You must take the group project module. The compulsory group project allows you to consolidate your mathematical knowledge and understanding whilst gaining experience of working collaboratively to solve complex problems. Completion of this project provides you with excellent preparation for your fourth year dissertation.
You will also have the option to choose some modules from outside mathematics if you wish.
Advanced Quantum Theory
In this module you will apply the quantum mechanics that you learned in Year 2 to more general problems. New topics will be introduced such as the quantum theory of the hydrogen atom and aspects of angular momentum such as spin.
Applied Statistical Modelling
During this module you will build on your theoretical knowledge of statistical inference by a practical implementation of the generalised linear model. You will progress to enhance your understanding of statistical methodology including the analysis of discrete and survival data. You will also be trained in the use of a high-level statistical computer program.
Classical and Quantum Dynamics
The module introduces and explores methods, concepts and paradigm models for classical and quantum mechanical dynamics. We explore how classical concepts enter quantum mechanics, and how they can be used to find approximate semi-classical solutions.
Coding and Cryptography
This module provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. You’ll learn cryptography, including classical mono- and polyalphabetic ciphers. There will also be a focus on modern public key cryptography and digital signatures, their uses and applications.
This is an excellent opportunity to gain first-hand experience of being involved with providing mathematical education. You will work in a local school alongside practising mathematics teachers in a classroom environment and improve your skills at communicating maths.
Typically, you will work within a class (or classes) for half a day a week for about sixteen weeks. You will be given a range of responsibilities, from classroom assistant to leading a self-originated mathematical activity or project.
The assessment is carried out in a variety of ways: an on-going reflective log, contribution to reflective seminar, oral presentation and a final written report.
This module introduces various analytical methods for the solution of ordinary and partial differential equations.
You will begin by studying asymptotic techniques, which can be used when the equations involve a small parameter, which is often the case. We will also study some aspects of dynamical systems theory, which has wide applicability to models of real world problems.
Discrete Mathematics and Graph Theory
In this module, a graph consists of vertices and edges, each edge joining two vertices. Graph theory has become increasingly important recently through its connections with computer science and its ability to model many practical situations.
Topics covered include:
- paths and cycles
- the resolution of Euler’s Königsberg Bridge Problem
- Hamiltonian cycles and many others
Elliptic curves are useful tools in many areas of number theory and can be used in cryptography. You'll study topics including:
- 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
The module provides an introduction to electromagnetism and the electrodynamics of charged particles. You will meet Maxwell’s equations and learn how they describe a wide variety of phenomena, including electrostatic fields and electromagnetic waves.
You will extend your understanding of fluid flow by introducing the concept of viscosity and studying the fundamental governing equations for the motion of liquids and gases.
Methods for solution of these equations are introduced, including exact solutions and approximate solutions valid for thin layers. A further aim is to apply the theory to model fluid dynamical problems of physical relevance.
Further Number Theory
Number theory concerns the solution of polynomial equations in whole numbers, or fractions. We will establish the basic properties of the Riemann zeta-function to find out how evenly the prime numbers are distributed.
This module will also present several methods to solve Diophantine equations including analytical methods using zeta-functions and Dirichlet series, theta functions and their applications to arithmetic problems. We will also cover an introduction to more general modular forms.
Game theory is relevant to many branches of mathematics (and computing); the emphasis here is primarily algorithmic. The module starts with an investigation into normal-form games, including strategic dominance, Nash equilibria, and the Prisoner’s Dilemma.
We look at tree-searching, including alpha-beta pruning, the ‘killer’ heuristic and its relatives. It then turns to the mathematical theory of games, exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.
This module builds on the basic ideas of group theory. It covers a number of key results such as:
- the simplicity of the alternating groups
- the Sylow theorems (of fundamental importance in abstract group theory)
- the classification of finitely generated Abelian groups (required in algebraic number theory, combinatorial group theory and elsewhere).
Other topics to be covered are group actions, used to prove the Sylow theorems, and series for groups, including the notion of solvable groups.
You will study an introduction to the ideas of functional analysis with an emphasis on Hilbert spaces and operators on them. Many concepts from linear algebra in finite dimensional vector spaces, for example, writing a vector in terms of a basis, eigenvalues of a linear map, diagonalization, have generalisations in the setting of infinite dimensional spaces.
This makes this theory a powerful tool with many applications in pure and applied mathematics.
You will explore the concepts of discrete time Markov chains to understand how they used. We will also provide an introduction to probabilistic and stochastic modelling of investment strategies, and for the pricing of financial derivatives in risky markets.
You will gain well-rounded knowledge of contemporary issues which are of importance in research and workplace applications.
Mathematical Medicine and Biology
Mathematics can be usefully applied to a wide range of applications in medicine and biology.
Without assuming any prior biological knowledge, this module describes how mathematics helps us understand topics such as:
- population dynamics
- biological oscillations
- pattern formation
- nonlinear growth phenomena
There is considerable emphasis on model building and development.
Metric and Topological Spaces
A metric space generalises the concept of distance familiar from Euclidean space. It provides a notion of continuity for functions between quite general spaces. The module covers:
- metric spaces
- topological spaces
- separation properties like Hausdorffness and normality
- Urysohn’s lemma
- quotient and product topologies, and connectedness
Finally, Borel sets and measurable spaces are introduced.
This module is concerned with the analysis of multivariate data, in which the response is a vector of random variables rather than a single random variable. A theme running through the module is that of dimension reduction.
Key topics to be covered include:
- principal components analysis
- modelling and inference for multivariate data
- classification of observation vectors into sub-populations using a training sample
- canonical correlation analysis
- factor analysis
- methods of clustering
- multidimensional scaling
In this module a variety of techniques of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming.
These techniques have a wide range of applications to real world problems, in which a process or system needs to be made to perform optimally.
You will be introduced to Einstein’s theory of general and special relativity. The relativistic laws of mechanics will be described within a unified framework of space and time. You’ll learn how to compare other theories against this work and you’ll be able to explain exciting new phenomena that occur in relativity.
Rings and Modules
Commutative rings and modules over them are the fundamental objects of what is often referred to as commutative algebra. 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.
Scientific Computation and Numerical Analysis
You'll learn how to use numerical techniques for determining the approximate solution of ordinary and partial differential equations where a solution cannot be found through analytical methods alone. You will also cover topics in numerical linear algebra, discovering how to solve very large systems of equations and find their eigenvalues and eigenvectors using a computer.
This module is concerned with the two main theories of statistical inference, namely classical (frequentist) inference and Bayesian inference.
You will explore the following topics in detail:
- estimating equations
- likelihood ratio tests
- best-unbiased estimators
There is special emphasis on the exponential family of distributions, which includes many standard distributions such as the normal, Poisson, binomial and gamma.
This module will develop your knowledge of discrete-time Markov chains by applying them to a range of stochastic models. You will be introduced to Poisson and birth-and-death processes. You will then move onto more extensive studies of epidemic models and queuing models, with introductions to component and system reliability.