Advanced Group Theory
This module looks at basic examples of finite groups and their actions on finite sets.
The material is focused on illustrating how the abstract algebraic concept of a group gives rise to numerous applications in a wide range of subjects, such as combinatorics, number theory, geometry, probability and mathematical physics.
Advanced Linear Analysis
This module gives an introduction into some basic ideas of functional analysis with an emphasis on Hilbert spaces and operators on them.
Many concepts from linear algebra in finite dimensional vector spaces (e.g. writing a vector in terms of a basis, eigenvalues of a linear map, diagonalisation etc.) have generalisations in the setting of infinite dimensional spaces making this theory a powerful tool with many applications in pure and applied mathematics.
Algebraic geometry is one of the great twentieth century achievements in mathematics.
This module describes geometrical structures using the language of algebra.
It presents and discusses affine and projective algebraic varieties over algebraically closed fields, and the associated algebraic structures: co-ordinate rings and function fields. Various notions are illustrated on algebraic curves, including elliptic curves.
The module further introduces the concept of the Zariski topology and the spectrum of rings. The structure sheaf on the spectrum of rings and its properties are motivated and explained. Birational geometry notions are discussed.
Algebraic Number Theory
This module presents the fundamental features of algebraic number theory, the theory in which numbers are viewed from an algebraic point of view.
So numbers are often treated as elements of rings, fields and modules, and properties of numbers are reformulated in terms of the relevant algebraic structures.
This approach leads to understanding of certain arithmetical properties of numbers (in particular, integers) from a new point of view.
Combinatorial Group Theory
This module is largely concerned with infinite groups, especially free groups, although their role in describing and understanding finite groups is emphasized.
Following the basic definitions of free groups and group presentations, the fundamental Nielson-Schreier Theorem is covered in some detail.
Methods for manipulating group presentations, and using them to read off properties of a given group, will be covered: for example, obtaining presentations for subgroups of finite index in a given group.
The interplay between presentations and standard group-theoretic construction (for example, direct products and group extensions) will also be covered.
There will also be an introduction to the advanced topics of free products with amalgamation and corresponding extensions.
Many of these methods may be automated and have been implemented on computers.
Complex analysis is one of the central areas of pure mathematics, with many remarkable, often surprising, theorems.
Whereas the introductory course emphasizes applications and computational techniques, differentiable functions of a complex variable display striking properties, and the course will investigate these.
The module will start with several topics from the perspective of what can be explicitly calculated with an emphasis on applications to geometry and number theory. Topics for this course include: basic notions of projective geometry; plane algebraic curves including elliptic curves; addition of points on elliptic curves; results on the group of rational points on an elliptic curve; properties of elliptic curves and their applications.
Foundations of Advanced Analysis
This module introduces and studies the abstract and more advanced concepts of metric and topological spaces, leading on to topics such as Functional Analysis.
The module covers metric spaces, topological spaces, compactness, separation properties like Hausdorffness and normality, Urysohn's lemma, quotient and product topologies, and connectedness. Finally, Borel sets and measurable spaces are introduced.
The coursework and presentation part of the assessment will deepen understanding and explore further related special topics.
Further Topics in Analysis
This module will solidify and deepen the student's knowledge and understanding of the concepts and theory of mathematical analysis.
It builds on the fundamental theory of metric and topological spaces and develops this to some more advanced settings.
At the same time, the student will learn further methods of rigorous proof in abstract analysis and will be trained in the use of concepts and techniques relevant to research in mathematicial analysis.
Further Topics in Rings and Modules
Commutative rings and course over them are the fundamental objects of what is often referred to as commutative algebra. Already encountered key examples of commutative rings are polynomials in one variable over a field and number rings such as the usual integers or the Gaussian integers. There are many close parallels between these two types of rings, for example the similarities between the prime factorization of integers and the factorization of polynomials into irreducibles.
In this module, these ideas are extended and generalized to cover polynomials in several variables and power series, and algebraic numbers.
Galois Theory concerns the study of the group of symmetries associated with polynomial equations (in modern language, the group of automorphisms of field extensions). It shows the impossibility of solving the general quintic equation by radicals, and it can be used to prove that once cannot square the circle, duplicate a cube, nor trisect an angle using only a ruler and a compass! We start by laying a theoretical foundation to the construction of splitting fields, and thence the factorisation of polynomials.
The Fundamental Theorem of Galois Theory lies at the heart of the course, and shows a close connection between field theory and group theory; we show how the question of solvability of a polynomial can be completely settled using group theory.
Another topic will concern finite field, which are of central importance in many applications including coding and cryptography.
Number fields form another large class of fields which, on the one hand, are sufficiently simple so as to allow an initial study by elementary methods, and, on the other hand, are sufficiently rich in structure to represent some major aspects of Galois theory.
Some number fields will be used for illustration purposes throughout, and some of them, e.g. cyclotomic fields, are studied in detail.
The principle objective of studying Galois Theory are not only its applications, but also the appreciation of the beauty of a single brilliant idea.
Higher Number Theory
This module develops some of the main aspects of the theory of numbers with emphasis on Dirichlet series and their applications to the study of prime numbers. It builds on elementary number theory and uses techniques of complex analysis.
This will provide the springboard for the study of more advanced topics which will be useful for further study of algebra, number theory and other areas of Mathematics.
Students should also learn to present and develop a mathematical argument on a self-directed basis.
Pure Mathematics Dissertation
In this module a substantial investigation will be carried out on a topic in Pure Mathematics. The study will be largely self-directed, although a supervisor will provide oversight and input where necessary.
The topic will be chosen from a list provided by the School. The topic will be in algebra, analysis or number theory.
The dissertation will consist of a review of some advanced theory and results in one of these areas, or an investigation into a research topic in algebra, analysis or number theory using research-based methodologies appropriate for the chosen topic.
Some projects may contain an element of computing using existing software packages.
The above is a sample of the typical modules that we offer but is not intended to be construed and/or relied upon as a definitive list of the modules that will be available in any given year. Due to the passage of time between commencement of the course and subsequent years of the course, modules may change due to developments in the curriculum and information is provided for indicative purposes only.