# Mathematical Sciences

*Course Co-ordinator:* Dr Stephen Theriault

*Pre-requisite(s):*
MA or BSc in Mathematics

Concrete examples of both Riemann and Lebesgue integration; Abstract theory - convergence theorems; Signed, product and Radon measures; Fractal sets and Hausdorff dimension; L^p spaces; Differentiation and Fourier Series

1 two-hour timetabled lecture and 1 one-hour tutorial (to be arranged) per week.

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

*Course Co-ordinator:* Dr Stephen Theriault

*Pre-requisite(s):*
MA or BSc in Mathematics

Basic concepts in group theory, including definitions and examples; construction of groups; generators and relations; simple groups; the Jordan-Holder theorem; soluble groups; group actions; conjugation; Sylow theorems and applications. The definitions and basic properties of rings and modules; chain conditions; Hilbert Basis theorem; PIDs, Euclidean domains and UFDs; some elementary background on algebraic numbers and algebraic integers.

1 timetabled two-hour lecture and 1 one-hour tutorial (to be arranged) per week.

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

*Course Co-ordinator:* Dr Stephen Theriault

*Pre-requisite(s):*
MA or BSc in Mathematics

Algebraic topology: elements of general topology; the fundamental group and covering spaces; free groups, group presentations and the Seifert-van Kampen theorem; compact orientable surfaces; CW-complexes; singular and simplicial homolgy. Differential geometry: smooth manifolds and smooth maps between them; vector bundles on smooth manifolds.

1 two-hour timetabled lecture and 1 one-hour tutorial (to be arranged) per week.

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

*Course Co-ordinator:* Professor Rob Archbold

*Pre-requisite(s):*
MA or BSc in Mathematics

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.

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

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

*Course Co-ordinator:* Dr Lacri Iancu

*Pre-requisite(s):*
MA or BSc in Mathematics

The roots of a quadratic polynomial are given by a formula involving the coefficients. Similar formulae exist for 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.

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

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

*Course Co-ordinator:*

*Pre-requisite(s):*
MA or BSc in Mathematics

*Note(s):* Not running in 2011/12.

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.

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

1 two hour written examination (80%; in-course assessment (20%)

*Course Co-ordinator:* Dr Stephen Theriault

*Pre-requisite(s):*
MA or BSc in Mathematics

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

1 one-hour meeting per week with the project supervisor

Assessed on the project report and the oral presentation (the oral presentation may be given in the second half-session.)

*Course Co-ordinator:* Marco Thiel

*Pre-requisite(s):*
MA or BSc in Mathematics

This course covers the fundamental mathematical concepts required for the description of dynamical systems, i.e., 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, i.e., fractal attractors.

Next to the theory of relativity and quantum mechanics, chaos and dynamical systems theory is 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 one-hour tutorial (to be arranged); 1 computer tutorial/project.

1 two hour written examination (50%); continuous assessment (50%).

*Course Co-ordinator:*

*Pre-requisite(s):*
MA or BSc in Mathematics; or at discretion of Head of Department of Mathematics

*Note(s):* Not running in 2011/12

Theories and proofs about mathematical structures are written in logical languages. Such structures are often organised into universes called categories which are widely used in research throughout the mathematical sciences. This course aims to introduce students to the fundamentals of both mathematical logic and category theory - important subjects which are usually not studied (in any depth) at the undergraduate level. Students will learn to formulate problems in logical languages and structures, and to understand the consequences and limitations of logical languages. They will also learn to work with categories, and about their relation to logic. Natural topics for inclusion in this course would be: basic mathematical logic; proof theory; intuitionism and its models; categories and categorical logic; monoidal categories and substructural logic.

1 one-hour tutorial per-week; approximately 11 hours per-week self-teaching (reading, exercises); and 4 2/3 hours per-week project work towards the final report.

Written report (100%).

*Course Co-ordinator:* Dr Stephen Theriault

*Pre-requisite(s):*
Analysis 1

Elementary notions in functional analysis; Banach and Hilbert spaces; Baire Category; Open Mapping and Uniform Boundedness Principle; Weak and weak* topologies; Compact operators; Spectral theory (C*-algebras).

1 two-hour timetabled lecture and 1 one-hour tutorial (to be arranged) per week.

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

*Course Co-ordinator:* Dr Stephen Theriault

*Pre-requisite(s):*
Algebra 1

Modules and Rings: finitely generated modules over a PID; Jordan canonical form of a matrix; the Artin-Wedderburn theorem; modules over semisimple Artinian rings. Ordinary representations theory of finite groups: Maschke's theorem; characters and character tables; tensor products; applications to groups such as Burnside's theorem.

1 timetabled two-hour lecture and 1 one-hour tutorial (to be arranged) per week.

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

*Course Co-ordinator:* Dr Stephen Theriault

*Pre-requisite(s):*
Geometry and Topology 1

Differential geometry: curvature and connections; geodesics: Riemannian curvature. Differential Topology: orientations and manifolds with boundary; differential forms and integration; Strokes' theorem and the Gauss-Bonnet theorem for surfaces; de Rham cohomology; the Mayer-Vietoris sequence; Poincare duality.

1 two-hour timetable lecture and 1 one-hour tutorial (to be arranged) per week.

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

*Course Co-ordinator:* Dr Jarek Kedra

*Pre-requisite(s):*
MA or BSc in Mathematics

*Note(s):* Not running in 2011/12

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

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

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

*Course Co-ordinator:*

*Pre-requisite(s):*
MA or BSc in Mathematics

*Note(s):* Not running in 2011/12

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

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

1 two hour written examination (80%); in-course assessment (20%)

*Course Co-ordinator:* Professor Ran Levi

*Pre-requisite(s):*
MA or BSc in Mathematics

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

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

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

*Course Co-ordinator:* Professor Alexey Bondal

*Pre-requisite(s):*
MA or BSc in Mathematics

*Note(s):* Not running in 2011/12.

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 formual 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.

2 one hour lectures and 1 one hour tutorial per week

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

*Course Co-ordinator:* Professor Meinolf Geck

*Pre-requisite(s):*
MA or BSc in Mathematics

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.

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

*Course Co-ordinator:* Professor Markus Linckelmann

*Pre-requisite(s):*
MA or BSC in Mathematics

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, and algebraic integers in quadratic number fields.

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

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

*Course Co-ordinator:* Dr Stephen Theriault

*Pre-requisite(s):*

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.

1 one-hour meeting per week with the project supervisor.

Assessed on the project report and the oral presentation (the oral presentation may be given in the second half-session).

*Course Co-ordinator:* Dr Marco Thiel

*Pre-requisite(s):*
MA or BSc in Mathematics; or at discretion of Head of Department of Mathematics

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, e.g. 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.

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

Continuous assessment (assignments & projects; 80%); oral exam (20%).

*Course Co-ordinator:* Dr Jarek Kedra

*Pre-requisite(s):*
None

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.

1 two hour written examination (80%); in-course assessment (20%)

*Course Co-ordinator:* Dr Stephen Theriault

*Pre-requisite(s):*

A specialist topic in mathematics will be chosen (with staff assistance). The topic will be studied in depth, with appropriate supervision. A dissertation will be written which is mathematically rigorous and of a high standard and the dissertation will be presented to a board of examiners.

Individual supervision sessions with an appropriate member of staff (normally one hour per week, although this may vary depending on the project).

Examination of dissertation (100%).