 MX5001  Analysis 1

 Credit Points
 20
 Course Coordinator
 Prof. Ran Levi
Prerequisites
MA or BSc in Mathematics
Overview
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
Structure
1 twohour timetabled lecture and 1 onehour tutorial (to be arranged) per week.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5002  Algebra 1

 Credit Points
 20
 Course Coordinator
 Dr J. B. Gramain
Prerequisites
MA or BSc in Mathematics
Overview
Basic concepts in group theory, including definitions and examples; construction of groups; generators and relations; simple groups; the JordanHolder 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.
Structure
1 timetabled twohour lecture and 1 onehour tutorial (to be arranged) per week.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5003  Geometry and Toplogy 1

 Credit Points
 20
 Course Coordinator
 Dr Richard Hepworth
Prerequisites
MA or BSc in Mathematics
Overview
Algebraic topology: elements of general topology; the fundamental group and covering spaces; free groups, group presentations and the Seifertvan Kampen theorem; compact orientable surfaces; CWcomplexes; singular and simplicial homolgy. Differential geometry: smooth manifolds and smooth maps between them; vector bundles on smooth manifolds.
Structure
1 twohour timetabled lecture and 1 onehour tutorial (to be arranged) per week.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5004  Measure Theory

 Credit Points
 20
 Course Coordinator
 Prof. Ran Levi
Prerequisites
MA or BSc in Mathematics
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
2 onehour lectures and 1 onehour tutorial per week.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5005  Galois Theory

 Credit Points
 20
 Course Coordinator
 Prof. Ran Levi
Prerequisites
MA or BSc in Mathematics
Overview
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.
Structure
2 onehour lectures and 1 onehour tutorial per week
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5006  Ordinary Differential Equations

 Credit Points
 20
 Course Coordinator
Prerequisites
MA or BSc in Mathematics
Notes
Not running in 2013/14.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
2 onehour lectures and 1 onehour tutorial per week
Assessment
1 two hour written examination (80%; incourse assessment (20%)
 MX5007  Reading Project

 Credit Points
 20
 Course Coordinator
 Dr Stephen Theriault
Prerequisites
MA or BSc in Mathematics
Overview
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.
Structure
1 onehour meeting per week with the project supervisor
Assessment
Assessed on the project report and the oral presentation (the oral presentation may be given in the second halfsession.)
 MX5008  Nonlinear Dynamics & Chaos Theory

 Credit Points
 20
 Course Coordinator
 Dr Marco Thiel
Prerequisites
MA or BSc in Mathematics
Overview
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.Structure
2 onehour lectures and onehour tutorial (to be arranged); 1 computer tutorial/project.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5009  Logic and Categories

 Credit Points
 20
 Course Coordinator
Prerequisites
MA or BSc in Mathematics; or at discretion of Head of Department of Mathematics
Notes
Not running in 2013/14.Overview
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.
Structure
1 onehour tutorial perweek; approximately 11 hours perweek selfteaching (reading, exercises); and 4 2/3 hours perweek project work towards the final report.
Assessment
Written report (100%).
 MX5501  Analysis 2

 Credit Points
 20
 Course Coordinator
 Prof. Ran Levi
Prerequisites
Analysis 1
Overview
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).
Structure
1 twohour timetabled lecture and 1 onehour tutorial (to be arranged) per week.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5502  Algebra 2

 Credit Points
 20
 Course Coordinator
 Prof. Ran Levi
Prerequisites
Algebra 1
Overview
Modules and Rings: finitely generated modules over a PID; Jordan canonical form of a matrix; the ArtinWedderburn 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.
Structure
1 timetabled twohour lecture and 1 onehour tutorial (to be arranged) per week.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5503  Geometry and Topology 2

 Credit Points
 20
 Course Coordinator
 Prof. Ran Levi
Prerequisites
Geometry and Topology 1
Overview
Differential geometry: curvature and connections; geodesics: Riemannian curvature. Differential Topology: orientations and manifolds with boundary; differential forms and integration; Strokes' theorem and the GaussBonnet theorem for surfaces; de Rham cohomology; the MayerVietoris sequence; Poincare duality.
Structure
1 twohour timetable lecture and 1 onehour tutorial (to be arranged) per week.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5504  Knot Theory

 Credit Points
 20
 Course Coordinator
 Dr Richard Wepworth
Prerequisites
MA or BSc in Mathematics
Overview
An introduction to knot theory. the course will include a study of knot invariants such as linking numbers, colourings, genus and some polynominal invariants.
Structure
2 onehour lectures and 1 onehour tutorial per week
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5505  Special Relativity

 Credit Points
 20
 Course Coordinator
Prerequisites
MA or BSc in Mathematics
Notes
Not running in 2013/14.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. 4vectors and Minkowski space. The dynamics of particles, momentum, energy and force. Relativistic optics, collision problems
Structure
2 onehour lectures and 1 onehour tutorial per week.
Assessment
1 two hour written examination (80%); incourse assessment (20%)
 MX5506  Algebraic Topology

 Credit Points
 20
 Course Coordinator
 Professor Ran Levi
Prerequisites
MA or BSc in Mathematics
Overview
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 Seifertvan 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.
Structure
2 onehour lectures and 1 onehour tutorial per week.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5507  Algebraic Geometry

 Credit Points
 20
 Course Coordinator
Prerequisites
MA or BSc in Mathematics
Notes
Not running 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. RiemannRoch 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. RiemannRoch theorem for surfaces and its applications. Rational maps between surfaces.
Structure
2 one hour lectures and 1 one hour tutorial per week
Assessment
1 two hour written examination (80%); incourse assessment (20%).
 MX5509  Number Theory

 Credit Points
 20
 Course Coordinator
 Dr Will Turner
Prerequisites
MA or BSC in Mathematics
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, pseudoprimes, primitive roots, Diophantine equations, the distribution of prime numbers, and algebraic integers in quadratic number fields.
Structure
2 onehour lectures and 1 onehour tutorial.
Assessment
One 3hour written examination (80%); one incourse assessment (20%).
 MX5510  Reading Project 2

 Credit Points
 20
 Course Coordinator
 Dr Stephen Theriault
Prerequisites
N/A
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
1 onehour meeting per week with the project supervisor.
Assessment
Assessed on the project report and the oral presentation (the oral presentation may be given in the second halfsession).
 MX5511  Mathematical Modelling

 Credit Points
 20
 Course Coordinator
 Dr Marco Thiel
Prerequisites
MA or BSc in Mathematics; or at discretion of Head of Department of Mathematics
Overview
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.Structure
2 one hour lectures, 1 one hour computer lab/lecture, and 1 one hour tutorial per week.
Assessment
Continuous assessment (assignments & projects: 80%); oral exam (20%).
 MX5512  Geometry

 Credit Points
 20
 Course Coordinator
 Dr Jarek Kedra
Prerequisites
None
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 onehour lectures and 1 onehour tutorial per week.
Assessment
1 three hour written examination (80%); incourse assessment (20%).
 MX5901  Dissertation in Mathematics

 Credit Points
 60
 Course Coordinator
 Prof. Ran Levi
Prerequisites
N/A
Overview
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.
Structure
Individual supervision sessions with an appropriate member of staff (normally one hour per week, although this may vary depending on the project).
Assessment
Examination of dissertation (100%).