## Department of Mathematics

# Postgraduate courses

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

## Semester One 2017

### MATHS 707

Special Topics in Mathematics Education: Mathematical Processes

This course includes hands-on mathematical problem-solving, problem-posing and modelling. Participants will reflect on their own mathematical processes, review research literature and consult mathematics researchers about their practice, to explore how such processes can be enhanced when teaching and learning mathematics.

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

### MATHS 715

Graph Theory and Combinatorics

This course covers theory and applications of combinatorial graphs (networks), plus discrete structures such as block designs or error-correcting codes.

Topics include graph connectivity, trees, colourings, embeddings, digraphs, matchings, adjacency and incidence matrices, Turan's theorem, eigenvalue methods, graph factorisations, plus other topics such as Steiner systems, perfect and linear codes, graph symmetries, and abstract polytopes.

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

### MATHS 720

Group Theory

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

### 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

**Recommended preparation**: Maths 333

### MATHS 769

Stochastic Differential and Difference Equations

Real world phenomena are often modeled with the stochastic (statistical) extensions of differential equations. For measurement data (observations), their discrete time counterparts are commonly used. This course considers the related theory, methods and applications.

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

### MATHS 770

Advanced Numerical Analysis

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

### MATHS 783

Advanced Topics in Mathematics - Foundations of Algebraic Geometry

Algebraic geometry is a branch of mathematics studying zeros of polynomials. The fundamental objects in algebraic geometry are algebraic varieties, i.e., solution sets of systems of polynomial equations.

Examples of the most studied classes of algebraic varieties are (plane) algebraic curves, such as lines, circles, parabolas, elliptic curves, etc. Algebraic geometry uses abstract techniques to solve geometrical problems about varieties and occupies a central place in modern mathematics with multiple connections with diverse fields inside and outside of mathematics.

**Prerequisites**: At least one of MATHS 320, 326, 328, 332, 333, or A+ in MATHS 255

## Semester Two 2017

### 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 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.

**Prerequisite**: MATHS 315 or PHIL 305

### MATHS 714

Number Theory

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**: MATHS 320 or B+ in MATHS 328

### MATHS 731

Functional Analysis

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 and MATHS 333

**Recommended preparation**: MATHS 730 and MATHS 750

### MATHS 740

Complex Analysis

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

**Recommended preparation**: MATHS 333 and MATHS 340.

### MATHS 750

Topology

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

**Strongly recommened**: Maths 333

### 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.

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

### 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, solutions and special traveling wave solutions.

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

### MATHS 764

Mathematical Biology

A course introducing central concepts in mathematical biology, with emphasis on modelling physiological systems. The course will cover, firstly, a selection of topics in cell physiology, including enzyme kinetics, transport across biological membranes, and action potentials in neurons. Secondly, the course will include a selection of topics in organ physiology, possibly including such topics as muscle physiology, neuroendocrine cells, photoreceptor physiology, and the gastrointestinal system. Usually, the exact selection of topics in the second part of the course will be determined after consultation with the students in the class.

**Prerequisites**: B- in both Maths 340 and Maths 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

## Other courses of interest

### STATS 708

Topics in Statistical Education

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.

**For advice**: Maxine Pfannkuch