618-391 Mathematics Topics A

Note:

Credit Points:

Coordinator:

Prerequisite/s:

Timetable:

Contact:

Objectives:

On completion of this subject, students should:

Comprehend:

• in some detail and depth the mathematical theory and techniques associated with some topic or topics of interest to a staff member.

Have developed:

• an ability to pursue a substantial mathematical theme at some depth;

• an ability to work with a certain amount of independence;

• a richer insight into major themes and applications of mathematics.

Appreciate:

• the methods and techniques required to complete an in-depth study of some mathematical topic, possibly at research level;

• integration of mathematical concepts and methods to solve problems.

Content:

The subject may be undertaken either as a project under the supervision of a staff member, or as an appropriate 400 - level subject, or as a course of 39 lectures from one of the following areas, by negotiation with the coordinator.

1. Geometry. Axiomatic systems: Euclidean, spherical, hyperbolic (non-Euclidean) geometry. Transformation and matrix groups. Isometry groups and tessellations. Projective and affine geometry.

2. Operations Research. Selected topics from linear programming, quadratic programming, dynamic programming, fractional programming, composite concave programming, nonlinear optimisation, parametric optimisation, global optimisation, combinatorial optimisation, branch and bound, and simulation.

3. Statistical Mechanics. Basic thermodynamics and statistical mechanics: ensembles and the partition function, thermodynamic limit. Phase transitions: first order and continuous transitions, singularities, critical exponents, universality, scaling and homogeneous functions, mean field theory, correlation functions. Enumeration: transfer matrices, generating functions, random walks. Applications: ideal gas, van der Waals-Maxwell fluid, Ising magnets, percolation.

4. Mathematical Modelling. The modelling process: some physical phenomena as case studies; empirical modelling versus model fitting/parameter estimation. Dimensional analysis: a tool for the physical sciences; stability and structural stability in systems of differential equations; limit cycles and nonlinear difference equations.

5. Mathematical Logic: First order theories and basic model theory, including completeness and compactness. Undecidability and incompleteness.

6. Selected topics in analysis, algebra, geometry and topology, methods and modelling, mathematical physics, operations research, and the history of Mathematics.

7. Project work. An in-depth study of one or more topics in analysis, algebra, geometry and topology, methods and modelling, mathematical physics, operations research, and the history of Mathematics, under the supervision of a staff member.

Assessment: