• AMA4005 Project (Full year)

Pre-requisite: This two-semester-long double module is only available to students on the Mathematics/Applied Mathematics and Physics/Theoretical Physics MSci pathways.

Co-ordinator: Dr G De Chiara

Introduction

The project involves a substantial investigation of a research problem incorporating literature survey, development of appropriate theoretical models and when necessary the construction of computer programs to solve specific stages of the problem, presentation of the work in the form of a technical report, a sequence of oral presentations culminating in a 30-minute presentation which is assessed. Each student will work under individual supervision of a member of staff.

Content

The mathematical contents of the project will depend on the nature of the research problem.

Assessment

Dissertation 80%

Presentation 20%

• MTH4011 Topology (Sem 1)

Pre-requisite: MTH3011 Measure and Integration, or
PMA3014 Set Theory and PMA3017 Metric and Normed Spaces [for 2021/22]

Lecturer:  Dr P Siegl

Introduction

Content

- Definition and examples (natural, geometric and pathological)
- Continuity and homeomorphisms
- Compact, Connected, Hausdorff
- Subspaces and product spaces
- Introduction to homotopy, calculations and applications

Assessment

Continual Assessment 30%

Exam 70%

•  MTH4021 Applied Algebra and Cryptography (Sem 2)

Pre-requisite: MTH3012 Rings and Modules, or PMA3012 Ring Theory [for 2021/22]

Lecturer: Dr T Huettemann

Introduction

Content

- (finite) fields and rings of polynomials over them.
- the division algorithm and splitting of polynomials.
- ideals and quotient rings, (principal) ideal domains, with examples from rings of polynomials.
- polynomials in several indeterminates, Hilbert’s basis theorem.
- applications of algebra to cryptography (such as affine Hill ciphers, RSA, lattice cryptography, Diophantine equations).
- optional topics may include Euclidean rings, unique factorisation domains, greatest common divisor domains.

Assessment

Continual Assessment 20%

Exam 80%

•  MTH4022 Information Theory (Sem 2)

Pre-requisite: None

Lecturer: Prof H van der Hart

Introduction

Content

Introduction to information theory. Basic modular arithmetic and factoring. Finite-field arithmetic. Random variables and some concepts of probabilities. RSA cryptography and factorisation.  Uniquely decipherable and instantaneous codes.  Optimal codes and Huffman coding.  Code extensions.  Entropy, conditional entropy, joint entropy and mutual information.  Shannon noiseless coding theorem.  Noisy information channels.  Binary symmetric channel.  Decision rules.  The fundamental theorem of information theory.  Basic coding theory.  Linear codes. A brief introduction to low-density parity-check codes.

Assessment

Exam 70%

Report 30%

•  MTH4023 Mathematical Methods for Quantum Information Processing (Sem 2)

Pre-requisite: MTH2011 Linear Algebra, or MTH2001 Linear Algebra and Complex Variables [for 2021/22]
Prior knowledge of Quantum Theory is not required.

Lecturer: Dr A Ferraro

Introduction

Content

1.  Operatorial quantum mechanics: review of linear algebra in Dirac notation; basics of quantum mechanics for pure states.
2.  Density matrix and mixed states; Bloch sphere; generalised measurements.
3.  Maps and operations: complete positive maps; Kraus operators.
4.  Quantum Communication protocols: quantum cryptography; cloning; teleportation; dense coding.
5.  Quantum computing: review of classical circuits and logic gates; quantum circuits and algorithms; implementation of quantum circuits on small prototypes of quantum computers (IBM Quantum Experience); examples of physical Hamiltonians implementing quantum gates.
6.  Theory of entanglement: basic notions and pure-state entanglement manipulation; detection of entanglement; measures of entanglement; entanglement and non-locality, Bell's inequality; multipartite entanglement.

Assessment

Exam 70%

Project 30%

• MTH4024 Practical Methods for Partial Differential Equations (Sem 1)

Pre-requisite: MTH2021 Mathematical Methods 2, or None [for 2021/22]

Lecturer: Dr C Ballance

Introduction

Content

Basics: solving first order ordinary differential equations, partial derivatives, surface, volume and line integrals, the Gauss theorem, Stokes' Theorem.
Partial differential equations (PDE) and their relation to physical problems: heat conduction, flow of a liquid, wave propagation, Brownian motion.
First order PDE in two variables: the method of characteristics, the transversality condition, quasilinear equations and shock waves, conservation laws, the entropy condition, applications to traffic flows.
Second order linear PDEs: classification and canonical forms.
The wave equation: d`Alembert’s solution, the Cauchy problem, graphical methods.
The method of separation of variables: the wave and the heat equations.
Numerical methods: finite differences, stability, explicit and implicit schemes, the Crank-Nicolson scheme, a stable explicit scheme for the wave equation.
Practical: the students are offered to solve a heat and a wave equation using the method of separation of variables and a finite difference scheme.
The Sturm-Liouville problem: a theoretical justification for the method of separation of variables.  Simple properties of the Sturmian eigenvalues and eigenfunctions.     
Elliptic equations: the Laplace and Poisson equations, maximum principles for harmonic functions, separation of variables for Laplace equation on a rectangle.
Green's functions: their definition and possible applications, Green’s functions for the Poisson equation, the heat kernel.

