# 2009 Postgraduate courses

### From MathsDept

## Contents |

# Semester 2 2009

### Mathematics Education

- In Semester 2 2009 this course can be taken either as 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.
- a taught course on

**Mathematical processes**

This course will consider the mathematical processes of problem-solving, modelling, conjecturing, argumentation and proving in the context of senior secondary and undergraduate mathematics

- 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. The topics will be 2007 Calculus, 2008 Algebra and 2009 Calculus.

### Pure Mathematics

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

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

- 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.

- 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.

- 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 project or as a course on **Lie Groups and Lie Algebras**.

** Lie Groups and Lie Algebras **

This course on Lie Groups and Lie Algebras will be given by Dr. Tom ter Elst.

Symmetries and invariants play a fundamental role in mathematics. Especially important are symmetries that depend continuously on various parameters. These form Lie groups. Closely related are structures called Lie algebras. Historically these structures have played an absolutely pivotal role in many areas which range from the theory of differential equations, and their solutions, to the classification of elementary particles. Strongly recommended for students advancing in theoretical physics and pure mathematics.

### Applied Mathematics

- 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.

- 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.

- Each of these courses (MATHS 786 - 789) deals with some special topic(s) from applied 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 project or as a course on **Advanced Topic in Non-linear PDEs**.

**Advanced Topics in Applied Mathematics: Nonlinear PDEs**

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.

# Semester 1 2009

### Mathematics Education

- The use of computers and calculators in mathematics education, with a 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.

### Pure Mathematics

- 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.

- 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.

- 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.

- Studies analytic functions and their properties, which often reflect the topology of the region on which the functions are defined. This relationship yields powerful conceptual and computational tools and results, including the uniformisation theorem of Riemann which is a cornerstone in conformal geometry. The concepts and objects have applications in many branches of mathematics, physics, and engineering.

- Presents the classical fractals of computer science and art such as Julia and Mandelbrot sets, iterated function systems and higher-dimensional strange attractors, and illustrates applications of chaos, fractals and bifurcation to areas including commerce, medicine, biological and physical sciences.

- 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.

### Applied Mathematics

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

- 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.