2008 Postgraduate courses

From MathsDept

1 Semester 2
2 Semester 1 2008

Semester 2

Algebra and Combinatorics

A broad introduction to various aspects of elementary, algebraic and computational number theory and its applications, including primality testing and cryptography.

MATHS 721 Representations and Structure of Algebras and Groups

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.

MATHS 782 Advanced Topic(s) in Mathematics 2

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.

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Analysis, Geometry, Topology

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Staff with interests in Analysis, Geometry and Topology

MATHS 731 Functional Analysis

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 (e.g. 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.

MATHS 735 Analysis on Manifolds and Differential Geometry

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.

MATHS 750 Topology

Unlike most geometries, topology models objects which may 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.

MATHS 784 Advanced Topic(s) in Mathematics 4

This course presents the currently developing theory of finite tight frames, and some of its applications such as those in signal analysis, quantum information theory and orthogonal polynomials of several variables.

Any vector in 2-dimensional space can be written as the sum of its orthogonal projections onto three unit vectors (times 2/3). This is the prototypical example of a finite tight frame expansion.

Over the last decade, it has become increasingly apparent that tight frames for finite dimensional spaces are highly useful for similar reasons. They can have desirable properties, such as good time--frequency localisation, small support (sparseness) and symmetries of the space, which may be impossible for an orthonormal basis. In addition, there are computational advantages of stability and robustness to erasures, and a special geometry (different for real and complex spaces) which has no analogue in the infinite dimensional setting.

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Applied Mathematics

MATHS 761 Dynamical Systems

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.

MATHS 769 Applied Differential Equations

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.

MATHS 787 Advanced Topic(s) in Applied Mathematics 2 : Numerical Methods for Differential Equations

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.

MATHS 789 'Inverse problems: Advanced Topic(s) in Applied Mathematics 4'

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, solitons and special travelling wave solutions.

PHYSICS 707 Inverse problems

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

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Mathematics Education

MATHS 702 Mathematics Curriculum

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.

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