Assessment

Exam 70%

Report 30%

• MTH4031 Advanced Quantum Theory (Sem 1)
  

Pre-requisite: MTH3032 Quantum Theory or PHY3011 Quantum Mechanics and Relativity, or AMA3002 Quantum Theory or PHY3011 Quantum Mechanics and Relativity [for 2021/22]

Lecturer: Dr G De Chiara

Introduction

Content

1.  Review of fundamental quantum theory (Postulates of quantum mechanics; Dirac notation; Schrödinger equation; spin-1/2 systems; stationary perturbation theory).
2.  Coupled angular momenta: spin-1/2 coupling; singlet and triplet subspaces for two coupled spin-1/2 particles; Coupling of general angular momenta;
3.  Spin-orbit coupling; fine and hyperfine structures of the hydrogen atom.
4.  Time-dependent perturbation theory.
5.  Elements of collisions and scattering in quantum mechanics.
6.  Identical particles and second quantisation; operators representation.
7.  Basics of electromagnetic field quantisation.
8.  Systems of interacting bosons: Bose-Einstein condensation and superfluidity.

Assessment

Exam 80%

Presentation 20%

• MTH4311 Functional Analysis (Sem 2)
  

Pre-requisite: MTH2013 Metric spaces and MTH3011 Measure and Integration, or PMA3014 Set Theory and PMA3017 Metric and Normed Spaces [for 2021/22]

Lecturer: Dr M Mathieu

Introduction

Content

A characterisation of finite-dimensional normed spaces; the Hahn-Banach theorem with consequences; the bidual and reflexive spaces; Baire’s theorem, the open mapping theorem, the closed graph theorem, the uniform boundedness principle and the Banach-Steinhaus theorem; weak topologies and the Banach-Alaoglu theorem; spectral theory for bounded and compact linear operators.

Assessment

Continual Assessment 30%

Exam 70%

• MTH4322 Topological Data Analysis (Sem 1)
  

Pre-requisite: MTH2011 Linear Algebra and MTH2013 Metric Spaces, or MTH2001 Linear Algebra and Complex Variables and PMA3017 Metric and Normed Spaces [for 2021/22]

Lecturer: Dr D Barnes and Dr F Pausinger

Introduction

Content

1. Simplicial complexes
2. PL functions
3. Simplicial homology
4. Filtrations and barcodes
5. Matrix reduction
6. The Mapper Algorithm
7. Learning with topological descriptors
8. Statistics with topological descriptors

Assessment

Continual Assessment 25%

Exam 75%

• MTH4332 Statistical Mechanics (Sem 2)
  

Pre-requisite: MTH3032 Quantum Theory or PHY3011 Quantum Mechanics and Relativity, or AMA3002 Quantum Theory or PHY3011 Quantum Mechanics and Relativity [for 2021/22]

Lecturer: Dr G Tribello

Introduction

Content

Fundamentals of classical thermodynamics: systems, phases, thermodynamic variables, equilibrium, equations of state, distinction between intensive and extensive thermodynamic variables, work and heat, Carnot cycle, Gibbs phase rule, first and second laws of thermodynamics, definitions of thermodynamic potentials, derivation of Maxwell relations, response functions, thermodynamic stability and Ehrenfest classification of phase transitions.
Equilibrium statistical mechanics: microstates and phase space, role of information and connection with entropy, method of Lagrange multipliers, generalised partition function, microcanoncial, canonical, isothermal-isobaric and grand-canonical partition functions, ensemble averages and connection between fluctuations and response functions.
Computer simulation: importance sampling, Monte Carlo algorithm, molecular dynamics.

Assessment

Class Test 10%

Portfolio 25%

Presentation 20%

Exam 45%

• PMA4001 Project (Full year)

Pre-requisite: This project is a compulsory component of the MSci pathway in Pure Mathematics. There is no specific pre-requisite for this module, but the student will need enough Level 3 background in Pure Mathematics to undertake an extended project at this level in some area of Pure Mathematics for which supervision can be offered.

Co-ordinator: Dr Y-F Lin

Introduction

This is an extended project designed to test the student's ability to work independently at a high level for a prolonged period of time with a restricted amount of supervision. This will give a taste of the kind of work expected of a mathematician in the commercial or academic world, unlike the relatively short bursts of work expected in most undergraduate modules. It will also provide an opportunity to develop those transferable skills that are sought by employers, including IT (both word-processing and data-base access), presentational and personal ones.

