Pre-requisite: This project is a compulsory component of the MSci pathway in Pure Mathematics. There is no specific pre-requisite for this module, but the student will need enough Level 3 background in Pure Mathematics to undertake an extended project at this level in some area of Pure Mathematics for which supervision can be offered.
This is an extended project designed to test the student's ability to work independently at a high level for a prolonged period of time with a restricted amount of supervision. This will give a taste of the kind of work expected of a mathematician in the commercial or academic world, unlike the relatively short bursts of work expected in most undergraduate modules. It will also provide an opportunity to develop those transferable skills that are sought by employers, including IT (both word-processing and data-base access), presentational and personal ones.
The project takes place during the first two terms of Level 4. It will normally involve study and exposition of a piece of mathematical work beyond the normal undergraduate syllabus and which will probably not be available in easily assimilated form. Originality of exposition will be expected, but not necessarily much in the way of original results. The main part of the assessment will consist of a word-processed report, but 20% of the marks for the project are awarded for an oral presentation of the work which will take place just after Easter. As preparation for this assessed oral presentation, the student will be expected to give two oral progress reports around the middle of each of the first two terms to a small group of staff and any other students undertaking this module. Constructive advice on these presentations will be provided after each one.
Near the beginning of the first semester, there will be computer-based workshops for students taking this module. These will cover topics including the use of LaTeX (the internationally accepted standard language for mathematical typesetting which is accepted by the majority of mathematical publishers, using the internet to access sources of mathematical information (including the use of MathSciNet, the on-line version of Mathematical Reviews which reviews almost every published paper in Pure Mathematics) and using the computer algebra package Sage to produce mathematical diagrams.
Students intending to take this module should seek advice and think about their choice of project during the summer. The selection of a project should be finalized no later than the start of the academic year, and it would be helpful to all involved if students actually did this even earlier.
80% by final word-processed report, 20% by oral presentation.
Pre-requisite: PMA3017 Metric and Normed Spaces, and PMA3014 Set Theory
Functional analysis arose in the early twentieth century when the need became apparent to study whole classes of functions rather than individual ones. For example, differential equations may be regarded as concerning maps from a set of functions into itself and (after some reformulation) looking for a solution of a differential equation is asking for a function that is left fixed under the action of a certain map. The proof that, in certain circumstances, such a function always exists also shows how to approximate such a function numerically when an analytic solution cannot be found. Further impetus to the development of functional analysis came when quantum mechanics was found to be describable within its ambit. This has been an active area of research ever since and remains so to this day.
To some extent you can regard linear functional analysis, to which this module is restricted, as an attempt to place linear algebra on a firm foundation within an infinite-dimensional context. The module will emphasize the topological tools (metric and non-metric ones) which are necessary for this. Familiarity with the Level 4 module Topology is desirable.
The topics will be chosen from:
A characterisation of finite-dimensional normed spaces.
The Hahn-Banach theorem with consequences.
The bidual and reflexive spaces.
The open mapping theorem, the closed graph theorem, the uniform boundedness principle and the Banach-Steinhaus theorem.
Weak topologies and the Banach-Alaoglu theorem.
Spectral theory for bounded linear operators.
Exam 60% Project & Presentation 20% Continuous Assessment 20%
PMA4003 Topology (1st semester)
Pre-requisite: PMA3017 Metric and Normed Spaces, and PMA3014 Set Theory
Topology (rather like Algebra or Analysis) is not so much a single branch of mathematics but a loose confederation of subject areas differing widely in their origins, techniques and motivation but united by sharing a common core of basic concepts and constructions. Problems of a topological nature include: how can we describe and classify knots? how can we describe and classify surfaces? to what extent is it possible to extend the ideas of analysis into sets that don't have metrics defined on them? what can be meant by saying that two objects are “fundamentally the same shape”, and how do we decide whether they are or not? what ‘models’ are available to describe certain aspects of theoretical computer science? Rather than attempting to supply answers to any such major questions, this module will concentrate on developing enough of the ‘common core’ to allow students to begin to appreciate how such issues can be tackled topologically.
