Mathematics

N.S. Trudinger, BSc, UNE, MS PhD Stan., FAA, FRS
Professor and Head of Department

Introduction

Mathematics is one of the oldest, most useful, and most vital intellectual disciplines. It is concerned with the formulation and solution of many important theoretical and practical questions. The Babylonians used mathematics to organise their understanding of the heavens; the Americans to land a man on the moon. Despite its long history in the culture of most civilisations, mathematics is having a golden age this century, with an explosion in the variety of structures and concepts which may be organised mathematically and set to myriad uses. The symbiosis between the process of abstraction on the one hand and the need to achieve practical ends on the other has always been crucial to the progress of mathematics, and leads to great diversity in what mathematicians do. The development of the modern computer, stemming directly from the work of John von Neumann, one of this century’s most brilliant mathematicians, has totally changed the face of the subject and of society. The increasing pervasiveness of mathematics in every area of human activity, for example, the biological, economic, social and technological sciences, together with the enormous advances in the subject itself, indicate that the next 100 years will be exciting indeed.

The Department of Mathematics offers a wide range of courses catering to students who wish to study the subject either for its intrinsic interest, or for its applicable and applied aspects. In addition, the department offers many courses which are designed to complement other fields of study in the University, such as computer science, the physical and biological sciences, statistics, engineering, economics and the social sciences. We provide, for undergraduate students:

• major studies in mathematics, appropriate to students with a range of backgrounds and abilities, for those with a significant interest in the subject in its own right, either from a vocational standpoint, or because of its intrinsic interest;
• major studies in mathematics, appropriate to students with a range of backgrounds and abilities, which support other major areas of study (eg computer science, physics, statistics);
• courses for those whose major interest is in another area (eg  biology, computing, economics, geography) for which they to need to make use of mathematical tools and techniques.

Graduate courses for the Graduate Diploma, Masters and PhD degrees are available to students with appropriate background.

The information contained in this entry is necessarily brief. More complete information, including a fuller description of courses and details of lecturers and text books, may be found in the Department handbook, which is available free of charge from the Department office. Course information may be obtained at any time from the relevant staff member.

First Year Courses

Students entering the Department of Mathematics may undertake their mathematics courses at several different levels. The choice of level, and the amount of mathematics studied, will depend on the student’s

• degree of interest in the subject
• background in mathematics
• need for one or more years of mathematics to support the study of other disciplines.

The mathematics topics available to students in their first year of study are as follows:

Mathematics AM1, AM3: Major in Advanced Mathematics in the ACT, 2 unit Mathematics in NSW, or equivalent.

Mathematics AM4: ACT Mathematics T major or equivalent.

These mathematical modelling units are designed for students with a wide variety of backgrounds and will cover important areas in mathematics and its applications. The three units will cover respectively, Calculus, Discrete Systems and Data Analysis. Students will be able to enrol in these courses independently. The courses are designed for students whose main area of study is in the application of mathematics to various areas (eg. social, biological, physical, environmental sciences, computational science and economics). Extensive use will be made of computer packages and the emphasis will be on the applicability of mathematics for solving interesting problems. It is possible to mix-and-match these units with semester units in other areas (eg statistics).

Mathematics AA1, AA2: A satisfactory pass in a major-minor in Advanced Mathematics Extended in the ACT, 3 unit Mathematics in NSW, or equivalent.

These units form the basic sequence for a study of mathematics which is both applicable to other disciplines (in particular to the physical sciences, computer science, statistics or economics) and introductory to a wide range of later year courses in mathematics itself. Students with excellent results in a major in Advanced Mathematics in the ACT, or NSW 2 unit Mathematics, or the equivalent from elsewhere may be permitted to enrol in Mathematics AA1.

Should you choose the AA1/AA2 stream or units from the modelling stream AM1/AM3/AM4?

Students with an interest in mathematically oriented disciplines such as the physical sciences, applied mathematics and the theoretical aspects of computer science, statistics and mathematical economics should enrol in the AA1/AA2 stream. These units will, however, assume a sound understanding of secondary level mathematics, so a good result in (for example) the ACT Advanced Mathematics Extended or NSW 3 Unit Mathematics will be required. The modelling units will assume less prior knowledge; they provide a suitable mathematical background, for example, for the biological and social sciences, economics and information technology, as well as leading to a full 3 year sequence in mathematics.

Mathematics AA1H, AA2H: Double-major in the ACT Advanced Mathematics Extended or NSW 4 unit Mathematics. Students with excellent results in either the ACT Advanced Mathematics Extended major- minor, NSW 3 unit Mathematics or equivalent from elsewhere may be permitted to enrol.