Content

The project takes place during the two terms of Level 4. It will normally involve study and exposition of a piece of mathematical work beyond the normal undergraduate syllabus and which will probably not be available in easily assimilated form. Originality of exposition will be expected, but not necessarily much in the way of original results. The main part of the assessment will consist of a word-processed report, but 20% of the marks for the project are awarded for an oral presentation of the work which will take place before or after Easter, depending on the academic calendar. As preparation for this assessed oral presentation, the student will be expected to give one oral progress report to a small group of staff and any other students undertaking this module. Constructive advice on this presentation will be provided.

Students intending to take this module should seek advice and think about their choice of project during the summer. The selection of a project should be finalized no later than the start of the academic year, and it would be helpful to all involved if students actually did this even earlier.

Assessment

Dissertation 80%

Presentation 20%

• PMA4004 Integration Theory (Sem 1 or Sem 2)

Pre-requisite: MTH2012 Analysis and PMA3014 Set Theory

Lecturer: Dr S Shkarin

Introduction

The theory of integration, developed by Lebesgue in the early part of the twentieth century in the context of the real line and subsequently extended to more general settings, is indispensable in modern analysis. The Lebesgue theory allows a very wide class of functions to be integrated and includes powerful convergence theorems which are not available in Riemann integration. In this module the theory is developed in the context of a general σ-algebra of sets. Special attention is given to the case of Lebesgue measure on the reals, and some applications of the integral to Fourier series are given.

Content

σ-algebras of sets, measurable spaces, measurable functions. Measures. Integrals of non-negative measurable functions: properties including Fatou's lemma and monotone convergence theorem. Integrable functions: Lebesgue dominated convergence theorem. Lebesgue integral on intervals: comparison with Riemann integral. L^p-spaces: inequalities of Hölder and Minkowski; Fourier series in L².

Assessment

Exam 70%

Assignment 10% (x 3)

Textbooks

R. G. Bartle, The Elements of Integration and Lebesgue Measure (Wiley, 1995).

• SOR4001 Project (Full year)

Pre-requisite: This two-semester-long double module is only available to students on the Mathematics and Statistics & Operational Research MSci pathway.

Co-ordinator: Dr L McFetridge

Introduction

A substantial investigation of a statistical/operational research problem incorporating literature survey,, use of relevant statistical packages and when necessary the construction of computer programs to solve specific stages of the problem, presentation of the work in the form of a technical report, a sequence of oral presentations culminating in a 30-minute presentation which is assessed. Each student will work under individual supervision of a member of staff.

Content

The mathematical contents of the project will depend on the nature of the research problem.

Assessment

Dissertation 80%

Oral Presentation 20%

• SOR4007 Survival Analysis (Sem 1)

Pre-requisite: SOR2004

Lecturer: Dr L McFetridge

Introduction

Survival analysis is an important tool for research in medicine and epidemiology. It is that part of statistics that deals with time-to-event data. For example, in a clinical study the data might consist of the post-treatment survival times of patients with hypernephroma (i.e., a malignant tumour of the kidney). Survival analysis might address questions such as:

• How does the patient's survival time depend of her age at treatment?

• What is the affect of kidney removal on the survival times of patients compared with others who are treated just with chemotherapy?

• Is the size of the tumour an equally good predictor of survival for patients under 60 years of age as for the over 60s?

The module introduces the student to the special features of survival data such as censoring (e.g. where a patient is lost to follow up but is known to have survived to a particular time) and positive skew in the distribution of survival times. Fundamental concepts of survival analysis will be introduced including the survivor function, the hazard function and the hazard ratio. The course will build from some elementary nonparametric techniques such as the Kaplan-Meier estimate of the survival curve to the Cox proportional hazards model - one of the most flexible and widely used tools for the analysis of survival data.

An important element of this module will be two hours a week of survival data analysis classes using statistical software packages, in particular SAS and R. These will be used to demonstrate the theory taught in the lectures. Previous experience of SAS is required.

Content

Introduction to survival data: Features of survival data, distribution of survival times, survivor function, hazard function, cumulative hazard function.
Some nonparametric procedures: Estimating the survivor function - life-table, Kaplan-Meier, Nelson-Aalen, confidence intervals. Estimating the hazard function, estimating median and percentile survival and confidence intervals. Comparing two groups of survival data, the log-rank and Wilcoxon tests. Comparison of k-groups of survival data.
The Cox proportional hazards model: The Cox proportional hazard model (Cox model), baseline hazard function, hazard ratio, including variates and factors, maximum likelihood for the Cox model. Treatment of ties in the Cox model. Confidence intervals for the Cox model regression parameters and hypothesis testing. Estimating the baseline hazard. Model building, Wald tests, likelihood ratio tests and nested models.
Evaluating the proportional hazards assumptions.
The stratified Cox procedure.
Extending Cox proportional hazards models for time dependent variables.
Recurrent events survival analysis.
Competing risks survival analysis.
Design issues for randomised trials.
Parametric models for survival data, time-dependent variables and non-proportional hazards, accelerated-failure-time models. Fitting parametric distributions.

Assessment

Exam 75%

Coursework 15%

Presentation 10%