MATHEMATICAL SCIENCES

Credit Points

15

Course Coordinator

Dr W Turner

Pre-requisites

MA 2004 and MA 2506

Overview

Numbers measure size, groups measure symmetry. Many groups occur naturally as symmetry groups of solids, patterns and other geometrical objects. This course will develop the basic ideas of group theory through such examples of groups acting on sets.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and in-course assessment (20%).

Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment).

Credit Points

15

Course Coordinator

Dr R Kessar

Pre-requisites

MA 2005

Overview

This course extends the rigorous study of sequences, series and functions that was introduced in MA 2005. In particular, the course will emphasise the interplay between power series and functions. More general sequences and series of functions will also be considered, and the advantages of uniform convergence over pointwise convergence will be discussed. Finally, the course will introduce improper integrals, such as those defining the Gamma and Beta functions, and will explore the links with summation.

Structure

12 week course - 5 one-hour lectures per fortnight and 1 one-hour tutorial per week (to be arranged).

Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment).

Credit Points

15

Course Coordinator

Professor V Gorbunov

Pre-requisites

MA 2506 and MA 2507

Overview

The course begins with some basic non-linear optimisation techniques for multivariable real valued functions, including the second derivative test, constrained optimisation and the method of Lagrange multipliers. The course will then continue with a short exposition of numerical methods of approximation, exemplified in approximation of eigenvalues of real-valued matrices. Following this, the course will specialise to linear optimisation problems. The simplex algorithm will be introduced and studied, including applications to matrix games. Computations will be done using a computer-based algebra package, as well as manually.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week (to be arranged).

Assessment

1st Attempt: 1 two-hour written examination (80%); continuous assessment (20%).

Resit: 1 two-hour written examination (maximum of 100% resit and 80% resit with 20% continuous assessment).

Credit Points

15

Course Coordinator

Professor G Hall

Pre-requisites

MA 2507

Overview

The course studies the Newtonian theory of the motion of a particle. The course begins with the revision of vectors and integration theory and an introduction to kinematics. Newton's laws of motion are introduced and illustrated through the study of dynamical problems such as projectile motion, air resistance and the theory of vibrations. Theoretical work is done on topics such as energy, linear and angular momentum and the role of inertial frames in Newtonian mechanics.

Structure

12 week course - 5 one-hour lectures per fortnight and 1 one-hour tutorial per week (to be arranged).

Assessment

1st Attempt: 1 two-hour examination (80%) and in-course assessment (20%).

Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment).

Credit Points

5

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

Available only to candidates for Honours Degrees involving Mathematics.

Notes

The assessment of this course does not count towards Honours classification. This course is not available in 2009/10.

Overview

The student will undertake a project specified by the department. The work may be done individually or in teams. The end result of the work is to be a report and presentation by the student or team. The work will be supervised by a member of the department and will be assessed on the quality of the report and its presentation.

Structure

12 week course - Classes as appropriate.

Assessment

Assessed on the report.

Credit Points

15

Course Coordinator

Dr R Reis

Pre-requisites

MX 3021

Overview

This is an introductory course on Complex Analysis. Holomorphic functions and power series, Cauchy’s throrem and its consequences, contour integration and the calculus of residues are discussed.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and in-course assessment (20%).

Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment).

Credit Points

15

Course Coordinator

Dr L Iancu

Pre-requisites

MX 3020

Overview

The familiar, simple and useful properties of the integers places the set of integers at the core of any study of algebraic objects. But many of these properties hold for other familiar mathematical objects; for polynomials, real numbers, matrices etc. This course develops the theory of rings and fields which unifies the study of many of these objects and, at the same time, clarifies the differences between them.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and in-course assessment (20%).

Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment).

Credit Points

15

Course Coordinator

Professor R J Archbold

Pre-requisites

MX 3021

Overview

The course gives an introduction to topology. In particular it introduces the concept of a topological space and the idea of continuity of mappings between topological spaces. An important class of topological spaces, metric spaces are studied in depth. The ideas and concepts are applied to the particular case of Euclidean space.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week (to be arranged).

Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment).

Credit Points

15

Course Coordinator

Dr A Sevastyanov

Pre-requisites

MA 2507

Overview

An introduction to the vector calculus leading to the divergence theorem and some of its applications; a brief treatment of Fourier series and their applications; an introduction to partial differential equations, their behaviour and solution.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour written examination (80%) and in-course assessment (20%).

Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment).

Credit Points

15

Course Coordinator

Professor R Levi

Pre-requisites

MX 3521 or permission of Head of Teaching (Mathematical Sciences).

Overview

The student will be given a mathematical topic on which to write and submit a report. The work will be supervised by a member of staff.

Structure

12 week course – Assessed on the project report and the oral presentation (the presentations are given during the second half-session).

Credit Points

15

Course Coordinator

Head of Mathematical Sciences

Pre-requisites

Available only to candidates for Honours in Mathematics, Mathematics with French, Mathematics with Gaelic, Mathematics with German, and Mathematics with Spanish.

Notes

Not available in session 2009/10.

Overview

The student will undertake a part-time placement in some company, external institution or other university department to work on a project approved by the department. The placement will extend over all or part of the half-session as appropriate. Both an on-site supervisor and a departmental supervisor will be appointed to monitor the student’s progress. The assessment of the course will be based on a report written by the student and on assessments by the supervisors. The course will only be available for selected students and if suitable placements can be found.

Structure

12 week course – Classes as appropriate.

Assessment

Assessed on the report and on the supervisors’ report.

Credit Points

30

Course Coordinator

Professor G Hall

Pre-requisites

The course is available only to students accepted into the Joint Honours Programme Mathematics-Physics (MA or BSc) or the single Honours Programmes Physics (BSc) or Natural Philosophy (MA).

Notes

This course is run over the full session.

Overview

The student will be given a Mathematical topic on which to write a report. The work will be supervised by a member of staff. The assessment of the project will be based on the report and an oral examination based on the material relevant to the assigned topic.

Structure

24 week course – 1 tutorial per week.

Assessment

Assessed on the project report and on the oral examination.

Credit Points

15

Course Coordinator

Dr A Potter

Pre-requisites

MX 3021

Overview

An introduction to the qualitative theory of systems of ordinary differential equations. Topics covered will include: existence and uniqueness theory, linear systems, equilibria and their stability, periodic solutions. Various particular examples will be analysed in detail.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st attempt: 1 two-hour examination (100%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Credit Points

15

Course Coordinator

Dr T Le

Pre-requisites

MX 3531

Overview

The roots of a quadratic polynomial are given by a formula involving the coefficients. Similar formulae exist for the roots of polynomial equations of degrees 3 and 4, but not for higher degrees. The precise relationship between a polynomial and the type of roots it has emerges as one of the consequences of Galois Theory, which is a unification of ideas embracing polynomials, fields and group theory. The course will also consider the classical ruler and compass constructions.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and in-course assessment (20%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Credit Points

15

Course Coordinator

Professor R Archbold

Pre-requisites

MX 3021, MX 3532

Overview

Measure Theory abstracts and makes precise the notions of "length" and "volume". In this course the basic concept of measure theory will be covered. The theory of Lebesgue integration will be introduced and various convergence results will be presented. The relevance to other branches of mathematics (for instance Probability Theory and Analysis) will be discussed.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week (to be arranged).

Assessment

1st Attempt: 1 two-hour written examination (80%), in-course assessment (20%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MX 3532

Notes

Special Option. Not available in session 2009/10.

Overview

A course on some mathematical aspects of the theories of fractals and discrete dynamical processes. It will normally include a treatment of fractal dimension and the use of iterated function systems to generate fractals.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MX 3531 and MX 3532

Notes

Special Option. Not available in session 2009/10.

Overview

An introduction to the topology associated to a variety of basic geometric spaces, including a discussion of topological invariants and applications to geometric problems.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and in-course assessment (20%).

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MX 3532

Notes

Special Option. Not available in session 2009/10.

Overview

Whereas earlier work in analysis tended to focus on single functions, this course deals with functions collectively, as elements of vector spaces or function algebras.
The course will cover topics from: normed spaces, Banach spaces, Hilbert spaces (with emphasis on sequence spaces and function spaces), linear functionals and operators, Hahn-Banach theorem, principle of uniform boundedness, open mapping and closed graph theorems, the algebra of continuous functions on a compact Hausdorff space, Stone-Weierstrass theorem and Gelfand theory.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination.

Credit Points

15

Course Coordinator

To be confirmed

Pre-requisites

The permission of the Head of Teaching (Mathematical Sciences).