These units are of a more advanced nature. They will appeal to students who are interested in why things are true, not simply in what is true. While they are also the first step towards an honours degree in mathematics, they are recommended for those with an appropriate background who intend doing advanced work in other mathematically based disciplines, such as physics or statistics, or in more mathematical areas of other sciences, engineering or economics. They are also recommended for students who, because of their interest and advanced background in mathematics from school, would not find the courses AA1, AA2 by themselves sufficiently challenging.

Later Year Courses

Most students who have completed 12 credit points of A units in mathematics have a wide range of options, including up to 32 credit points of B and C units in mathematics. By way of example, several possible continuations of mathematics into second and third year are outlined below, according to the first year units completed. These are examples only, and many other viable sequences can be constructed, depending on the student’s interests. Students should consult the Departmental handbook, or speak to the coordinators of second and third year courses, for more detailed information and examples.

Mathematics AM4/AM3

Mathematics B63 and B16 are now available.

Mathematics AM1/AM3

Mathematics B16, B61, B62, B63, C03, C07 are now available. B63 and B16 are particularly recommended for students interested in computer science.

Mathematics AA1/AA2

At least 24 credit points of mathematics at B level, followed by at least 24 credit points at C level may now be taken, provided appropriate prerequisites are observed. Note that a grade of credit or better in Mathematics AA2 is required for progression to B level applied mathematics honours units.

• Mathematics B16, B61, B63; C03, C07: students of computer science or mathematics;
• Mathematics B13, B14, B16, B27; C03, C07, C10, C11, C15: students of the physical sciences, mathematics or statistics. (Those with interests in mathematical statistics and probability should include B27, if you are interested in the applications of mathematics to finance and the stock market, include B14, B27 and C15).
• Mathematics B13, B27, B14, B16; a selection of C01, C03, C07, C10, C11: a strong program with a pure or applied orientation for a student doing physics and/or at least 44 credit points of mathematics in their degree.
• Mathematics B23H, B14H, B34H; C11H, C31H, C32H, C34H: the basis of an honours program in applied mathematics.
• Mathematics B13, B27 (or B23H), B14 (or B14H), B67; C52H, C53H, C54H: the basis of an astrophysics program.

Mathematics AA1H/AA2H

• These students have almost the full range of second and third year courses of mathematics to choose from.
• Mathematics B21H and Mathematics B23H are recommended for all students wishing to progress to honours in mathematics. Students who complete only one of these will be able to continue, but may find their choice of units in later years restricted.
• Students whose main interest is pure mathematics should also complete at least Mathematics B28H and B30H; and students whose main interest is in applied mathematics should also complete at least Mathematics B14H and 34H. Other combinations are possible, and students are encouraged to consult the second year coordinator.

A diagram of prerequisite links is shown on the next page; note that it is not totally exhaustive. Note that these formal prerequisites may be waived by the Head of Department on the basis of cognate background knowledge: in this regard, students should consult the year coordinator in the first instance.

Distinguished Scholar Program

Distinguished Scholars and other outstanding undergraduate students may be invited to participate in special courses which extend and develop their particular interests. Units offered in this category will be approved for one year only under the unit entries Mathematics B41H-49H and Mathematics C41H-49H below.

Honours Level Courses

Honours-level courses in the first three years may be taken as part of the pass degree, but are also preparatory to the degree with honours. An honours degree is of great value to those who wish to go more deeply into the subject; in particular to those who wish to proceed to professional work in mathematics or the theoretical sciences. Note also that many students completing an honours degree in mathematics over recent years have proceeded to graduate work in a wide range of fields including biology, computer science, economics, engineering, linguistics, philosophy and physics.

Honours Program in Mathematics

This program is organised within the School of Mathematical Sciences, with support from the Centre for Mathematics and its Applications (CMA). The Centre for Resource and Environmental Studies (CRES) also provides support.

Entry to the fourth (honours) year is at the discretion of the Head of Department. Faculty requirements for the pass degree must be met and, in addition, 48 Group B and C credit points in mathematics at honours level, with a minimum grade of credit, must have been completed. Students must also have completed sufficient in the way of prerequisites in mathematics to enable them to pursue an approved course of study in their fourth year.

Proposals for combined honours degrees in mathematics and in another subject (such as computer science, philosophy, physics, statistics, or theoretical physics) will be considered in consultation with the department concerned.

Honours Program in Astronomy and Astrophysics

Students may enrol in an honours program in Astronomy and Astrophysics in either of the Mathematics or the Physics departments. Theoretically-oriented students will do so by enrolling for Mathematics IV honours, and will pursue a program under the same rules as other Mathematics IV honours students, although parts of the program will be provided by staff from the Astrophysics Theory Centre and the Mount Stromlo and Siding Springs Observatory. More experimentally-oriented students will enrol in Physics IV honours, and should consult the Department of Physics for further details.

