MATH2970 Optimisation and Financial Mathematics Adv**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour computer laboratory per week (lectures given in common with MATH2070). Prerequisites: [MATH19X1 or MATH1906 or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] Prohibitions: MATH2010 or MATH2033 or MATH2933 or MATH2070 or ECMT3510 Assumed knowledge: MATH19X3 or MATH1907 or a mark of 65 or above in MATH1003 or MATH1023 Assessment: One 2 hour exam, assignments, quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Students may enrol in both MATH2970 and MATH3975 in the same semester

The content of this unit of study parallels that of MATH2070, but students enrolled at Advanced level will undertake more advanced problem solving and assessment tasks, and some additional topics may be included.

**STAT2011 Probability and Estimation Theory**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour computer laboratory week. Prerequisites: (MATH1X21 or MATH1931 or MATH1X01 or MATH1906 or MATH1011) and (MATH1XX5 or STAT1021 or ECMT1010 or BUSS1020) Prohibitions: STAT2901 or STAT2001 or STAT2911 Assessment: One 2 hour exam, assignments and/or quizzes, and computer practical reports (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit provides an introduction to univariate techniques in data analysis and the most common statistical distributions that are used to model patterns of variability. Common discrete random models like the binomial, Poisson and geometric, continuous models including the normal and exponential will be studied along with elementary regression models. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The unit will have weekly computer classes where candidates will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

**STAT2911 Probability and Statistical Models (Adv)**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour computer laboratory per week. Prerequisites: [MATH19X3 or MATH1907 or (a mark of 65 in MATH1023 or MATH1003)] and [MATH1905 or MATH1904 or (a mark of 65 in MATH1005 or ECMT1010 or BUSS1020)] Prohibitions: STAT2001 or STAT2901 or STAT2011 Assessment: One 2 hour exam, assignments and/or quizzes, and computer practical reports (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit is essentially an advanced version of STAT2011, with an emphasis on the mathematical techniques used to manipulate random variables and probability models. Common distributions including the Poisson, normal, beta and gamma families as well as the bivariate normal are introduced. Moment generating functions and convolution methods are used to understand the behaviour of sums of random variables. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The notions of conditional expectation and prediction will be covered as will be distributions related to the normal: chi^2, t and F. The unit will have weekly computer classes where candidates will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

**STAT2912 Statistical Tests (Advanced)**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour computer laboratory per week. Prerequisites: MATH1905 or Credit in MATH1005 or Credit in ECMT1010 or Credit in BUSS1020 Prohibitions: STAT2012 or STAT2004 or DATA2002 Assumed knowledge: STAT2911 Assessment: One 2-hour exam, assignments and/or quizzes, computer practical reports and one computer practical exam (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit is essentially an advanced version of STAT2012 with an emphasis on both methods and the mathematical derivation of these methods: Tests of hypotheses and confidence intervals, including t-tests, analysis of variance, regression - least squares and robust methods, power of tests, non-parametric methods, non-parametric smoothing, tests for count data, goodness of fit, contingency tables. Graphical methods and diagnostic methods are used throughout with all analyses discussed in the context of computation with real data using an interactive statistical package.

**DATA2002 Data Analytics: Learning from Data**

Credit points: 6 Teacher/Coordinator: Jean Yang Session: Semester 2 Classes: lecture 3 hrs/week; computer tutorial 2 hr/week Prerequisites: [DATA1001 or ENVX1001 or ENVX1002] or [MATH10X5 and MATH1115] or [MATH10X5 and STAT2011] or [MATH1905 and MATH1XXX (except MATH1XX5)] or [BUSS1020 or ECMT1010 or STAT1021] Prohibitions: STAT2012 or STAT2912 Assumed knowledge: (Basic Linear Algebra and some coding) or QBUS1040 Assessment: written assignment, presentation, exams Mode of delivery: Normal (lecture/lab/tutorial) day

Technological advances in science, business, engineering has given rise to a proliferation of data from all aspects of our life. Understanding the information presented in these data is critical as it enables informed decision making into many areas including market intelligence and science. DATA2002 is an intermediate course in statistics and data sciences, focusing on learning data analytic skills for a wide range of problems and data. How should the Australian government measure and report employment and unemployment? Can we tell the difference between decaffeinated and regular coffee ? In this course, you will learn how to ingest, combine and summarise data from a variety of data models which are typically encountered in data science projects as well as reinforcing their programming skills through experience with statistical programming language. You will also be exposed to the concept of statistical machine learning and develop the skill to analyze various types of data in order to answer a scientific question. From this unit, you will develop knowledge and skills that will enable you to embrace data analytic challenges stemming from everyday problems.