Overview

The student is given a mathematical topic on which to write and submit a report. The work will be supervised by a member of staff.

Structure

12 week course - Assessed on the project report and oral presentation.

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MA 2506 and MX 3532

Notes

Special Option. Not available in session 2009/10.

Overview

A Hilbert space is a vector space which is complete with respect to the metric arising from a given inner product. This setting permits the development of geometric ideas, taken from Euclidean space, which can then be applied to spaces of functions arising naturally in the theory of differential equations. The course will cover topics from: norms, inner products and Hilbert spaces, orthogonality, orthogonal expansions and Fourier series, dual spaces, linear operators.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination.

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MX 3532

Notes

Special Option. Not available in session 2009/10.

Overview

The course is concerned with the analysis of functions of several variables, in particular the differentiability and integrability of such functions. Appropriate background material will be discussed in order to prove some important theorems of analysis, for instance the inverse and implicit function theorems, Fubini’s theorem and convergence theorems of integration.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination.

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MX 3531

Notes

Special Option. Not available in session 2009/10.

Overview

Traditional applied mathematics is centred in the area where calculus and its developments are used to solve problems in the physical sciences. This course looks at another and more recent set of problems deriving from such things as digital communication and the design of efficient statistical experiments. The course is primarily an introduction to the algebraic theory of error-correcting linear codes.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination.

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MX 3532

Notes

Special Option. Not available in session 2009/10.

Overview

This course studies Fourier Series and their applications to the solution of boundary value problems associated with certain linear partial differential equations. In particular the wave equation, heat equation and Laplace’s equation will be studied using the technique of separation of variables. Various aspects of the theory of Fourier series will be discussed, for instance Bessel’s inequality, Parseval’s formula and the convergence and differentiability of Fourier series.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt:1 two-hour examination.

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MX 3023

Notes

Special Option. Not available in session 2009/10.

Overview

This course is a continuation of Mechanics A (MX 3023). The ideas and methods of that course are extended to study such topics as: Galilean transformations, systems of particles, the kinematics and dynamics of rigid bodies, analytic mechanics.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt: 1 two-hour examination.

Credit Points

15

Course Coordinator

Professor G Hall

Pre-requisites

Either: (a) MA 2507 and MA 2506; or: (b) MA 2507 and PX 2014.

Notes

(i) Special Option. Available in session 2009/10.

Overview

The failure of the Newtonian model of physics. The basic principles of the Special Theory of Relativity. The Lorentz transformation and its applications, including length and time dilation. The kinematics of particles. 4-vectors and Minkowski space. The dynamics of particles, momentum, energy and force. Relativistic optics. Collision problems.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%); in-course assessment (20%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MX 3522

Notes

Special Option. Not available in session 2009/10.

Overview

This course is concerned with the application of the Laplace and Fourier transformations to differential and integral equations. It begins with a brief discussion of differential equations. Then the theories of Laplace and Fourier transforms are developed and applied to various problems arising in the study of ordinary differential, partial differential and integral equations.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt: 1 two-hour examination.

Credit Points

15

Course Coordinator

Head of Teaching (Mathematical Sciences)

Pre-requisites

MX 3021

Notes

Special Option. Not available in session 2009/10.

Overview

An introduction to the differential geometry of surfaces. The emphasis will be on explicit local co-ordinate descriptions of surfaces, allowing the introduction of explicit examples throughout the course. The course will include Gauss’s Theorema Egregium, that the Gaussian Curvature, originally defined in terms of a particular embedding of the surface in space, is an intrinsic property of the surface.

Structure

12 week course - 2 one-hour lectures per week and 1 one-hour tutorial per fortnight.

Assessment

1st Attempt: 1 two-hour examination paper.

Credit Points

15

Course Coordinator

Dr D Benson

Pre-requisites

MX 3021

Notes

Special Option. Available in session 2009/10.

Overview

An introduction to knot theory. The course will include a study of knot invariants such as linking numbers, colourings, genus and some polynomial invariants.

Structure

12 week course - 2 one-hour lectures and 1 tutorial per week.

Assessment

1st Attempt: 1 two-hour written examination.

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination.

Credit Points

15

Course Coordinator

Mr A G Duncan / Mrs H Martin

Pre-requisites

Pass on 60 credits at level 2 mathematics.

Notes

Special Option. Available in 2009/10.