Details of Units

Mathematical Modelling with Calculus    MATH1003

(Mathematics AM1)    (6cp) Group A

First semester
48 lectures and ten 2-hour laboratory sessions

Prerequisite: ACT Advanced Mathematics major or NSW 2 unit Mathematics or equivalent.

Syllabus: Introduction to differential equations and related mathematics and their use in mathematical modelling. Emphasis will be placed on developing facility, technique and use in applications. Modelling of processes and phenomena which occur in economics and the physical, environmental and life sciences will be used as a vehicle throughout.

Topics to be covered include: Functions and graphs, the transcendental functions, approximation, differential equations, techniques and uses of differentiation and integration.

Mathematical Modelling with Discrete Systems    MATH1005

(Mathematics AM3)    (6cp) Group A

Second semester
48 lectures and ten 2-hour laboratory sessions

Prerequisite: ACT Advanced Mathematics major or NSW 2 unit Mathematics or equivalent.

Syllabus: Introduction to finite mathematics and its use in mathematical modelling. Emphasis will be placed on developing facility, technique and use in applications. Modelling of processes and phenomena which occur in economics and the physical, environmental and life sciences will be used as a vehicle throughout.

Topics to be covered include: Combinatorics and counting, recurrence relations and generating functions, graph theory, matrix arithmetic, logic and finite set theory, relations and functions.

Mathematical Modelling of Data    MATH1006

(Mathematics AM4)    (6cp) Group A

First Semester
48 lectures and ten 2-hour laboratory sessions.

Prerequisite: ACT Mathematics T major or equivalent.

This course assumes a basic knowledge of algebra, and develops the mathematical tools and understanding necessary for data analysis. Extensive use will be made of mathematical packages and there will be computer-aided examples of data acquisition and interpretation. Applications will be drawn from a wide range of disciplines including astronomy, biology, engineering, geography, meteorology, physics, psychology and sociology.

Syllabus: Functions: polynomials, exponential, logarithm, and trigonometric functions. Introduction to calculus: finding maxima and minima and graphing functions of one variable. Functions of two variables: level curves and surfaces. Elementary probability theory: binomial, Poisson and normal distributions. Error of observation: accidental and systematic error, least squares, error in compound quantities. Model comparison, distribution function fitting. Representation of data by Fourier Series.

Advanced Mathematics and its Applications I    MATH1013

(Mathematics AA1)    (6cp) Group A

First semester
48 lectures and ten 2-hour laboratory sessions
Incompatible with Mathematics AM1, AM2, A01, A11, A21H and ENGN1212

Prerequisite: A satisfactory result in ACT Advanced Mathematics Extended major-minor or the NSW 3 unit course or equivalent.

Syllabus
Calculus: (approx 24 lectures) Differentiation techniques such as chain rule and implicit differentiation, applications including extreme value problems and linear approximation. Taylor polynomials and power series-calculation of transcendental functions, Intermediate and Mean Value theorems. Differential equations — particularly solution and use of first order equations such as the logistic equation, predator-prey models. Numerical techniques for integration. Curves and surfaces in three dimensional space.

Linear Algebra: (approx 24 lectures) Complex numbers. Solution of linear equations, vector and matrix equations and matrix algebra. Emphasis on understanding and using algorithms and applications.

Advanced Mathematics and its Applications II    MATH1014

(Mathematics AA2)    (6cp) Group A

Second semester
48 lectures and ten 2-hour laboratory sessions
Incompatible with Mathematics AM1, AM2, A12, A22H and ENGN1222

Prerequisite: Mathematics AA1

Syllabus
Calculus: Integration and techniques of integration. Functions of several variables — visualisation, continuity, partial derivatives and directional derivatives.

Linear Algebra: Theory and application of the vector space Rn. Vector spaces. Linear independence, bases and dimension, eigenvalues and eigenvectors, orthogonality and least squares.

Modelling: Mathematical techniques from Mathematics AA1 and this course will be applied to processes and phenomena which occur in economics and the physical, environmental and life sciences.

Advanced Mathematics and its Applications Honours I    MATH1115

(Mathematics AA1H)    (6cp) Group A

First semester
48 lectures and 20 hours of laboratory and tutorial sessions
Incompatible with Mathematics AA1, A21H, A11, AM1, AM2, A01 and ENGN1212

Prerequisite: A satisfactory pass in the ACT Advanced Mathematics Extended double major or NSW 4 unit Mathematics or the equivalent. Students with excellent results in either the ACT Advanced Mathematics Extended major-minor, NSW 3 unit Mathematics, or equivalent, may be permitted to enrol.

Syllabus: There will be 4 lectures each week, of which 2 will be from the calculus component of AA1H. This is the more applied part of the course. In addition, there will be 2 lectures a week on more theoretical material, basic to advanced work in mathematics and to applications of mathematics at a sophisticated level in other disciplines. The foundations of calculus and an introduction to general vector spaces will be treated.

