Units that may be offered in 2010

MAT4CI - Computability and Intractability
Marcel Jackson
Prerequisite: (MAT1DM and MAT2PDM and MAT1CLA) or any third year mathematics unit.

When does a problem have an effective algorithmic solution? What does it mean for an algorithm to be effective? In this component we attempt to give rigorous meaning to questions of this type and investigate some possible answers.  Algorithmic problems in mathematics are the primary focus.

MAT4DS - Chaos and Order in Dynamical Systems (Not available 2009)
Reinout Quispel
Prerequisite: 30 credit points of second or third year mathematics units.

What is chaos? How does it arise in dynamical systems? What other dynamical phenomena exist, or, to put it slightly differently: what are the different kinds of dynamical systems? If one has a differential equation that exhibits chaos, how should one solve it? These are some of the ingredients of this unit.

MAT4DT - Duality Theory
Brian Davey
Prerequisites: MAT3DS and MAT3TA.

The unit will begin with a primer on category theory, general algebra and topology. We shall cover the general theory of dualities between classes of algebras and classes of topological relational structures. Applications of duality theory will also be presented.

Note: At most one of MAT4DT and MAT4GA will be offered in any year.

MAT4GA - General Algebra
Brian Davey
Prerequisite: MAT3DS (with MAT3TA highly recommended).   

General algebra, otherwise known as universal algebra, provides a theory within which to study the common features of all algebraic systems such as vector spaces, groups, rings, lattices and semigroups. The component will present all of the basic results in the theory. The close relationship between general algebra and lattice theory will be emphasized throughout the unit.

Note: At most one of MAT4DT and MAT4GA will be offered in any year.

MAT4GG - Group Actions
Grant Cairns
Prerequisites: MAT3TA. 

This unit studies the foundations of the theory of group actions. In doing so, it touches on a selection of topics which display interconnections between geometry, group theory, topology and calculus. The unit is problem based.

MAT4GM - Geometric Methods for Differential Equations
Geoff Prince
Prerequisite: MAT3AC.   

This course aims to show how geometric symmetry of the solutions of differential equations can be used to find those solutions using integrating factors. These integrating factors exist for all ordinary differential equations, not just the ones you learnt about in first year Mathematics.

Note: At most one of MAT4GM and MAT4GR will be offered in any year.

MAT4GR - General Relativity
Geoff Prince
Prerequisite: MAT3AC.

This is unit is not a comprehensive first course in general relativity. Instead it covers tensor and exterior calculus, metric differential geometry (following from MAT31AAC) and the geometry of geodesics. After studying curvature, Einstein's field equations are developed and the spacetime of the solar system is studied, including two of the famous experimental tests of general relativity.

Note: At most one of MAT4GM and MAT4GR will be offered in any year.

MAT4MP – Applied Mathematics Projects
(TBA)
Prerequisites: MAT2AM and MAT2APD; MAT3NA or (CSE1IPC and CSE1OOP or equivalent).

This component introduces students to mathematical modelling using some of the computer-based tools available to the professional applied mathematician. Models in various areas of application, such as heat and mass transport, financial mathematics, biomathematics, statistical mechanics and dynamic systems, are considered.

Note: Not available to students who have taken MAT3AMP.

MAT4NT –Introduction to Number Theory
Dr Peter Van Der Kamp
Prerequisites: MAT1DM and MAT2PDM and MAT1CLA; or 30 credit points of MAT2 units; or any MAT3 unit.

We will start this course with a gentle introduction to number theory, including tantalizing connections to cryptography.  If time permits, in the later parts of the unit we may touch on one or more of the recent analytic and algorithmic advances in the areas of Mersenne primes, the Riemann hypothesis, and primality proving.  My aim will be to make this course accessible and interesting to a varied audience, including students with interests in applied mathematics, pure mathematics, or computer science.

MAT4TD - Topology and Dynamics
John Banks
Prerequisites: MAT3TA      

As well as deepening your knowledge of point set topology, this unit shows how the theory of point set topology may be applied to the study of discrete dynamical systems. We will see that most of the ideas behind the definition of "chaos" are essentially topological and as such can be analysed using point set topological methods.

Some of the units that may be available to students in 2009 through the Access Grid Room (AGR) include:

• Financial Time Series
• Advanced Optimization
• Waves in Fluids
• Topological Groups
• Cryptography, Computer and Network Security
• Methods in Advanced Electromagnetic Theory
• And many more (in 2008 there were 17 AGR units in total)

STA4AMD –Analysis of Medical Data
Robert Staudte
Recommended Prior Studies: STA3SI     

This unit considers calibrating evidence in a test, variance stabilizing transformations, one- and two-sample Binomial models, evaluating and comparing Poisson rates, evidence in one and two-sample Welch t-tests and  compensating for publication bias. The evidence obtained by variance stabilization will be the basis for confidence intervals for effects, which are demonstrably more accurate than those obtained by traditional large-sample methods.  Further, variance stabilizations facilitates a meta-analysis of results from different studies.

STA4SA - Spatial Analysis
Andriy Olenko
Recommended Prior Studies: STA3SI, STA2RSP

The unit surveys the theory of random fields, spatial statistics models, and their applications to a wide range of areas, including image analysis and GIS (geographic information system). The course will cover the methodology and modern developments for spatial-temporal modelling, estimation and prediction, and spectral analysis of spatial processes. All the methods presented will be introduced in the context of specific real datasets with GRASS and R software.

STA4SI – Statistical Inference
Paul Kabaila
Recommended Prior Studies: STA3SI     

This unit covers a selection of topics in classical statistical inference at the fourth year level.  It consists of a selection of material from the following chapters of Casella and Berger (2002): Chapter 6 (Principles of Data Reduction), Chapter 7 (Point Estimation), Chapter 8 (Hypothesis Testing), Chapter 9 (Interval Estimation) and Chapter 10 (Asymptotic Evaluations).  A knowledge of this material is helpful in almost any statistical endeavour.

Reference: Statistical Inference, 2nd Edition, Duxbury (2002), by G. Casella and R. L. Berger.

STA4RA – Regression Analysis
Luke Prendergast
Recommended Prior Studies: STA3AS and STA3LM     

The topics for this unit include - multiple linear regression; classical estimation and testing; residual analysis; diagnostics; variable selection; robust regression and some modern dimension reduction techniques.