Overview

• Theories of learning: Piaget, Bruner, Gardner (multiple intelligences), Learning styles, constructivism (radical and social)


• Theories of learning mathematics: Dienes, Skemp (relational, instrumental understanding), Thompson (mental arithmetic strategies)


• Methods of teaching: direct interactive, exposition, investigative approach, problem solving, group work and discussion


• Contribution of technology (graphic calculators, graph drawing software, CAS, dynamic geometry, PowerPoint animation, internet)


• Lesson planning and preparation, presentation skills


• Research on learning and teaching school mathematics


• Project (choice of subject matter)

Structure

8 two-hour lectures/workshops and 4 one hour tutorials (total 20 hours) plus tutor directed activities.

Period of school experience - ideally four half mornings over two weeks. Presentation sessions.

Assessment

1st Attempt: Assessment will have three components:

• report on the School Project


• essay related to topics drawn from the lectures


• presentation to the class (peers)

Credit Points

15

Course Coordinator

Professor A Bondal

Pre-requisites

MX 3531

Notes

Special Option. This course is available in 2009/10.

Overview

Definition of Lie algebras; first properties and examples. Nilpotent, solvable and semisimple Lie algebras. The Killing form. Cartan subalgebras and the Jordan-Chevalley decomposition of linear transformation. Representations of sl(2). Root systems and Dynkin diagrams. The classification of complex semisimple Lie algebras. Elements of representation theory: highest weight modules. Weyl's character formula and applications.

Structure

12 week course - 2 one-hour lectures and one 1-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Credit Points

15

Course Coordinator

Professor G Robinson

Pre-requisites

MX 3020

Notes

Special Option. This course is not available in session 2009/10.

Overview

Some revision of group homomorphisms, vector spaces, and linear transformations. The complex group algebra of a finite group. Modules and representations, equivalence of matrix representations, Irreducibility. Maschke's Theorem on complete reducibility, Schur's Lemma. Complex characters; the ring of generalized characters of a finite group and its natural inner product. Irreducible characters, character tables, and orthogonality relations for group characters. Examples of construction of small character tables.

Algebraic integers, divisibilty of the group order by degrees of irreducible characters. Burnside's p^aq^b-theorem and other sample applications to group structure.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Credit Points

15

Course Coordinator

Dr A Bondal

Pre-requisites

MX 4082

Notes

Special Option. Available in session 2009/10.

Overview

Number theory is the study of integers and has three main branches: Elementary, Analytical and Algebraic. This course consists of a selection of topics from these branches. The topics will include some of the following: the theory of quadratic congruences, continued fractions, pseudo-primes, primitive roots, Diophantine equations, the distribution of prime numbers, algebraic integers in quadratic number fields.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week.

Assessment

1st Attempt: 1 two-hour examination (80%) and in-course assessment (20%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Credit Points

15

Course Coordinator

Professor R Levi

Pre-requisites

MX 3532 and MX 3020

Notes

Special Option. Available in 2009/10.

Overview

Elementary concepts of homotopy theory. The fundamental group and its naturality properties. Fundamental groups and covering spaces. Free groups and subgroups of free groups. The Seifert-VanKampen theorem. Presentations of groups. The concept of a surface. Triangulations. The classification of compact surfaces without boundary. If time allows, an introduction to homology theory.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week (to be arranged).

Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Credit Points

15

Course Coordinator

Professor A Bondal

Pre-requisites

MX 3531 and MX 3021

Notes

Special Option. This course is not available in 2009/2010.

Overview

The classical concept of an algebraic variety and the modern definition. Examples of algebraic varieties: curves, surfaces, projective spaces, quadrics. Methods of algebraic geometry 1: algebra vs. geometry. Projective curves. Parameterisation of curves and rational curves. Elliptic curves. The genus of curves. Methods of algebraic geometry 2: linear systems of divisors and projective embeddings. Linear systems on curves and line bundles. Riemann-Roch formula for curves. Methods of algebraic geometry 3: local vs. global. Maps between algebraic varieties. Singularities of algebraic varieties. If time allows, methods of algebraic geometry 4: coherent sheaves and cohomology. Intersection theory for divisors on surfaces. Riemann-Roch theorem for surfaces and its applications. Rational maps between surfaces.

Structure

12 week course - 2 one-hour lectures and 1 one-hour tutorial per week (to be arranged).

Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).