Ordered Sets: Partial order, maximal and maximum, minimal and minimum, upper and lower bounds, infima and suprema, lattices.
Topological Spaces: Topologies, open set, closed set, closure, neighbourhood, base, subbase, lattice of topologies, subspaces.
Continuity and Sequential Convergence: Continuous functions between topological spaces, composites, homeomorphism, topological invariants, contractive and expansive invariants. Convergence of a sequence in a topological space; the inadequacy of sequential convergence in topological spaces, contrasting with the adequacy of sequential convergence in metric spaces. First-countable spaces, separable spaces, second-countable spaces.
Separation Axioms: T0-spaces, T1-spaces, Hausdorff or T2-spaces, regular or T3-spaces, normal or T4-spaces. Every second-countable T3-space is metrizable and separable.
Compactness: Open covers, compact subsets, KC-spaces, a compact T2-space is T4, maximal compact and minimal T2-spaces.
Connectedness: Connected spaces, connected subsets, components, totally disconnected spaces, locally connected spaces.
Exam 70% 2x Mini Project 30% (15% each)
The libraries contain several texts on topology which include sections relevant to this module, but be aware that a lot of them are written at postgraduate level. More accessible are:
Mansfield, M. J., Introduction to Topology (Van Nostrand).
Mendelson, B., Introduction to Topology (Allyn & Bacon).
Moore, T. O., Elementary General Topology (Prentice Hall).
Simmons, G. F., Introduction to Topology and Modern Analysis (McGraw-Hill).
Lipschutz, S., General Topology (Schaum).
PMA4004 Integration Theory (2nd semester)
Pre-requisite: PMA2002 Analysis and PMA3014 Set Theory
The theory of integration, developed by Lebesgue in the early part of the twentieth century in the context of the real line and subsequently extended to more general settings, is indispensable in modern analysis. The Lebesgue theory allows a very wide class of functions to be integrated and includes powerful convergence theorems which are not available in Riemann integration. In this module the theory is developed in the context of a general σ-algebra of sets. Special attention is given to the case of Lebesgue measure on the reals, and some applications of the integral to Fourier series are given.
Contents: σ-algebras of sets, measurable spaces, measurable functions. Measures. Integrals of non-negative measurable functions: properties including Fatou's lemma and monotone convergence theorem. Integrable functions: Lebesgue dominated convergence theorem. Lebesgue integral on intervals: comparison with Riemann integral. Lp-spaces: inequalities of Hölder and Minkowski; Fourier series in L2.
Exam 70% 3x Assignment 30%
R. G. Bartle, The Elements of Integration and Lebesgue Measure (Wiley, 1995).
PMA4010 Algebraic Topology (2nd semester)
Pre-requisite: PMA4003 Topology
This module is an introduction to algebraic topology. Given two to[pological spaces, one may want to know if these spaces are homeomorphic (that is, indistinguishable). Surprisingly, this question is particularly hard to deal with if the answer is negative. Algebraic topology provides a toolkit for this situation, and for many far-reaching generalisations as well. Surprising and important consequences of our investigations will be that every self-map of a disc (a solid ball) must leave at least one point fixed, and that any self-homeomorphism of a disc must map the boundary of the disc to the boundary.
Outside of topology and geometry, these results have applications in many mathematical disciplines: for example, the fundamental theorem of algebra (that every non-constant polynomial with complex coefficients has at least one root) will be proved by algebro-topological means. Another example: the Frobenius-Perron theorem in linear algebra.
After discussing some point-set topological notions, most notably that of identification maps, and fundamental geometric constructions (cone and suspension), the central notion of homotopy is introduced. Building on this, the two-dimensional theory is developed using the so-called fundamental group of a space, and the general theory without restrictions is introduced using combinatorial descriptions of spaces (simplicial complexes and triangulations).
Exam 70% Assignment & Presentation 20% Participation 10%