Algebra

Description: This honours unit devotes approximately half of its time to ring theory and half to representation theory,
over the complex numbers, for finite groups. The ring theory half provides a grounding in non-commutative
ring theory leading, in a somewhat round-about way, to the Wedderburn Structure Theorem for semi-simple
algebras. It does not take the most direct path but rather develops some general radical theory before focusing
on the nil-radical. The Wedderburn theorem is applied to (classical) representation theory, establishing the
orthogonality of irreducible characters that was taken as a given in the MATH337 introduction. The theory of
characters is then extended, with a brief excursion into the theory of algebraic integers, to include suchmethods
as inducing characters from subgroups. Finally this character theory is applied to show that groups of order
pa.qb are soluble. As well as gaining a good theoretical knowledge of representation theory students develop
considerable skill in calculating character tables of groups with few normal subgroups, not so much for its own
sake but as a way of integrating their knowledge of the theory.

Pre-requisite: MATH337 AlgebraIIIA

Description: This unit develops the basic ideas of modern abstract algebra by concentrating on the many facets of group theory. As well as the standard material leading to the isomorphism theorems, we cover combinational aspects such as presentations of groups, the Todd-Coxeter algorithm, and subgroups of free groups via groupoids. Also studied are permutation groups, finitely generated abelian groups, soluble groups and group representations. MATH337 is especially suitable for students majoring in the theoretical aspects of physics or computing science.

Lie Groups

Description: Topology is the study of continuity. The definition of topological space was conceived in order to say what it
means for a function between such spaces to be continuous. There are several different ways of defining topological
structure and the proofs that these are equivalent abstract many concrete results about specific kinds of
spaces. Different ways of expressing continuity are obtained. Sequences are not adequate for general topological
spaces, they need to be replaced by nets or filters, and we discuss convergence of those. Particular properties
of topological spaces are analyzed in detail: these include separation properties, compactness, connectedness,
countability conditions, local properties, metrizability, and so on. Applications to basic calculus are emphasized.
A little bit of algebraic topology may be included by discussing the Poincare or fundamental group of a
space.

Pre-requisite: MATH300 Geometry and Topology

Description: 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. MATH300 is especially recommended for those students preparing to become teachers of high-school mathematics.

Applied Functional Analysis

Description: This unit prepares you to use differential and integral equations to attack significant problems in the physical
sciences, engineering and applied mathematics. The concepts of functional analysis provide a suitable frameworkfor
the development of effective analytical and computational methods to solve such problems. A selection
of material will be drawn from the following topics.
1. Four alternative formulations of physical problems:conservation laws, boundary value problems, weak
formulations and variational principles.
2. Green's functions and integral equations.
3. One dimensional boundary value problems and the Fredholm alternative.
4. Operators on Hilbert space and conditions for the solvability of equations.
5. Integral Equations.
6. NumericalMethods: Galerkin's method, least squares.
7. Ill-posed problems and their regularisation (stabilisation)
8. Effective treatments of potential theory problems and the scattering of waves by obstacles.

Pre-requisite: MATH336 Partial Differential Equations & MATH339 Real Functional Analysis

Description: Partial differential equations form one of the most fundamental links between pure and applied mathematics. Many problems that arise naturally from physics and other sciences can be described by partial differential equations. Their study gives rise to the development of many mathematical techniques, and their solutions enrich both mathematics and their areas of origin.

This unit explores how partial differential equations arise as models of real physical phenomena, and develops various techniques for solving them and characterising their solutions. Especial attention is paid to three partial differential equations that have been central in the development of mathematics and the sciences -- Laplace's equation, the wave equation and the diffusion equation.

Real Functional Analysis: This unit is concerned with a review of the limiting processes of real analysis and an introduction to functional analysis. Through the discussion of such abstract notions as metric spaces, normed vector spaces and inner product spaces, we can appreciate an elegant and powerful combination of ideas from analysis and linear algebra.