MATHEMATICAL SCIENCES | 2013 Undergraduate Catalogue of Courses

MATHEMATICAL SCIENCES

(see also Mathematics(MA), Statistics(ST)) NOTES(S): FOR ALL COURSES AT LEVEL 3 WHICH ARE EXAMINED IN PART BY CONTINUOUS ASSESSMENT: STUDENTS MAY IN EXCEPTIONAL CIRCUMSTANCES BE REQUIRED TO ATTEND AND ORAL EXAMINATION. NOT ALL THE LEVEL 4 MATHEMATICAL SCIENCES SPECIAL OPTIONS WILL BE AVAILABLE IN ANY ONE ACADEMIC SESSION

Level 3

MX 3010 - STOCHASTIC PROCESSES

Credit Points: 15
Course Coordinator: Dr Francesco Ginelli

Pre-requisites

For Physics students: PX 3012, PX 3508 and PX 3509
For Applied Mathematics: Completion of second year.

Overview

This course provides an introduction to statistical physics and simple stochastic processes.
A brief review of thermodynamics is offered in the first lectures to set up the background needed to understand the foundations of statistical mechanics. The microcanonical and canonical ensembles are discussed in details and it is shown as thermodynamical irreversibility emerges statistically from reversible microscopic dynamics. Quantum statistics are also discussed.
Applications are limited to non-interacting systems and include computing the specific heats of solids and of monoatomic and simple diatomic gases, blackbody radiation, Fermi gases and ferromagnetism. Ising magnetism is briefly discussed in mean field approximation.
In addition, the course discusses Brownian motion and random walks, introducing the concepts of Master and Fokker-Planck equations for one-dimensional random walks. This offers a simple context in which the central limit theorem can be introduced.

Structure

Two lectures and one tutorial per week

Assessment

1st Attempt:70 % final examination and 30% continuous assessment exercises.

Resit: 100% Examination
Only the mark obtained at first attempt can count towards Honours classification

Formative Assessment

By means of class tutorials and dialogue with the lecturer.

Feedback

Feedback on assessments will be given within two weeks of receipt and immediately during classroom exercises.

MX 3020 - GROUP THEORY

Credit Points: 15
Course Coordinator: Dr W Turner

Pre-requisites

MA 2004 and MA 2506

Overview

Group axioms, subgroups, examples of groups.

Cosets of a subgroup
Lagrange's Theorem.

Homomorphisms, isomorphisms, normal subgroups, quotient groups.

Calculations in symmetric and alternating groups.

Group actions.

Sylow's Theorems.

Structure

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

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).
Only the marks obtained on first sitting can be used for Honours classification.

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials.

Students will be invited to contact Course Coordinators for feedback on the final examination.

MX 3021 - FURTHER REAL ANALYSIS

Credit Points: 15
Course Coordinator: Dr D Quinn

Pre-requisites

MA 2005

Overview

Cauchy sequences, superior and inferior limits.

Standard series tests, absolute and conditional convergence.

Power series, Taylor's theorem and Taylor series.

Uniform convergence of sequences and series of functions.

Improper integrals.

Structure

5 one-hour lectures per fortnight and 1 one-hour tutorial per week.

Assessment

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

MX 3022 - OPTIMISATION AND NUMERICAL ANALYSIS

Credit Points: 15
Course Coordinator: Professor V Gorbunov

Pre-requisites

MA 2506 and MA 2507.

Overview

Introduction to the analysis in several variables.

Convex sets, convex functions.

Linear optimisation.

Simplex method.

Two phase simplex method.

Structure

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

Assessment

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

MX 3023 - MECHANICS A

Credit Points: 15
Course Coordinator: Professor V Gorbounov

Pre-requisites

MA 2507

Overview

Air Resistance Revision of the basic ideas of Newtonian mechanics and application to particles moving under gravity.

The effect of air resistance and the idea of terminal velocity.