Advanced Mathematics and its Applications Honours II    MATH1116

(Mathematics AA2H)    (6cp) Group A

Second semester
48 lectures and 20 hours of laboratory and tutorial sessions
Incompatible with Mathematics AA2, A22H, A12, AM2, A02 and ENGN1222

Prerequisite: Mathematics AA1H

Syllabus: About half of the lectures will be in common with the AA2 class. There will also be more theoretical material on the development of calculus and some more advanced material related to that treated in the AA2 part of the course.

Engineering Mathematics 3    ENGN2212

(6cp) Group B

First Semester
36 lectures and 10 tutorials

Prerequisite: Engineering Mathematics 2.

Incompatible with Mathematics B11, B13, B23H.

Syllabus
Ordinary differential equations: (24 lectures in common with Mathematics B13) An introduction to the theory of differential equations and systems of differential equations. Techniques of solution, Laplace transforms. Qualitative behaviour of solutions, applications.

Vector calculus: (12 lectures) Scalar and vector fields, vector differential operators, line, surface and volume integrals, theorems of Gauss, Green and Stokes. Application to simple problems.

Ordinary Differential Equations    MATH2013

(Mathematics B13)    (4cp) Group B

First Semester
24 lectures and 5 tutorials

Prerequisite: MATH1014 or equivalent. Incompatible with Mathematics B23H and Engineering Mathematics 3.

Syllabus: An introduction to the theory of differential equations, difference equations and systems of differential equations. Techniques of solution, Laplace transforms. Qualitative behaviour of solutions, applications.

Partial Differential Equations    MATH2014

Partial Differential Equations Honours    MATH2114

(Mathematics B14/H)    (4cp) Group B

Second Semester
24 lectures and 5 tutorials
These units will be taught in the same lectures. They will be assessed independently. Those doing MATH2114 will be expected to do extra work.

Prerequisite: MATH2013 (at credit or better for MATH2114). Incompatible with Mathematics C09, C09H.

An introduction to some of the important partial differential equations of mathematics, and methods for their solution and use.

Syllabus: Heat equation, Laplace’s equation, wave equation. Separation of variables in cartesian coordinates, Fourier series. Solution of boundary value problems for each class of equation. Separation in polar coordinates, Bessel functions and spherical harmonics.

Introduction to Algebraic Systems    MATH2016

(Mathematics B16)    (4cp) Group B

Second Semester
24 lectures and 5 tutorials

Prerequisite: Two A points = 12 credit points of Mathematics. Incompatible with Mathematics B03, B04.

An introduction to the use of abstract methods in mathematics, using algebraic systems which play an important role in many applications of mathematics.

Syllabus: Groups: definitions, examples, abelian and cyclic groups. Subgroups and Lagrange’s theorem. Normal subgroups and quotient groups. Fields and their properties, examples. Finite fields. Abstract vector spaces, subspaces. Linear dependence, linear transformations and matrices. Eigenvalues and diagonalisation of square matrices.

Analysis and Algebra Honours    MATH2021

(Mathematics B21H)    (8cp) Group B

First Semester
48 lectures, tutorials by arrangement

Prerequisite: Mathematics AA2H at credit level or better.

Syllabus
Analysis: (32 lectures) Introductory logic and set theory. Axiomatic approach to the field of real numbers. Metric spaces and basic topological notions. Convergence and completeness. Contraction mapping theorem. Bolzano-Weierstrass theorem and sequential compactness. Continuity of functions. Uniform convergence.

Algebra: (16 lectures) Definition of group, examples. Consequences of the group axioms, exponents, index laws, order of an element. Subgroups, cosets, Lagrange’s theorem. Normal subgroups and quotient groups. Homomorphisms, kernels, isomorphisms, automorphisms. Cayley’s theorem. Permutation groups, cycle structure, alternating groups. Conjugacy and the class equation, Cauchy’s theorem. Sylow’s theorems.

Applied Mathematics Honours    MATH2023

(Mathematics B23H)    (8cp) Group B

First Semester
48 lectures, tutorials by arrangement
The lectures are in common with those for MATH2013 and MATH2027. Assessment will be independent. Extra work will be required of MATH2023 students.

Prerequisite: Mathematics AA2 at credit level or better. Incompatible with Mathematics B11, B13, Engineering Mathematics 3, B27, B31H, B32H and B65.

Syllabus
Ordinary differential equations: As for Mathematics B13.
Advanced Calculus and differential equations: As for Mathematics B27.

Advanced Calculus and Differential Equations    MATH2027

(Mathematics B27)    (4cp) Group B

First semester
24 lectures, tutorials by arrangement

Prerequisite: Mathematics AA2. Incompatible with Mathematics B11, B23H, B31H and Engineering Mathematics 3

Corequisite: Mathematics B13.

