Postgraduate courses

This course prepares students for postgraduate study in mathematics and statistics education. Its emphasis is on workshops in the key research skills required by students working at this level. It will cover a range of research issues and techniques.

Recommended preparation
MATHS 302: Teaching and Learning Mathematics and one of MATHS 702-709 is recommended.

Availability: SS 2012 C
Points: 15
Coordinator: Bill Barton

This course considers such issues as the historical development of mathematics and statistics curricula, current New Zealand and international trends, the relationship between curriculum and assessment, and the politics of curriculum development.

Availability: tbc
Points: 15
Coordinator: Judy Paterson

An analysis of theoretical perspectives that inform research in mathematics education, with a focus on learning theories, both social and psychological, and their implications for teaching and learning in mathematics.

Recommended preparation
MATHS 302: Teaching and Learning Mathematics is recommended.

Availability: S1 2012 C
Points: 15
Coordinator: Judy Paterson

This course will examine mathematics teaching and learning from a sociological perspective. Topics covered will include gender differences in mathematics, grouping students by ability versus mixed ability teaching and the performance of students from working class and ethnic minority backgrounds. Equity issues will be a central focus, and we will discuss the ways in which sociological ideas complement other approaches to research in mathematics education.

Recommended preparation
MATHS 302: Teaching and Learning Mathematics is recommended.

Availability: tba
Points: 15
Coordinator: Judy Paterson

This course focuses on the use of computers and calculators in mathematics education, with a strong focus on both theoretical and practical aspects of the use of computers in the mathematics classroom. The pedagogical implications of computers for the present and the future of mathematics education are discussed.

Availability: tbc
Points: 15
Coordinator: Michael Thomas

A new course designed to examine mathematical processes such as problem-solving, argumentation and proving, conjecturing, abstracting and generalising, and modelling. The focus will be on senior secondary and undergraduate mathematics.

Availability: tbc
Points: 15
Coordinator: Judy Paterson

This course is a special topic 15-point paper that is available for personal study in a particular area of interest in mathematics education. You will have a supervisor for your topic. These courses require students do individual study in a particular area of interest in mathematics education. They are offered in every semester, including Summer School. Please consult the Department of Mathematics before enrolment.

Availability: By arrangement
Points: 15 for MATHS 707-710, 30 for MATHS 711
Coordinator: Judy Paterson

Recent theoretical perspectives on the teaching and learning of school and university mathematics are linked to the learning of either calculus or algebra. The focus is on the mathematics content, applications, and effective learning at school and university.

Recommended preparation
Students taking this course should normally have studied mathematics or statistics at Stage II.

Availability: S2 2012 C
Points: 15
Coordinator: Michael Thomas

This course covers a wide range of research in statistics education at school and tertiary level. An examination of the issues involved in statistics education in the curriculum, teaching, learning, technology and assessment areas is covered.

Availability: 2013
Points: 15
Coordinator: Maxine Pfannkuch

This course covers the foundations of pure mathematics, formalising the notions of a “mathematical proof” and “mathematical structure” through predicate calculus and model theory. It explores the limits of these formalisations such as those posed by Gödel’s Incompleteness theorems, and it includes a study of axiomatic set theory.

Prerequisites
MATHS 315: Mathematical Logic or PHIL 305: Advanced Logic is required.

Availability: 2013
Points: 15
Coordinator: Sina Greenwood

This course gives a broad introduction to various aspects of elementary, analytic, algebraic and computational number theory and its applications. The material includes classical topics such as divisibility, prime numbers and their distribution, primitive roots modulo p, linear and quadratic congruencies and representing integers as sums of squares. The course also covers basic algebraic number theory and diophantine equations.

Prerequisites
B+ in either MATHS 328: Algebra and Applications or MATHS 320: Algebraic Structures is required.

Availability: S2C
Points: 15
Coordinator: Steven Galbraith

This course covers theory and applications of combinatorial graphs (networks), block designs, and error-correcting codes. Topics include graph connectivity, trees, colourings, embeddings, digraphs, matchings, incidence matrices, eigenvalue methods, Steiner systems, perfect and linear codes.

Prerequisites
B+ pass in MATHS 326: Combinatorial Computing or MATHS 320: Algebraic Structures is required.

Availability: S1C
Points: 15
Coordinator: Dimitri Leemans

This course covers the study of groups focusing on basic structural properties, presentations, automorphisms and actions on sets, illustrating their fundamental role in the study of symmetry, for example in crystal structures in chemistry and physics, topological spaces, and manifolds.

Prerequisites
MATHS 320: Algebraic Structures is required.