#### 3000-level units of study

###### Selective

**MATH3061 Geometry and Topology**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3001 or MATH3006 Assessment: One 2 hour exam, tutorial tests, assignments (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

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, emphasising 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). Topics include the classification of surfaces, map colouring, decomposition of knots and knot invariants.

**MATH3063 Nonlinear ODEs with Applications**

Credit points: 6 Teacher/Coordinator: Prof Leon Poladian Session: Semester 1 Classes: Three lectures, one tutorial per week Prerequisites: 12 credit points of Intermediate mathematics Prohibitions: MATH3003 or MATH3923 or MATH3020 or MATH3920 or MATH3963 Assumed knowledge: MATH2061 or [MATH2X21 and MATH2X22] Assessment: Class tests, Assignments, Final examination Mode of delivery: Normal (lecture/lab/tutorial) day

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.

**MATH3066 Algebra and Logic**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 6 credit points of Intermediate Mathematics Prohibitions: MATH3062 or MATH3065 Assessment: One 2 hour exam (60%), two assignments (15% each), peer review of each assignment (5% each). Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study unifies and extends mathematical ideas and techniques that most participants will have met in their first and second years, and will be of general interest to all students of pure and applied mathematics. It combines algebra and logic to present and answer a number of related questions of fundamental importance in the development of mathematics, from ancient to modern times. Classical and novel arithmetics are introduced, unified and described abstractly using field and ring axioms and the language of field extensions. Applications are presented, in particular the unsolvability of the celebrated classical construction problems of the Greeks. Quotient rings are introduced, culminating in a construction of the real numbers, by factoring out rings of Cauchy sequences of rationals by the ideal of null sequences. Axiomatics are placed in the context of reasoning within first order logic and set theory.

The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including sketches of proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem and the undecidability of First Order Logic.

The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including sketches of proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem and the undecidability of First Order Logic.

**MATH3076 Mathematical Computing**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour laboratory per week. Prerequisites: 12 credit points of MATH2XXX and 6 credit points from (MATH1021 or MATH1001 or MATH1023 or MATH1003 or MATH19X1 or MATH19X3 or MATH1906 or MATH1907) Prohibitions: MATH3976 or MATH3016 or MATH3916 Assessment: One 2 hour exam, assignments, quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit of study provides an introduction to Fortran 95/2003 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.

**MATH3078 PDEs and Waves**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3018 or MATH3921 or MATH3978 Assumed knowledge: [MATH2X61 and MATH2X65] or [MATH2X21 and MATH2X22] Assessment: One 2 hour exam, assignments, quizzes (100%). To pass MATH3078/3978, students must achieve satisfactory performance in the in-semester assessment component. Mode of delivery: Normal (lecture/lab/tutorial) day

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.

To pass MATH3078, students must achieve satisfactory performance in the in-semester assessment component in order to pass the unit of study.

To pass MATH3078, students must achieve satisfactory performance in the in-semester assessment component in order to pass the unit of study.

**MATH3961 Metric Spaces (Advanced)**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: Credit average or greater in 12 credit points of Intermediate Mathematics units Prohibitions: MATH3001 or MATH3901 Assumed knowledge: MATH2923 or MATH2962 Assessment: 2 hour exam, assignments, quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

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 (Adv)**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: Credit average or greater in 12 credit points of Intermediate Mathematics Prohibitions: MATH3062 or MATH3902 or MATH3002 Assumed knowledge: MATH2922 or MATH2961 Assessment: One 2 hour exam, homework assignments (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Students are advised to take MATH2968 before attempting this unit.