Oscillations The theory of oscillations of one-dimensional systems. Hooke's law for springs. Free vibrations, damped vibrations and forced vibrations. The concept of resonance.

Momentum and Angular Momentum Definitions and basic concepts. Rate of change of angular momentum equals moment.

Conservation laws. Motion under a central force. Newton's law of gravity.

Energy and Potentials Force fields. Gradient of a scalar function. Consevative force fields and potentials.

Consevation of energy.

Applications.

Inertial frames The basic ideas of inertial frames and a brief discussion of Galilean transformations.

Systems of Particles Centre of mass, the motion of the centre of mass. Total angular momentum of a system
of particles. Closed systems.

Two body problem.

Collisions Elementary theory. Conservation of momentum. Elastic collisions and conservation of energy.

Simple applications.

Structure

5 one-hour lectures per fortnight and 1 one-hour tutorial per week.

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).
Only marks gained on first sitting can be used for Honours classification.

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

MX 3522 - COMPLEX ANALYSIS

Credit Points: 15
Course Coordinator: Dr J Elmer

Pre-requisites

MX 3021

Overview

Revision of complex numbers, roots of unity, polynomials.

Elementary functions, differentiation, Cauchy-Riemann equations.

Path integrals, Cauchy's Theorem and Cauchy's Integral Formulae.

Liouville's Theorem and the Fundamental Theorem of Algebra.

Taylor Series, Laurent Series, Cauchy's Residue Theorem and applications to real integrals.

Structure

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

Assessment

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

MX 3531 - RINGS AND FIELDS

Credit Points: 15
Course Coordinator: Dr D Quinn

Pre-requisites

MX 3020

Overview

Basic concepts and examples. Ideals, factor rings, isomorphism theorems.

Rings of polynomials.

Field of fractions of a domain.

Unique Factorization Domains, Principal Ideal Domains, Euclidean Domains.

Passage from R to R[X]. Gauss's Theorem. Eisenstein's criterion.

Fields : characteristic, prime subfield.

Finite fields, construction.

Algebraic and transcendental elements, algebraic closure.

Structure

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

Assessment

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

MX 3532 - METRIC AND TOPOLOGICAL SPACES

Credit Points: 15
Course Coordinator: Professor V Gorbounov

Pre-requisites

MX 3021

Overview

Metric spaces and topological spaces.

Compactness, connectedness.

Subspace and product space topology.

Complete metric spaces.

Structure

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

Assessment

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

MX 3533 - METHODS OF MATHEMATICAL PHYSICS

Credit Points: 15
Course Coordinator: Dr A Sevastyanov

Pre-requisites

MA 2507

Overview

Revision of chain rule.

Curves and surfaces.

Scalar and vector fields.

Directional derivative and the gradient of a scalar field.

Divergence and curl of a vector field.

Some coordinate systems in space.

Integrals over a curve.

Integrals over a surface.

Volume integrals and the divergence theorem.

Partial differential equations (an introduction).

Fourier series.

Structure

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

Assessment

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

Level 4

MX 4022 - EXTERNAL PROJECT

Credit Points: 15
Course Coordinator: Dr M Boyle

Pre-requisites

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

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

Weekly meetings with a member of staff and external representatives (to be arranged).

Assessment

1st Attempt: Assessed on the project report (40 pages approximately) and the oral presentation (the presentations are given during the second half-session).

Resit: Assessed on the revised project report. Only the marks gained on first attempt will count towards Honours classification.

Feedback

Students contact their project supervisors and/or the course coordinator for feedback.

MX 4023 - PROJECT

Credit Points: 15
Course Coordinator: Dr M Boyle

Pre-requisites

None

Notes

Subject to availability. Available only to students in the 4th year of a maths-related programme or to non-graduating students with permission of the Head of Discipline.