Availability: S1 C
Points: 15
Coordinator: Eamonn O’Brien

Representation theory studies properties of abstract groups and algebras by representing their elements as linear transformations of vector spaces or matrices, thus reducing many problems about the structures to linear algebra, a well-understood theory. The course is roughly divided into two parts. The first part covers the structure and representations of finite-dimensional associative algebras and the second covers representation theory of finite groups.

Prerequisites
MATHS 320: Algebraic Structures is required.

Availability: S2 C
Points: 15
Coordinator: Arkadii Slinko

Symmetries and invariants play a fundamental role in mathematics. Especially important in their study are the Lie groups and the related structures called Lie algebras. These structures have played a pivotal role in many areas, from the theory of differential equations to the classification of elementary particles. Strongly recommended for students advancing in theoretical physics and pure mathematics.

Prerequisites
MATHS 320 Algebraic Structures and MATHS 332 Real Analysis

Recommended preparation: MATHS 333 Analysis in Higher Dimensions
Availability: S2 2013 C
Coordinator: Tom ter Elst

Presenting the modern elegant theory of integration as developed by Riemann and Lebesgue, it includes powerful theorems for the interchange of integrals and limits so allowing very general functions to be integrated, and illustrates how the subject is both an essential tool for analysis and a critical foundation for the theory of probability.

Prerequisites
MATHS 332: Real Analysis is required.

Recommended preparation
MATHS 333: Analysis in Higher Dimensions is strongly recommended.

Availability: S1C
Points: 15
Coordinator: Shayne Waldron

This course provides the mathematical foundations behind some of the techniques used in applied mathematics and mathematical physics in particular. For example, many phenomena in physics can be described by the solution of a partial differential equation (eg, the Heat equation, the Wave equation and Schrödinger's equation, etc). This course presents some of the fundamental ideas that under-pin the modern treatment of these topics.

Prerequisites
MATHS 332: Real Analysis and MATHS 333: Analysis in Higher Dimensions is required.

Recommended preparation
MATHS 730 Measure Theory and Integration and MATHS 750 Topology are recommended.

Availability: S2C
Points: 15
Coordinator: Shayne Waldron

Studies surfaces and their generalisations, smooth manifolds, and the interaction between geometry, analysis and topology, it is a central tool in many areas of mathematics, physics and engineering. Topics include Stokes’ theorem on manifolds and the celebrated Gauss Bonnet theorem.

Prerequisites
MATHS 332: Real Analysis is required.

Recommended preparation
MATHS 333: Analysis in Higher Dimensions and MATHS 340: Real and Complex Calculus are strongly recommended.

Availability: S2  2012 C
Points: 15
Coordinator: Rod Gover

An introductory course to functions of one complex variable, including Cauchy’s integral formula, the index formula, Laurent series and the residue theorem. Many applications are given including a three line proof of the fundamental theorem of algebra. Complex analysis is used extensively in engineering, physics and mathematics.

Prerequisites
MATHS 332: Real Analysis is required.

Recommended preparation
MATHS 333: Analysis in Higher Dimensions is strongly recommended as preparation for this course. The course MATHS 340 Real and Complex Calculus is also recommended for this course.

Availability: S1C
Points: 15
Coordinator: Tom ter Elst

Unlike most geometries, topological models can be stretched non-uniformly. Its ideas have applications in other branches of mathematics as well as physics, chemistry, economics and beyond. Its results give a general picture of what is possible rather than precise details of when and where. The course covers aspects of general and algebraic topology.

Prerequisites
MATHS 332: Real Analysis and MATHS 353: Geometry and Topology are required.

Recommended preparation
MATHS 333: Analysis in Higher Dimensions is strongly recommended preparation for this course.

Availability: S1 C
Points: 15
Coordinator: David Gauld

Mathematical models of systems that change are frequently written in the form of nonlinear differential equations, but it is usually not possible to write down explicit solutions to these equations. This course covers analytical and numerical techniques that are useful for determining the qualitative properties of solutions to nonlinear differential equations.

Prerequisites
MATHS 340: Real and Complex Calculus and MATHS 361: Partial Differential Equations are required.

Availability: S2C
Points: 15
Coordinator: Claire Postlethwaite

A study of exact and numerical methods for non-linear partial differential equations. The focus will be on the kinds of phenomena which only occur for non-linear partial differential equations, such as blow up, shock waves, solutions and special travelling wave solutions.

Prerequisites
MATHS 340: Real and Complex Calculus and MATHS 361: Partial Differential Equations are required.

