## Department of Mathematics

# Postgraduate courses

Thinking about postgraduate study? Explore your postgraduate course options at the Department of Mathematics.

## Semester One 2020

### MATHS 712

Teaching and Learning in Algebra

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. Students taking this course should normally have studied mathematics or statistics at 200 level.

**Prerequisite:** MATHS 302 or significant teaching experience or department approval.

### MATHS 715

Graph Theory and Combinatorics

A study of combinatorial graphs (networks), designs and codes illustrating their application and importance in other branches of mathematics and computer science.

**Prerequisites**: B+ in MATHS 326 or 320

### MATHS 720

Group Theory

A 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

### MATHS 730

Measure Theory and Integration

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

**Strongly recommended:** MATHS 333

### 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. Strongly recommended: MATHS 333 and 340.

**Prerequisite:** MATHS 332

### 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.

**Prerequisite:** B- in both MATHS 340 and 361

### MATHS 763

Advanced Partial Differential Equations

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

**Prerequisites**: B- in both MATHS 340 and 361

### MATHS 769

Stochastic Differential and Difference Equations

Differential and difference equations are often used as preliminary models for real world phenomena. The practically relevant models that can explain observations are, however, often the stochastic extensions of differential and difference equations. This course considers stochastic differential and difference equations and applications such as estimation and forecasting.

**Prerequisites**: B- in both MATHS 340 and 361

## Semester Two 2020

### MATHS 708

Special Topic

**Prerequisite**: MATHS 302 or significant teaching experience or department approval

### MATHS 713

Logic and Set Theory

A study of the foundations of pure mathematics, formalising the notions of a 'mathematical proof' and 'mathematical structure' through predicate calculus and model theory. It includes a study of axiomatic set theory.

**Prerequisites**: MATHS 315 or PHIL 305

### MATHS 714

Number Theory

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

**Prerequisites**: B+ in MATHS 328 or 320

### 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.

**Prerequisite:** MATHS 320

### MATHS 731

Functional Analysis

Provides the mathematical foundations behind some of the techniques used in applied mathematics and mathematical physics; it explores how many phenomena in physics can be described by the solution of a partial differential equation, for example the heat equation, the wave equation and SchrÃ¶dinger's equation.

**Prerequisites**: MATHS 332 and 333

**Recommended preparation**: MATHS 730 and 750

### MATHS 750

Topology

Aspects of point-set, set-theoretic and algebraic topology including: properties and construction of topological spaces, continuous functions, axioms of separation, countability, connectivity and compactness, metrization, covering spaces, the fundamental group and homology theory. Strongly recommended: MATHS 333.

**Prerequisite:** MATHS 332

**Restriction:** MATHS 350

### MATHS 762

Nonlinear Partial Differential Equations

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.

**Prerequisites**: B- in both MATHS 340 and MATHS 361

### MATHS 765

Mathematical Modelling

Advanced topics in mathematical modelling, including selected topics in a range of application areas, principally taken from the physical and biological sciences.

**Prerequisite: **At least B- or better in both MATHS 340 and 361

### MATHS 766

Inverse Problems

Covers the mathematical and statistical theory and modelling of unstable problems that are commonly encountered in mathematics and applied sciences.

**Prerequisite: **At least B- in both MATHS 340 and 363, or PHYSICS 701

### MATHS 770

Advanced Numerical Analysis

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.

**Prerequisite:** B- in MATHS 270, 340 and 361