Overview

Upon registration for the course, the student will be asked to see the course
coordinator, who will normally have a list of topics for students to choose from. The course coordinator will discuss
preferences with the student and then assign a topic and a supervisor. Requests to work on a certain topic of the
student's own choice are acceptable, but this request will only be granted if the topic is regarded as appropriate and if proper supervision is conveniently available.

Structure

Weekly meetings with a member of staff (to be arranged).

Assessment

1st Attempt: Assessed on the project report (40 pages approximately) and the oral presentation (the presentations are given during the second half-session).

Resit: Assessed on the revised project report. Only the mark obtained on first attempt will count towards Honours classification.

Feedback

Students contact their project supervisor and/or the course coordinator for feedback.

MX 4082 - GALOIS THEORY

Credit Points: 15
Course Coordinator: Dr J Elmer

Pre-requisites

MX 3531

Notes

This course will run in 2013/14.

Overview

Field Theory, Field Extensions.

Constructible Numbers.

The Galois Group of a Field Extension.

Cyclotomic Fields.

Splitting Fields of Polynomials.

Normal Extensions, Separable Extensions.

Simple Fields Extensions.

Counting Field Homomorphisms.

Galois Extensions.

The Galois Correspondence.

Cyclic Galois Groups.
Radical Extensions and Solvable Galois Groups.

The Galois Group of a Polynomial. Applications.

Structure

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%).

Only the marks obtained on first sitting can be used for Honours classification.

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

MX 4083 - MEASURE THEORY

Credit Points: 15
Course Coordinator: Dr J Kedra

Pre-requisites

MX 3021, MX 3532

Notes

This course will run in 2013/14.

Overview

Sigma-algebras, measures, measurable functions.

The integral of simple, positive, measurable functions and hence of positive measurable functions.

Monotone Convergence Theorem.

Integrable functions, Fatou's Lemma, the Dominated Convergence Theorem and applications.

Comparison of Riemann, Cauchy-Riemann and Lebesgue integrals.

The Lebesgue L^p-spaces.

Discussion of the Tonelli and Fubini theorems, with applications to continuous functions of two real variables.

Structure

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

Assessment

Only the marks obtained on first sitting can be used for Honours classification.

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

MX 4085 - NONLINEAR DYNAMICS & CHAOS THEORY I

Credit Points: 15
Course Coordinator: Dr Marco Thiel

Pre-requisites

For students taking MSc in systems biology or students having completed 3rd year Mathematics/Physics or at discretion of Head of Department.

Notes

Part of the Applied Mathematics Degree; Option for Physics. (Option for the MSc in systems biology). This course will run 2013/14.

Overview

This course covers the fundamental mathematical concepts required for the description of dynamical systems, ie., systems that change in time. It discusses ordinary differential equations and nonlinear systems, for which typically no analytical solutions can be found; these systems are pivotal for the description of natural systems in physics, engineering, biology etc. Some emphasis will be on the study of chaotic systems and strange, ie., fractal attractors.

Next to the theory of relativity and quantum mechanics, chaos and dynamical systems theory is considered as one of three major advances in the natural sciences. This course covers the mathematics behind this paradigm changing theory.

Structure

2 one-hour lectures and 1 one-hour tutorial (to be arranged).

Assessment

1st Attempt: 1 two-hour written examination (70%); continuous assessment in the form of weekly assignments (30%).

Written Exam (100%).
Only the marks obtained at the first attempt can count towards Honours classification.

Formative Assessment

By weekly tutorials and dialogue with lecturer. Toward the end of the course a mock examination will help to gauge the development of the students.

Feedback

Within two weeks of a continuous assessment exercise - immediate feedback in class tutorials.

MX 4520 - PROJECT

Credit Points: 15
Course Coordinator: Dr M Boyle

Pre-requisites

Notes

Subject to availability. Available only to students in the 4th year of a maths-related programme or to non-graduating students with permission of the Head of Discipline.