Syllabus
Advanced Calculus: Differentiability of functions of several variables, differentials, chain rule for partial differentiation. Multiple integrals, coordinate transformations, Jacobians. Scalar and vector fields, vector differential operators, line, surface and volume integrals, conservative fields and path independence, theorems of Gauss, Green and Stokes.

Differential Equations: Solution of differential equations using power series, Bessel and Legendre functions. Qualitative behaviour of solutions, Sturm theorems. Autonomous differential equations, critical points, linear stability theory, limit cycles, bifurcation of solutions.

Abstract Algebra Honours    MATH2028

(Mathematics B28H)    (4cp) Group B

Second Semester
24 lectures, tutorials by arrangement

Prerequisite: Mathematics B21H at credit level or better.

Syllabus: Definition of a ring, examples of rings. Special types of rings, integral domains, fields. Homomorphisms, ideals, quotient rings, isomorphism theorems. Commutative rings: maximal ideals and their quotients, field of fractions of an integral domain; Euclidean domains, principal ideal domains, division algorithm, unique factorisation; the ring of Gaussian integers, Fermat’s Little Theorem; polynomial rings over fields, division algorithm, unique factorisation; polynomials over the rational field, primitive polynomials, Eisenstein’s criterion; polynomial rings over commutative rings, unique factorisation domains. Modules, submodules, quotient modules, homomorphisms, isomorphism theorems; modules over Euclidean rings, cyclic decomposition theorem.

Real Analysis and Metric Spaces Honours    MATH2030

(Mathematics B30H)    (4cp) Group B

Second Semester
24 lectures, tutorials by arrangement

Prerequisite: Mathematics B21H at credit level or better.

Syllabus: Existence and uniqueness of solution for first order differential equations (Picard’s theorem). Connectedness. Compactness in terms of open coverings. Ascoli-Arzela theorem, Peano theorem for existence of solution of first order differential equations. Functions of several variables: Taylor’s theorem, chain rule, inverse and implicit function theorems.

Introduction to Computational Mathematics    MATH2034

(Mathematics B34H)    (4cp) Group B

Second Semester
24 lectures, tutorials by arrangement

Prerequisite: Mathematics B23H.

Much scientific computing is concerned with the solution of differential equations. The objective of this course is to use differential equations as a focus for an introduction to computational mathematics.

Syllabus: Numerical methods for solving ordinary and partial differential equations: theoretical treatment of questions of accuracy, efficiency and stability. Runge-Kutta methods, polynomial interpolation and multistep methods, boundary value problems and associated algorithms for the solution of linear systems. Finite difference methods and an introduction to algorithms for large linear systems.

Mathematics B41H-49H    MATH2041-2049

Topics may be offered under these code numbers from time to time for the benefit of distinguished scholars and other outstanding undergraduate students. They will be given at second year honours level. Entry will be by invitation of the Head of Department.

Applications of Mathematical Modelling    MATH2061

(Mathematics B61)    (4cp) Group B

First Semester
24 lectures and 5 tutorials

Prerequisite: Two A points = 12 credit points of Group A Mathematics units including either AM1 or AA1.

Syllabus: This course is designed for students from any discipline who are interested in an understanding of mathematical models. Drawing on a basic knowledge of calculus, models from many subjects (eg biology, archaeology, geography, physics) will be formulated and then analysed and validated. The use of Maple software will be integrated into these procedures. Case studies will be included and students with particular interests will have the opportunity to study one from their own discipline.

Non-linear Modelling and Chaotic Behaviour    MATH2062

(Mathematics B62)    (4cp) Group B

Second Semester
24 lectures and 5 tutorials

Prerequisite: Two A points = 12 credit points of Mathematics including either AM1 or AA1. Incompatible with Mathematics C08.

This course is an introduction to non-linear phenomena, using mathematics already developed in first year, particularly those aspects which are ubiquitous in the modern understanding of dynamical systems.

Syllabus: Regular and chaotic behaviour in non-linear systems, characterisation and measures of chaos, stability and bifurcation, the period doubling and intermittency routes to chaos. Relation of fractal structures to simple non-linear systems.

Graphs, Games and Machines    MATH2063

(Mathematics B63)    (4cp) Group B

First Semester
24 lectures and 5 tutorials

Prerequisite: Mathematics AM3 or Mathematics AA2. Incompatible with Mathematics B01, B06.

Syllabus: Algebraic operations on sets, laws in general and important particular laws, consequences. Particular algebraic systems: Boolean algebra (logic), set theory, matrix algebra, integers mod n, permutations, floating point arithmetic. Game theory. Graph theory: problems arising from applications, algorithms and their specification, examples from the sciences, social sciences, engineering and computer science.

