# Undergraduate Catalogue of Courses 2012/2013

# 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

*Course Co-ordinator:* Dr W Turner

*Pre-requisite(s):*
MA 2004 and MA 2506

- 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.

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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.

*Course Co-ordinator:* Dr D Quinn

*Pre-requisite(s):*
MA 2005

- 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.

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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.

*Course Co-ordinator:* Professor V Gorbunov

*Pre-requisite(s):*
MA 2506 and MA 2507.

- Introduction to the analysis in several variables.
- Convex sets, convex functions.
- Linear optimisation.
- Simplex method.
- Two phase simplex method. ,/ul>
- 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.
- 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.
- 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.
- Metric spaces and topological spaces.
- Compactness, connectedness.
- Subspace and product space topology.
- Complete metric spaces.
- 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.
- 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.
- 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.
- Knots and links.
- Linking number.
- Colouring of links.
- The Alexander polynomial.
- The Jones polynomial.
- Genus.
- The Jones polynomial.
- 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)
- report on the School Project
- essay related to topics drawn from the lectures
- presentation to the class (peers)
- 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.
- 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, divisibility of the group order by degrees of irreducible characters.
- Burnside's p^aq^b-theorem and other sample applications to group structure.
- 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.
- 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.
- 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.

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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.

*Course Co-ordinator:* Dr C Lopez

*Pre-requisite(s):*
MA 2507

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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

*Course Co-ordinator:* Dr J Elmer

*Pre-requisite(s):*
MX 3021

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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

*Course Co-ordinator:* Dr D Quinn

*Pre-requisite(s):*
MX 3020

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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

*Course Co-ordinator:* Dr A Gonzalez

*Pre-requisite(s):*
MX 3021

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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

*Course Co-ordinator:* Dr C Lopez

*Pre-requisite(s):*
MA 2507

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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

**PLEASE NOTE: Resit: (for Honours students only): Candidates achieving a CAS mark of 6-8 may be awarded compensatory level 1 credit. Candidates achieving a CAS mark of less than 6 will be required to submit themselves for re-assessment and should contact the Course Co-ordinator for further details.**

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
Available only to candidates for Honours in Mathematics, Mathematics with French, Mathematics with Gaelic, Mathematics with German, and Mathematics with Spanish.

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.

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

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.

## Formative Assessment and Feedback Information

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

*Course Co-ordinator:* Dr A Sevastyanov

*Pre-requisite(s):*
None

*Note(s):*
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.

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.

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

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.

## Formative Assessment and Feedback Information

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

*Course Co-ordinator:* Professor G Hall

*Pre-requisite(s):*
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).

*Note(s):*
This course is run over the full session.

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.

24 week course – 1 tutorial per week.

Assessed on the project report and on the oral examination.

*Course Co-ordinator:* TBC

*Pre-requisite(s):*
MX 3021

*Note(s):*
Not available in 2012/13.

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.

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

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%).

*Course Co-ordinator:* Dr J Elmer

*Pre-requisite(s):*
MX 3531

*Note(s):*
This course will run in 2012/13.

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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

*Course Co-ordinator:* Dr A Sevastyanov

*Pre-requisite(s):*
MX 3021, MX 3532

*Note(s):*
This course will run in 2012/13.

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

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 and Feedback Information

Informal assessment of weekly homework through discussions in tutorials.

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

*Course Co-ordinator:* Dr Marco Thiel

*Pre-requisite(s):*
For students taking MSc in systems biology or students having completed 3rd year Mathematics/Physics or at discretion of Head of Department.

*Note(s):*
Part of the Applied Mathematics Degree; Option for Physics. (Option for the MSc in systems biology). This course will run 2012/13.

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.

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

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 and Feedback Information

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.

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

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3532

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

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

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3531 and MX 3532

*Note(s):*
Special Option. Not available in session 2012/13.

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

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

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

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3532

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

1st Attempt: 1 two-hour examination.

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*

*Note(s):*
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.

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.

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

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.

## Formative Assessment and Feedback Information

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

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MA 2506 and MX 3532

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

1st Attempt: 1 two-hour examination.

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3532

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

1st Attempt: 1 two-hour examination.

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3531

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

1st Attempt: 1 two-hour examination.

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3532

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

1st Attempt: 1 two-hour examination.

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3023

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

1st Attempt: 1 two-hour examination.

*Course Co-ordinator:* TBC

*Pre-requisite(s):*
Either: (a) MA 2507 and MA 2506; or: (b) MA 2507 and PX 2015.

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

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%).

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3522

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

1st Attempt: 1 two-hour examination.

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3021

*Note(s):*
Special Option. Not available in session 2012/13.

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.

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

1st Attempt: 1 two-hour examination paper.

*Course Co-ordinator:* Dr R Hepworth

*Pre-requisite(s):*
MX 3532.

*Note(s):*
Special option. Available in 2012/13.

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:

2 hours of lectures and 1 tutorial each week.

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 and Feedback Information

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.

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

*Course Co-ordinator:* TBC

*Pre-requisite(s):*
Pass on 60 credits at level 2 mathematics.

*Note(s):*
Special Option. Available in 2012/13.

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.

1st Attempt: Assessment will have three components:

Reassessment on the first two of the above.

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3531

2 hour lectures and 1 hour of tutorial each week.

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 at the first attempt can count towards Honours classification.

## Formative Assessment and Feedback Information

Tutorials contain practice questions allowing formative assessment from the tutor.

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

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3020

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

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 at the first attempt can count towards Honours classification.

## Formative Assessment and Feedback Information

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.

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

*Course Co-ordinator:* Dr M McLean

*Pre-requisite(s):*
MX 4082

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.

3 hour lectures and 1 hour tutorial each week.

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 and Feedback Information

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.

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

*Course Co-ordinator:* Professor R Levi

*Pre-requisite(s):*
MX 3532 MX 3030

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

1st attempt: 1 two-hour written 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 and Feedback Information

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.

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

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
MX 3020

*Note(s):*
This course will not run in 2012/13.

The classical concept of an algebraic variety and the modern definition. Examples of algebraic varieties: curves, surfaces, projective spaces, quadrics.

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

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 and Feedback Information

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.

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

*Course Co-ordinator:* Professor M Geck

*Pre-requisite(s):*
MX 3020

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

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 and Feedback Information

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.

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

*Course Co-ordinator:* Dr J Kedra

*Pre-requisite(s):*
None.

*Note(s):*
Special Option: Available in 2011-2012.

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

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

1st Attempt: 1 two-hour written 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%).

## Formative Assessment and Feedback Information

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.

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

*Course Co-ordinator:* Head of Mathematical Sciences

*Pre-requisite(s):*
Available only to candidates for Honours in Mathematics, Mathematics with French, Mathematics with Gaelic, Mathematics with German, and Mathematics with Spanish.

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.

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

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 and Feedback Information

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

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

*Course Co-ordinator:* Dr M Thiel

*Pre-requisite(s):*
Level 3 in Engineering, Physics or Mathematics.

*Note(s):*
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.

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.

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

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

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

## Formative Assessment and Feedback Information

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.

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.

*Course Co-ordinator:* Dr M Thiel

*Pre-requisite(s):*
MX 4085 or at discretion of Head of Department.

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.

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

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 and Feedback Information

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.

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