Availability: S2C
Points: 15
Coordinator: Graham Donovan

A study of exact and approximate methods of solution for the linear partial differential equations that frequently arise in applications.

Prerequisites
MATHS 340: Real and Complex Calculus and MATHS 361: Partial Differential Equations are required.

Availability: S1C
Points: 15
Coordinator: Steve Taylor

A course introducing central concepts in mathematical biology, with emphasis on modelling physiological systems. The course will cover, among other things, enzyme kinetics, transport across biological membranes, and action potentials. The first half of the course will be taught by James Sneyd, the second half by Edmund Crampin.

Availability: to be announced
Points: 15
Coordinator: James Sneyd

Systems taken from a variety of areas such as financial mathematics, fluid mechanics and population dynamics can be modelled with partial differential equations and stochastic differential equations. This course uses such applications as the context to learn about these two important classes of differential equations.

Prerequisites
MATHS 340: Real and Complex Calculus and MATHS 340: Real and Complex Calculus are required.

Availability: S1C
Points: 15
Coordinator: Shixiao Wang

This course covers the use, implementation and analysis of efficient and reliable numerical algorithms for solving several classes of mathematical problems. The course assumes students have done an undergraduate course in numerical methods and can use Matlab or other high-level computational language.

Prerequisites
MATHS 270: Numerical Computation and one of MATHS 340: Real and Complex Calculus, MATHS 361: Partial Differential Equations or MATHS 363: Advanced Modelling and Computation is required.

Availability: S1C
Points: 15
Coordinator: Philip Sharp

Each of these courses (MATHS 781 - 784) deals with some special topic(s) from pure mathematics. Not all of them are offered every year; further information may be obtained from the Department of Mathematics. This course can be taken either as a supervised reading paper or as a project.

Availability: by arrangement
Points: 15
Coordinator: Steven Galbraith

This course involves a selection of topics from discrete and combinatorial geometry, which is a relatively modern area of mathematics with wide-ranging applications in engineering, crystallography, computer-aided design, graphics, and pattern recognition, for example. Topics will be chosen from the following: abstract and convex polytopes, point-line arrangements symmetries of discrete geometric structures, regular maps, hyperplanes in discrete spaces, applications.

Recommended preparation
Students should have a good background in at least two of algebra, geometry and combinatorics, as gained from courses such as MATHS 320 or MATHS 328, and MATHS 326.

Availability: to be announced
Points: 15
Coordinator: Arkadii Slinko

Each of these courses (MATHS 786 - 789) deals with some special topic(s) from applied mathematics. Not all of them are offered every year.

A special topic course can be taken as a supervised reading paper or as a supervised project, in either Semester One or Two, but not during Summer semester.

Sometimes a new course will be offered under a special topic number (see entries below for details).

Further information can be obtained from the Mathematics Postgraduate Adviser.

Availability: by arrangement
Points: 15
For further information: Claire Postlethwaite

This course will cover concepts and examples in mathematical modelling. We will begin with the underlying concepts of why we build models, what makes a good model, and what models can teach us. The course will also cover a range of models in practice to demonstrate these ideas; these topics may include models from mathematical biology and physiology, engineering, finance, population dynamics, and more.

Recommended preparation
MATHS 340: Real and Complex Calculus and MATHS 361: Partial Differential Equations is required.

Availability: S1C
Points: 15
Coordinator: Graham Donovan

This course is intended for students who are familiar with standard methods for solving ordinary differential equations, such as Runge–Kutta methods and linear multistep methods, or who have an interest in learning about these methods.

The content will be divided into three parts. First we will consolidate and formalise existing knowledge of the traditional methods. We will then introduce some new and more specialised topics, some of which are associated with so called “General linear methods”, which are generalisations of both Runge-Kutta and linear multistep methods. Finally, we will go more seriously into some of the new topics, with the actual selection based on interests that will have developed amongst members of the class.

Throughout the course the emphasis will be balanced between theoretical and practical considerations.

Availability: S2 2012 C
Points: 15
Coordinator: John Butcher

Inverse problems involve making inferences about physical systems from experimental measurements. Topics in this course include, the linear inverse problem, regularisation, introduction to multi-dimensional optimisation, Bayes theorem, prior and posterior probabilities, physically-based likelihoods, inference and parameter estimation, sample based inference, computational Markov chain Monte Carlo and output analysis.

Prerequisites
PHYSICS 701: Linear systems, or MATHS 340: Real and Complex Calculus and MATHS 361: Partial Differential Equations is required.

Availability: S2C
Points: 15
Coordinator: Jari Kaipio