Theoretical Astrophysics    MATH2067

Theoretical Astrophysics Honours    MATH2167

(Mathematics B67/H)    (4cp) Group B

Second Semester
24 lectures and 5 tutorials

Prerequisite: Either MATH2013 and MATH2027, or MATH2023. PHYS2023 is strongly recommended.

MATH2067 and MATH2167 will be taught in the same lectures. They will be assessed independently. Students doing MATH2167 will be expected to do extra work. Both are suitable preparation for subsequent astrophysics units but students wishing to keep open the option of doing honours in maths should do MATH2167.

Syllabus: This unit shows how it is possible to mathematically model a wide range of astrophysical objects. The emphasis is on approximation methods and basic physical principles, rather than detailed mathematical simulations. Topics include: an historical account of the nature of the universe, gravitational potential theory, elementary statistical mechanics of gases, the Maxwell-Boltzmann distribution, the pressure of non-degenerate and degenerate gasses, relativistic effects, polytropic models of stars, the structure of white dwarfs, Newtonian cosmology, curved spaces and redshift.

Scientific Computation    MATH3103

(Mathematics C03)    (4cp) Group C

First semester
24 lectures and ten 2-hour laboratory sessions.

Prerequisite: 8 credit points of B level mathematics. Incompatible with COMP3067.

Syllabus: The object of this course is to introduce the use of high quality mathematical software in the solution of sophisticated, yet standard, computational problems. The algorithms underlying appropriate, computational techniques will be described. Emphasis will be placed on the development of efficient techniques for using standard, commercially available software. Advantages and limitations of using such mathematical software will be demonstrated by using it on real-life problems.

The mathematics needed to model the problems discussed will include: solving large systems of linear equations, optimisation techniques and systems of differential equations, alone and in combination.

Number Theory    MATH3001

Number Theory Honours    MATH3101

(Mathematics C01/H)    (4cp) Group C

First semester
24 lectures and 5 tutorials
These units will be taught in the same lectures. They will be assessed independently. Those doing C01H will be expected to do extra work.

Prerequisite: Mathematics B16 (for C01) or B28H (for C01H).

Syllabus: Euclidean algorithm; congruences; prime numbers, highest common factor, prime factorisation; diophantine equations; sums of squares; chinese remainder theorem, Euler’s function; continued fractions, Pell’s equation; quadratic residues, reciprocity.

An Introduction to Modern Geometries    MATH3106

Coding Theory    MATH3006

(Mathematics C06)    (4cp) Group C

Second semester
24 lectures and 5 tutorials
Modern Geometries will be offered in even numbered years and Coding Theory in odd numbered years, subject to staff availability and student demand.

Prerequisite: Mathematics AA2 and Mathematics B16.

Syllabus: Geometry This unit will review basic Euclidean plane geometry and its axioms. Hyperbolic geometry: axioms, basic theorems and a model. Transformations giving rise to other non-Euclidean geometries. Projective geometry: axioms and a model; perspectivities and projectivities; theorems of Desargues, Pascal and Pappus.

Coding Theory: The theory of polynomial rings and fields, with applications to error correcting codes.

Optimisation and Linear Programming    MATH3107

(Mathematics C07)    (4cp) Group C

First semester
24 lectures and 5 tutorials

Prerequisite: Mathematics AA2 or B16. Incompatible with Mathematics C04.

Syllabus: A treatment of mathematical optimisation techniques with emphasis on linear programming. Extensions and applications may include transportation and transhipment problems and game theory. The linear programming package MAPLE will be used extensively.

Systems and Control Theory    MATH3110

Systems and Control Theory Honours    MATH3210

(Mathematics C10/H)    (4cp) Group C

Second semester
24 lectures and 5 tutorials
These two units will be taught in the same lectures. They will be assessed independently. Those doing MATH3210 will be expected to do extra work.

Prerequisite: B13 (for C10), or B23H (for C10H). Incompatible with Mathematics C13 and C33H.

Syllabus: An introduction to systems and control theory. Topics may include: classical linear systems described by differential and difference equations, feedback systems, stability, optimal LQP-control, dynamic programming.

Complex Calculus    MATH3011

Complex Analysis Honours    MATH3111

(Mathematics C11/H)    (4cp) Group C

First semester
24 lectures and 5 tutorials
These units will be taught in the same lectures. They will be assessed independently. Those doing MATH3111 will be expected to do extra work.

Prerequisites: Mathematics B27 or B65 (for C11), B30H and B23H (for C11H). Incompatible with Mathematics B20 and B24H.

Syllabus: This unit introduces the calculus of functions of one complex variable. The content will include: analytic functions, contour integration, power series, Laurent series, calculus of residues, conformal mapping.

Mathematics of Finance     MATH 3015

Mathematics of Finance Honours     MATH3115

(Mathematics C15/H)    (4cp) Group C

Second semester
24 lectures, tutorials by arrangement
These units will be taught in the same lectures. They will be assessed independently. Those doing C15H will be expected to do extra work.

