You will choose from a wide range of advanced optional modules, one of which may involve project work. You will be able to specialise in one of the three main subject areas, or choose a broad course which may include modules from all three areas. 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 general theory you learnt in Introduction to Mathematical Physics 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
In this module you will build on your theoretical knowledge of statistical inference by a practical implementation of the generalised linear model. You will move on 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.
Coding and Cryptography
This course provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. It also provides an introduction to to cryptography, including classical mono- and polyalphabetic ciphers as well as modern public key cryptography and digital signatures, their uses and applications.
Classical and Quantum Dynamics
The course introduces and explores methods, concepts and paradigm models for classical and quantum mechanical dynamics exploring how classical concepts enter quantum mechanics, and how they can be used to find approximate semi-classical solutions.
In classical dynamics we discuss full integrability and basic notions of chaos in the framework of Hamiltonian systems, together with advanced methods like canonical transformations, generating functions and Hamiltonian-Jacobi theory. In quantum mechanics we recall Schrödinger's equation and introduce the semi-classical approximation. We derive the Bohr-Sommerfeld quantization conditions based on a WKB-approch to the eigenstates. We will discuss some quantum signatures of classical chaos and relate them to predictions of random-matrix theory. We will also introduce Gaussian states and coherent states and discuss their semi-classical dynamics and how it is related to the corresponding classical dynamics. An elementary introduction to complete descriptions of quantum mechanics in terms of functions on the classical phase space will be given.
Data Analysis and Modelling
This module involves the application of probability and statistics to a variety of practical, open-ended problems, typical of those that statisticians encounter in industry and commerce. Specific projects are tackled through workshops and student-led group activities.
The real-life nature of the problems requires students to develop skills in model development and refinement, report writing and teamwork. Students will have an opportunity to apply a variety of statistical methods and knowledge learned in previous modules.
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Differential Equations
This course introduces various analytical methods for the solution of ordinary and partial differential equations, focussing on asymptotic techniques and dynamical systems theory. Students taking this course will build on their understanding of differential equations covered in Modelling with Differential Equations.
Elliptic Curves
The course 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 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.
Electromagnetism
The course provides an introduction to electromagnetism and the electrodynamics of charged particles. The aims of this course are:
- to develop an appropriate mathematical model of electromagnetic phenomena that is informed by observations
- to understand electromagnetic configurations of practical importance and to relate predictions made to everyday phenomena
- to illustrate the use of solutions of certain canonical partial differential equations for determining electrostatic fields and electromagnetic waves in vacuum and in matter
- to illustrate the interplay between experimental input and the development of a mathematical model, and the use of various mathematical techniques for solving relevant problems.
Fluid Dynamics
This course aims to extend previous knowledge 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. For example, the cubic equation x3 + y3 = z3 with x, y, z non-zero has infinitely many real solutions yet not a single solution in whole numbers.
We shall establish the basic properties of the Riemann zeta-function to find out how evenly these primes are distributed in nature. This course will present several methods to solve Diophantine equations including analytical methods using zeta-functions and Dirichlet series, theta functions and their applications to arithmetic problems, and an introduction to more general modular forms.
Game Theory
Game theory contains 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 mathematical theory of games; exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.
Graph Theory
A graph (in the sense used in Graph Theory) 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 in the course include:
- paths and cycles
- the resolution of Eulers Königsberg Bridge Problem
- Hamiltonian cycles
- trees and forests
- labelled trees,
- the Prüfer correspondence
- planar graphs
- Demoucron et al. algorithm
- Kruskal's algorithm
- the Travelling Salesman's problem
- the statement of the four-colour map theorem
- colourings of vertices
- chromatic polynomial
- colourings of edges.
Group Theory
This course 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), and 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 that will be used in Galois theory.
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
Mathematical Finance
In this module the concepts of discrete time Markov chains are explored and used to provide an introduction to probabilistic and stochastic modelling for 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 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 course describes how mathematics helps us understand topics such as population dynamics, biological oscillations, pattern formation and nonlinear growth phenomena. There is considerable emphasis on model building and development.
Mathematics Project
This module consists of a self-directed investigation of a project selected from a list of projects or, subject to prior approval of the School, from elsewhere.
Project modules are carried out in the Autumn and Spring semesters.
The project will be supervised by a member of staff and will be based on a substantial mathematical problem, an application of mathematics or investigation of an area of mathematics not previously studied by the student. The course includes training in the use of IT resources, the word-processing of mathematics and report writing.
Metric and Topological Spaces
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, compactness, separation properties like Hausdorffness and normality, Urysohn’s lemma, quotient and product topologies, and connectedness. Finally, Borel sets and measurable spaces are introduced.
Multivariate Analysis
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, whose purpose is to identify the main modes of variation in a multivariate dataset; modelling and inference for multivariate data, including multivariate regression data, based on the multivariate normal distribution; classification of observation vectors into sub-populations using a training sample; canonical correlation analysis, whose purpose is to identify dependencies between two or more sets of random variables. Further topics to be covered include factor analysis, methods of clustering and multidimensional scaling.
Optimisation
In this module a variety of techniques and areas of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming. You’ll develop techniques for application which can be used outside the mathematical arena.
Relativity
In this module you’ll have an introduction 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 new phenomena which 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. 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.
Scientific Computation and Numerical Analysis
You will 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.
Stochastic Models
In this module you 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 and then you will move onto more extensive studies of epidemic models and queuing models with introductions to component and system reliability.
Statistical Inference
In this module you will explore two main concepts of statistical inference; classical (frequentist) and Bayesian. Topics such as sufficiency, estimating equations, likelihood ratio tests and best-unbiased estimators will be discussed in detail. You will gain knowledge of the theory and concepts underpinning contemporary research in statistical inference and methodology.
Vocational Mathematics
This module involves the application of mathematics to a variety of practical, open-ended problems, typical of those that mathematicians encounter in industry and commerce. Specific projects are tackled through workshops and student-led group activities. The real-life nature of the problems requires students to develop skills in model development and refinement, report writing and teamwork.