Overview

Upon registration for the course, the student will be asked to see the course coordinator, who will normally have a list of topics for students to choose from. The course coordinator will discuss preferences with the student and then assign a topic and a supervisor. Requests to work on a certain topic of the student's own choice are acceptable, but this request will only be granted if the topic is regarded as appropriate and if proper supervision is conveniently available.

Structure

Weekly meetings with a member of staff (to be arranged).

Assessment

1st Attempt: Assessed on the project report and the oral presentation (the presentations are given during the second half-session).

Resit: Assessed on the revised project report.

Only the marks obtained at the first attempt can count towards Honours classification.

Feedback

Students contact their project supervisor and/or the course coordinator for feedback.

MX 4540 - KNOTS

Credit Points: 15
Course Coordinator: Dr R Hepworth

Pre-requisites

MX 3532.

Notes

Special option. Available in 2013/14.

Overview

Knots have been studied mathematically since the 19th century, and knot theory connects with many other areas of pure maths and even theoretical physics. This course concentrates on knot invariants: numbers, polynomials or groups that try to "measure" properties of the knots. Classic invariants such as the colouring group will lead to more modern ones like the Jones polynomial, which was only discovered in the mid-1980s.

Topics will include the following:

Knots and links.
Linking number.
Colourings,determinants and the colouring group.
The Alexander polynomial.
The Jones polynomial.
The Genus.

Structure

2 hours of lectures and 1 tutorial each week.

Assessment

1st Attempt: 1 two-hour written examination.

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

Only the marks obtained at the first attempt can count towards Honours classification.

Formative Assessment

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.
Students do practice questions in tutorials allowing formative assessment and feedback from the tutor.

Feedback

In-course assessment will be marked and feedback provided to the students.

MX 4545 - NUMBER THEORY

Credit Points: 15
Course Coordinator: Dr W Turner

Pre-requisites

MX 4082

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

2 hour lectures and 1 hour tutorial each 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%).

Only the marks obtained at the first attempt can count towards Honours classification.

Formative Assessment

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.
Students undertake practice questions in tutorials allowing formative assessment and feedback from tutors.

Feedback

In-course assessment will be marked and feedback provided to the students.

MX 4546 - ALGEBRAIC TOPOLOGY

Credit Points: 15
Course Coordinator: Dr R Hepworth

Pre-requisites

MX 3532, MX 3030

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

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%).

Only the marks obtained at the first attempt can count towards Honours classification.

Formative Assessment

Feedback

In-course assessment will be marked and feedback provided to the students.

MX 4547 - ALGEBRAIC GEOMETRY

Credit Points: 15
Course Coordinator: Head of Mathematical Sciences

Pre-requisites

MX 3020

Notes

This course will not run in 2013/14.

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

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

Assessment

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

Only the marks obtained at the first attempt can count towards Honours classification.

Formative Assessment

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.

Students do practice questions in tutorials allowing formative assessment and feedback from tutors.

Feedback

In-course assessment will be marked and feedback provided to the students.

MX 4548 - REFLECTION GROUPS

Credit Points: 15
Course Coordinator: Head of Mathematical Sciences

Pre-requisites

MX 3020

Notes

This course is not available in 2013/14.

Overview

Review of scalar product spaces and group actions.

Root vectors and reflections in real Euclidean space.

Regular polygons in two dimensions.

Root systems and groups generated by reflections.
Presentations and Coxeter groups.

The classification in terms of Dynkin diagrams.

Structure

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

Assessment

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

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

Only the marks obtained on first attempt can be used for Honours classification.

Formative Assessment

Feedback

In-course assessment will be marked and feedback provided to the students.

MX 4549 - GEOMETRY

Credit Points: 15
Course Coordinator: Dr J Kedra

Pre-requisites

None.

Notes

Special Option: Available in 2013-2014.

Overview

The geometry of polygonal complexes; manifolds, curves on manifolds, vector fields, simple mechanical systems, Riemannian metric, geodesics, curvature; examples from modern physics.