Prerequisite: Mathematics B14 for C15, Mathematics B14H at credit or better for C15H.

Syllabus: Heuristic introduction to discrete and continuous random processes. Basic notions of options and the need for a theory. Stochastic processes and associated differential equations, Ito’s lemma. The Black-Scholes model. The diffusion equation and its use in Black-Scholes. Variations on the Black-Scholes model, American and European options. Numerical procedures including binomial methods.

Measure Theory and Functional Analysis Honours I    MATH3021

(Mathematics C21H)    (4cp) Group C

First semester
24 lectures, tutorials by arrangement

Prerequisite: Mathematics B30H at credit level or better.

Syllabus: Fundamental concepts necessary for studying integration theory and probability: abstract measures, measurable functions and integration, convergence theorems, Lp-spaces. An introduction to Banach and Hilbert spaces.

Measure Theory and Functional Analysis Honours II    MATH3022

(Mathematics C22H)    (4cp) Group C

Second semester
24 lectures, tutorials by arrangement

Prerequisite: Mathematics C21H at credit level or better.

Syllabus: Measure theory: Fubini’s theorem, Radon-Nikodym theorem, Riesz representation theorem. Functional analysis: Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, duality theory, introduction to spectral theory.

General Topology and Set Theory     MATH3223

(Mathematics C23H)    (4cp) Group C

First semester
24 lectures, tutorials by arrangement

Prerequisite: Mathematics B30H at credit level or better.

Syllabus: Set theory, topological spaces, continuity, compactness and connectedness. Applications to contemporary mathematics.

Associative Algebras and Modules    MATH3025

Galois Theory    MATH3125

(Mathematics C25H)    (4cp) Group C

First semester
24 lectures, tutorials by arrangement
Associative Algebras and Modules will be offered in even numbered years and Galois Theory in odd numbered years, subject to staff availability and student demand.

Prerequisite: Mathematics B28H at credit level or better.

Syllabus: Associative algebras: Further group theory. The structure of associative algebras and modules, with applications to the representation theory of finite groups.

Galois Theory: Fields, field extensions, normal extensions, splitting fields, separable extensions. Abelian groups, soluble groups. Galois Theorem, solubility of equations by radicals.

Differential Geometry    MATH3027

Partial Differential Equations    MATH3127

(Mathematics C27H)    (4cp) Group C

A course in differential geometry or partial differential equations, normally alternating between the two in successive years but depending on student demand and staff availability. It is anticipated that partial differential equations will be offered in 1999.
First semester
24 lectures, tutorials by arrangement
Offered in association with CMA.

Prerequisite: Mathematics B30H at credit level or better. MATH3127 is incompatible with MATH3024.

Syllabus (Differential Geometry): Surfaces, tangent spaces, vector fields, differential forms, Gauss-Bonnet theorem, Riemannian manifolds. Applications to geometric problems.

Syllabus (Partial Differential Equations): Introduction to the theory of Partial Differential Equations. The course will discuss the three main classes of equations, elliptic, parabolic and hyperbolic.

Topics will include fundamental solutions, maximum principles, regularity (smoothness) of solutions, variational problems, Holder and Sobolev spaces.

Complex Analysis    MATH3228

Foundations of Mathematics    MATH3128

(Mathematics C28H)    (4cp) Group C

Second semester
24 lectures, tutorials by arrangement
Complex Analysis will be offered in even numbered years, and Foundations of Mathematics in odd numbered years, subject to staff availability and student demand. Offered in association with CMA.

Prerequisite: Complex Analysis: Mathematics B20 and Mathematics B30H, at credit level or better. Foundations: Mathematics B28H and B30H at credit level or better.

Syllabus: Foundations: First order logic, Turing machines, Gdels incompleteness theorem, axiomatization of set theory, model theory.

Complex Analysis: Review of complex numbers and complex differentiability. Elementary conformal mapping, integration, theorem and formula, convergence of holomorphic function, Cauchy integral theorems, representation of a holomorphic function by its Taylor series, isolated singularities, residues, residue theorem and its application to real integration. Argument principle, Runge’s Theorem, monodromy theorem, Riemann surfaces, theorems of Picard, Weierstrass and Mittag-Leffler.

Probability Honours 1    MATH3029

(Mathematics C29H)    (4cp) Group C

First semester
Offered in 1999 subject to staff availability and student demand
24 lectures, tutorials by arrangement
Offered in association with CMA.

Prerequisite: Mathematics B30H at credit level or better.

Syllabus: Probability spaces and random variables, modes of convergence, weak and strong laws of large numbers, the central limit theorem.

