This unit is designed to widen geometric intuition and horizons by studying topics such as projective geometry, topology of surfaces, graph theory, map colouring, ruler-and-compass constructions, knot theory and isoperimetric problems. This unit is especially recommended for those students preparing to become teachers of high school mathematics
| Credit Points: | 3 |
| Contact Hours: | 4 |
| When Offered: | D1 - Day; Offered in the first half-year |
| Staff Contact(s): | Mathematics Staff |
| Prerequisites: | MATH235(P) |
| Corequisites: | |
| NCCW(s): | |
| Unit Designation(s): | Science |
| Assessed As: | Graded |
| Offered By: | Department of Mathematics |
This unit introduces the theory of waves by a systematic study of the underlying partial differential equations. Waves involve the transfer, without bulk motion, of both energy and information. Fundamental properties of waves are first examined in the simplest one dimensional setting. The treatment is then broadened to two dimensional and three dimensional waves, particularly for acoustic and electromagnetic waves. Resonators and waveguides provide some examples of how waves behave in confined regions. In contrast, the scattering and diffraction of waves by obstacles in free space carries information about the scatterer itself; this is the basis of many imaging technologies.
| Credit Points: | 3 |
| Contact Hours: | 4 |
| When Offered: | D1 - Day; Offered in the first half-year |
| Staff Contact(s): | Mathematics staff |
| Prerequisites: | MATH235(P) and (MATH232(P) or MATH236(P)) |
| Corequisites: | |
| NCCW(s): | |
| Unit Designation(s): | Science Technology |
| Assessed As: | Graded |
| Offered By: | Department of Mathematics |
The remarkable fact that determinism does not guarantee regular or predictable behaviour is having a major impact on many fields of science and engineering, as well as mathematics. The discovery of chaos, or of chaotic motions, in simple dynamical systems changes our understanding of the foundations of physics and has many practical applications as well, shedding new light on the workings of lasers, fluids, mechanical structures and chemical reactions. Dynamical systems involve the study of maps and systems of differential equations. In this unit, the diversity of nonlinear phenomena is explored through the study of second-order differential equations, and one dimensional and two dimensional maps. Chaotic motions are introduced by a study of the driven pendulum, a second-order system that includes nonlinear aspects usually ignored in simpler treatments. An appropriate balance between forcing and damping leads to irregular, but bounded, motions that do not repeat themselves, even approximately - truly chaotic motion in a simple deterministic system.
| Credit Points: | 3 |
| Contact Hours: | 4 |
| When Offered: | D2 - Day; Offered in the second half-year |
| Staff Contact(s): | Mathematics staff |
| Prerequisites: | MATH235(P) and (MATH232(P) or MATH236(P)) |
| Corequisites: | |
| NCCW(s): | |
| Unit Designation(s): | Technology Science |
| Assessed As: | Graded |
| Offered By: | Department of Mathematics |
This unit develops the ideas and techniques of analysis important in many branches of pure and applied mathematics. Topics include the theory of ordinary differential equations, including linear and nonlinear systems and their stability. Some special functions are also discussed, together with important applications in various branches of mathematics.
| Credit Points: | 3 |
| Contact Hours: | 4 |
| When Offered: | D1 - Day; Offered in the first half-year |
| Staff Contact(s): | Mathematics Staff |
| Prerequisites: | MATH235(P) and (MATH232(P) or MATH236(P)) |
| Corequisites: | |
| NCCW(s): | |
| Unit Designation(s): | Science Technology |
| Assessed As: | Graded |
| Offered By: | Department of Mathematics |
This unit further develops the theory of algebraic structures commenced in MATH337 Algebra IIIA, and involves the study of a selection of topics in Ring Theory and Field Theory. The Ring Theory strand develops the basic theory, including the study of integral domains, ideals, quotient rings, principal ideal domains, unique factorisation domains and Euclidean domains, followed by a study of one or two topics related to ring theory such as ideals in quadratic fields, the first case of Fermat's last theorem, Hopf algebras or the Wedderburn Structure Theorem. The Field Theory strand also develops the basic theory, including the notion of irreducibility, simple, algebraic and transcendental extensions, and the tower law. The ideas of group theory studied inMATH337 are then applied to the study of field extensions via the notion of automorphisms, culminating in the study of the Galois correspondence theorem.
| Credit Points: | 3 |
| Contact Hours: | 4 |
| When Offered: | D2 - Day; Offered in the second half-year |
| Staff Contact(s): | Mathematics Staff |
| Prerequisites: | MATH337(P) |
| Corequisites: | |
| NCCW(s): | |
| Unit Designation(s): | Science |
| Assessed As: | Graded |
| Offered By: | Department of Mathematics |