Structure

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%).

Formative Assessment

Feedback

In-course assessment will be marked and feedback provided to the students.

MX 4552 - EXTERNAL PROJECT

Credit Points: 15
Course Coordinator: Dr M Boyle

Pre-requisites

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

Overview

Structure

Weekly meetings with a member of staff and external representatives (to be arranged).

Assessment

1st Attempt: Assessed on the project report (40 pages approximately) and the oral presentation (the presentations are given during the second half-session).

Resit: Assessed on the revised project report. Only the marks obtained on first sitting can be used for Honours classification.

Formative Assessment

Supervisors will normally give feedback during the first stages of writing up and on the first draft of the report.

Feedback

Students contact their project supervisors and/or the course coordinator for feedback.

MX 4553 - MODELLING THEORY

Credit Points: 15
Course Coordinator: Dr M Thiel

Pre-requisites

Level 3 in Engineering, Physics or Mathematics.

Notes

Physical Sciences intend to describe natural phenomena in mathematical terms. This course bridges the gap between standard courses in physical sciences, where successful mathematical models are described, and scientific research, where new mathematical models have to be developed. Students will learn the art of mathematical modelling, which will enable them to develop new mathematical models for the description of natural systems. Examples from a wide range of phenomena will be discussed, eg from biology, ecology, engineering, physics, physiology and psychology.
A focus will be the critical interpretation of the mathematical models and their predictions. The applicability of the models will be assessed and their use for the respective branch of the natural sciences will be discussed.

Overview

Physical Sciences intend to describe phenomena in mathematical terms. This course bridges the gap between standard courses in physical sciences, where successful mathematical models are described, and scientific research, where new mathematical models have to be developed. Students will learn the art of mathematical modelling, which will enable them to develop new mathematical models for the description of natural systems. Examples from a wide range of phenomena will be discussed, eg from biology, ecology, engineering, physics, physiology and psychology.

A focus will be the critical interpretations of the mathematical models and their predictions. The applicability of the models will be assessed and their use for the respective branch of the natural sciences will be discussed.

Structure

2 one-hour lectures, 1 one-hour computer lab/lecture, and 1 one-hour tutorial per week.

Assessment

1st attempt: Continuous assessment (assignments & projects (80%); oral exam (20%)).

Resit: Mini modelling project (80%) + oral exam (20%).

Formative Assessment

Formative assessment will be by means of a continuous dialogue with the lecturer and interaction with the same during the problem solving exercises and the developement of models.

Feedback

Due to the nature of the (primarily) continuous assessment of the course - summative assessment will be on a continuous ongoing basis as project work is marked.

MX 4555 - NONLINEAR DYNAMICS & CHAOS THEORY II

Credit Points: 15
Course Coordinator: Dr M Thiel

Pre-requisites

MX 4085 or at discretion of Head of Department.

Overview

This course covers advanced mathematical concepts required for the description of dynamical systems, ie., systems that change in time. It discusses nonlinear systems, for which typically no analytical solutions can be found; these systems are pivotal for the description of natural systems in physics, engineering, biology etc. Some emphasis will be on the study of chaotic systems and strange, ie., fractal attractors.

Next to the theory of relativity and quantum mechanics, chaos and dynamical systems theory has been considered as one of three major advances in the natural sciences. This course covers the mathematics behind this paradigm changing theory.

Structure

2 one-hour lectures and 1 one-hour tutorial (to be arranged).

Assessment

1st Attempt: 1 two-hour written examination (70%); continuous assessment in the form of weekly assignments (30%).

Resit: Written Exam (100%).

Only the marks obtained at the first attempt can count towards Honours classification.

Formative Assessment

By weekly tutorials and dialogue with lecturer. Toward the end of the course a mock examination will help to gauge the development of the students.

Feedback

Within two weeks of a continuous assessment exercise. Immediate feedback in class tutorials.