Non Linear Dynamics and Chaos    MATH3131

Fluid Dynamics    MATH3231

(Mathematics C31H)    (4cp) Group C

First semester
24 lectures, tutorials by arrangement
Non Linear Dynamics and Chaos will be offered in even numbered years, and Fluid Dynamics in odd numbered years, subject to staff availability and student demand.
Not offered in 1999

Prerequisite: 16 credit points of B level mathematics units at credit level or better, including Mathematics B23H.

Corequisite: Mathematics C11.

Syllabus
Fluid Dynamics: General equations of motion: conservation laws, Euler’s and Bernoulli’s equations, boundary conditions. Flow in two dimensions: potential functions, complex variable methods. Irrotational flow in three dimensions: general principles. Dynamics of real fluids: viscosity, Reynold’s number, boundary layers. Introduction to compressible flow: sound waves, shock waves, normal shock relations, hodograph equations.

Non Linear Dynamics: and Chaos: one dimensional and two dimensional maps, chaos and fractals. Chaos in two dimensional maps, chaotic attractors. Differential equations, periodic orbits, limit sets, chaos in differential equations. Stable manifolds, bifurcations, cascades and crises.

Introduction to the Finite Element Method    MATH3032

Computational Fluid Dynamics    MATH3132

(Mathematics C32H)    (4cp) Group C

Second semester
24 lectures, tutorials by arrangement
The Finite Element Method will be offered in even numbered years and Computational Fluid Mechanics in odd numbered years, subject to staff availability and student demand.

Prerequisite: 16 credit points of mathematics at B level including MATH2023 and MATH2034, or credit or better in MATH2015.

Syllabus
Introduction to the Finite Element Method: This method is the most widely used computational technique for engineering design and analysis. This course will develop the basic mathematical theory of the method. Numerical computations will be undertaken using MATLAB.

Computational Fluid Dynamics: Mathematical structure essential to the development, analysis and use of numerical methods to solve the equations of compressible fluid flow and non-linear systems of conservation laws will be introduced. Particular attention will be given to the effect of shock waves and other non-linear behaviour.

Mathematics C41H-C49H    MATH3041-49

(Topics in Mathematics)    (4cp) Group C

Under these codes topics may be offered from time to time to take advantage of the expertise of visitors to the university or the special interests of staff in the wider university. They will be taught at third year honours level. Entry will be at the discretion of the Head of Department.

Relativity    MATH3050

(Mathematics C50H)    (4cp) Group C

First Semester
24 lectures, tutorials by arrangement

Prerequisite: Either MATH2013 and MATH2027, or MATH2023.

Syllabus: Development of the mathematical theory of special and general relativity with applications to compact stars and black holes.

Galaxies and Cosmology    MATH3052

(Mathematics C52H)    (4cp) Group C

Second semester
Offered in association with MSSSO
24 lectures, tutorials by arrangement

Prerequisite: PHYS2023; MATH2067 or MATH2167. PHYS3001 would also be useful.

Syllabus: Classification of galaxies. Elementary galactic dynamics. Mass distribution in galaxies. Dark matter. The expanding universe. Cosmological models.

Stellar Astrophysics    MATH3053

(Mathematics C53H)    (4cp) Group C

First semester
Offered in association with MSSSO
24 lectures, tutorials by arrangement

Prerequisites: PHYS2023; MATH2067 or MATH2167. PHYS2020 or PHYS2022.

Syllabus: Properties of radiation. The radiative transfer equation. Stellar atmospheres. Opacities and nuclear energy sources. Stellar structure and evolution. Stellar pulsations.

High Energy Astrophysics    MATH3054

(Mathematics C54H)    (4cp) Group C

Second Semester
Offered in association with MSSSO
24 lectures, tutorials by arrangement

Prerequisite: PHYS2023; MATH2067 or MATH2167. PHYS2016. PHYS3001 would also be useful.

Syllabus: The gas dynamic equations for compressible flows. Eulerian and Lagrangian formulation. Solutions in simple cases. The Bernoulli integral. Astrophysical shocks. Magnetic fields. The Kelvin-Helmholtz and Rayleigh-Taylor instabilities. Wind theory. Astrophysical jets.

Environmental Mathematics    MATH3134

(Mathematics C34H)    (4cp) Group C

Second semester
24 lectures, tutorials by arrangement
Offered in association with CRES.

Prerequisite: 16 credit points of B level mathematics units at credit level or better, including Mathematics B23H.

Syllabus: Examination of the major types of models used to represent environmental systems and how they are constructed. The major emphasis will be on hydrological systems and the basic processes and physical processes will be examined. Case studies and project assessment will cover catchment hydrology, soil physics, subsurface hydrology and land surface- atmosphere interactions.

Algebraic Topology    MATH3060

(Mathematics C60H)    (4cp) Group C

Second semester
24 lectures, tutorials by arrangement

Prerequisites: MATH3223

Syllabus: An introduction to homotopy and homology theory, in which algebraic structures are employed to study topological problems.