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.

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.

**MATH3963 Nonlinear ODEs with Applications (Adv)**

Credit points: 6 Teacher/Coordinator: Dr Robert Marangell Session: Semester 1 Classes: Three lectures, one tutorial per week Prerequisites: 12 credit points of Intermediate mathematics Prohibitions: MATH3003 or MATH3923 or MATH3020 or MATH3920 or MATH3063 Assumed knowledge: (MATH2961 or [MATH2921 and MATH2922]) and (MATH2962 or MATH2923) Assessment: Class tests, Assignments, Final examination Mode of delivery: Normal (lecture/lab/tutorial) day

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 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, linearisation, and analysis of asymptotic behaviour. The applications in this unit will be drawn from predator-prey systems, population models, chemical reactions, and other equations and systems from mathematical biology.

**MATH3968 Differential Geometry (Advanced)**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: A mark of 65 or above in MATH2961 or MATH2921 Prohibitions: MATH3903 Assumed knowledge: At least 6 credit points of Intermediate Advanced Mathematics or Senior Advanced Mathematics units Assessment: One 2 hour exam and 2 assignments (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

This unit is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, we 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 course aims 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.

**MATH3969 Measure Theory and Fourier Analysis (Adv)**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorials per week. Prerequisites: Credit average or greater in 12 credit points Intermediate Mathematics Prohibitions: MATH3909 Assumed knowledge: At least 6 credit points of (Intermediate Advanced Mathematics or Senior Advanced Mathematics units) Assessment: One 2 hour exam, assignments, quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

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. The Radon-Nikodyn Theorem provides a representation of measures in terms of a density. Probability theory is then discussed with topics including distributions and conditional expectation.

**MATH3974 Fluid Dynamics (Advanced)**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: Credit average or greater in 12 credit points of Intermediate Mathematics Prohibitions: MATH3914 Assumed knowledge: [MATH2961 and MATH2965] or [MATH2921 and MATH2922] Assessment: One 2 hour exam (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

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)**

Credit points: 6 Session: Semester 1 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: 12 credit points of MATH2XXX and [6 credit points from (MATH1923 or MATH1903 or MATH1933 or MATH1907), or a mark of 65 or above in (MATH1023 or MATH1003)] Prohibitions: MATH3076 or MATH3016 or MATH3916 Assessment: One 2 hour exam, assignments, quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

See entry for MATH3076 Mathematical Computing.

**MATH3977 Lagrangian and Hamiltonian Dynamics (Adv)**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: Credit average or greater in 12 credit points of Intermediate Mathematics Prohibitions: MATH2904 or MATH2004 or MATH3917 Assessment: One 2 hour exam and assignments and/or quizzes (100%) Mode of delivery: Normal (lecture/lab/tutorial) day

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.

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 PDEs and Waves (Advanced)**

Credit points: 6 Session: Semester 2 Classes: Three 1 hour lectures and one 1 hour tutorial per week. Prerequisites: Credit average or greater in 12 credit points of Intermediate Mathematics Prohibitions: MATH3078 or MATH3018 or MATH3921 Assumed knowledge: [MATH2X61 and MATH2X65] or [MATH2X21 and MATH2X22] Assessment: One 2 hour exam, assignments, quizzes (100%). To pass MATH3078 or MATH3978, students must achieve satisfactory performance in the in-semester assessment component. Mode of delivery: Normal (lecture/lab/tutorial) day

As for MATH3078 PDEs and Waves but with more advanced problem solving and assessment tasks. Some additional topics may be included.

MATH3X20, MATH3X10, MATH3X70, MATH3979, MATH3X11, MATH3X21, MATH3X22, MATH3X23, MATH3X24, MATH3X25, MATH3X26, MATH3X27, MATH3X28, MATH3X29, MATH3X12, MATH3X13, MATH3X14, MATH3X15, MATH3X16, MATH3X17, MATH3X18, MATH3X19, STAT3021, STAT3X22, STAT3X23, STAT3024 to be developed for offering in 2019.