• 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. The assessment consists in an oral presentation and a written dissertation. 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

Topology (rather like Algebra or Analysis) is not so much a single branch of mathematics but a loose confederation of subject areas differing widely in their origins, techniques and motivation but united by sharing a common core of basic concepts and constructions. Problems of a topological nature include: how can we describe and classify knots? how can we describe and classify surfaces? to what extent is it possible to extend the ideas of analysis into sets that don't have metrics defined on them? what can be meant by saying that two objects are “fundamentally the same shape”, and how do we decide whether they are or not? what ‘models’ are available to describe certain aspects of theoretical computer science? Rather than attempting to supply answers to any such major questions, this module will concentrate on developing enough of the ‘common core’ to allow students to begin to appreciate how such issues can be tackled topologically. 

 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

Continuous assessment 30% 

Written examination 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

Any two numbers can be added, subtracted and multiplied, and in many cases divided, to produce a new number. This is very familiar for real numbers, or for complex numbers, or for rational numbers. However, there are more general “number systems” that behave “like the reals” in most respects. Some of these number systems (or to use the technical term: fields) have only finitely many members, which may come as a surprise initially.

These finite fields are fascinating from a theoretical point of view, but also have important practical applications, for example in the area of cryptography (as used in “secure” internet connections nowadays, and by spies throughout all history). One of the reasons is that finite fields, as opposed to the field of real numbers, lend themselves to efficient calculations on computers.

Many applications are built on the theory of polynomials. After developing the foundational material to some extent, we will construct finite fields and discuss various applications; this may include encryption algorithms or generation of pseudo-random numbers.

 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

Continuous assessment 20% 

Written examination 80% 

•  MTH4022 Information Theory (Sem 2)

Pre-requisite: None

Lecturer: Prof H van der Hart

Introduction

Information theory concerns itself with the mathematics of communication. It underpins mobile phone technology, digital broadcasting, and many other aspects of modern technology, and as such is fundamental to current society. Information theory is a relatively new topic in mathematics, with many of the fundamental ideas formulated between 1940 and 1955. However, significant breakthroughs in communication theory were achieved far more recently: the most modern schemes for communication only started their development in the 1990s.   

Information theory thus forms an important basis of the development of information technology in the 21st century. It is an area of mathematics with direct applications, and it is an area where mathematical ideas can be taken up very quickly in new technology. These mathematical ideas can be based on any area of mathematics, and so information theory is connected to a wide range of mathematics. But information theory is not merely an application of mathematics. Information theory has had an important and significant influence on mathematics and has given profound contribution to pure mathematics.

Information transfer occurs through so-called "codes". These codes can take various forms, ranging from the English language to the Morse code to the ASCII code for computer data storage. In this module, we will define what we mathematically mean with a code. We will then develop measures for the amount of information stored within a message, and introduce the key quantity of information theory, entropy. We will investigate how communication affects entropy, leading to the fundamental theorem of information theory, relating information transfer and communication capacity. We will then introduce the basics of coding theory, and study the principles of error correction. To start the module, however, we will look at cryptology.

The module combines both mathematical theory and application. The module not only highlights how the mathematical theory is developed from the underpinning principles, but also explains how this theory is then applied in a practical context.  

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

We are currently witnessing an information revolution: digital devices are everywhere around us. If the incredible level of miniaturisation of electronic devices continues at the current pace, in a few years the elementary components will be made of a few atoms. At this level, physical effects ruled by quantum mechanics will start playing a major role. Inspired by this change of perspective, quantum information processing has been developed as a new framework for future computers. The logic of these quantum computers is different from the traditional computers, as the elementary unit of information, the quantum bit, can be in a superposition of two states 0 and 1. 

The aim of this module is to develop the mathematical theory that underpins most application of this emerging field, including quantum computing, quantum communication (teleportation), and entanglement. Quantum mechanics is not a requisite and will be introduced at the beginning of the module as an abstract framework in linear algebra. 

Students completing this module will acquire knowledge of the mathematical concepts of quantum information processing with possible applications in theoretical physics, applied, pure maths, and computer science. 

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

Many problems in applied mathematics reduce to solving either ordinary or, more usually, partial differential equations subject to certain boundary conditions. In many cases of practical interest exact analytical solutions are not available. 

In non-relativistic quantum mechanics for example the problems involve finding numerical solutions to the Schrödinger equation. In fluid mechanics solutions of the Navier-Stokes equation are needed in the context for example of weather systems, or of hypersonic flow around a space shuttle, or the flow of blood through arteries. In financial mathematics solutions of the Black-Scholes equation that describes the pricing of options are required. A familiar partial differential equation arising in many problems in physics and engineering is the diffusion equation, which describes how heat flows from hot to cold regions and many other processes. Another familiar example is the wave equation that governs phenomena all around us: electromagnetic waves, the vibrations of guitar strings, and the propagation of the sound they produce. 

In order to solve these or other problems we need to understand first the conditions that give rise to a unique solution. Since analytical solutions are only rarely available we examine numerical techniques such as finite difference and finite element to solve particular problems. Creating these programs that solve multi-dimensional problems subject to specific boundary conditions is a large aspect of the course. 

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

Quantum mechanics is one of the greatest intellectual achievements of the last century. In this course we will recap the mathematical framework of quantum mechanics, and show how it describes the structure of atoms and their collisions. Atomic, molecular and optical physics is a burgeoning field, with numerous Nobel prizes awarded in the area in the last 20 years. Intense research is ongoing internationally, aimed at obtaining ever more precise understanding of atomic structure and atomic interactions, and how they can be controlled by external fields. Such understanding is key to, e.g., developing new quantum-based technologies for metrology and cryptography, for precision tests of fundamental laws of physics, for the study of astrophysical phenomena, and for many other applications. 

Students completing this module will expand the toolkit of mathematical methods of quantum mechanics, and obtain basic theoretical knowledge of atomic structure and introductory many-body quantum physics. 

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: Prof M Mathieu

Introduction

At the beginning of the 20th century, the new discipline of Functional Analysis grew out of joint efforts by mathematicians from various countries to unify their understanding of spaces of functions and spaces of operators. In conjunction with the emerging area of Topology, novel tools were developed that allowed for the treatment of infinite dimensional vector spaces, which was hitherto out of reach. It quickly became apparent that this new area of Pure Mathematics was fundamental for many parts of modern Physics (such as Quantum Theory, for example) and many other sections of mathematics. For instance, in the study of partial differential equations it provides indispensable techniques.  
This module introduces and discusses in detail the fundamental concepts of (linear) Functional Analysis such as Banach spaces and their (bounded) operators. As there is a strong research group in Functional Analysis at QUB, the module also provides the basis for further academic work (as part of a PhD, for example).

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

Assignment 1 15%  

Assignment 2 15%  

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

Topological data analysis (TDA) stands for a collection of powerful mathematical and computational tools that can quantify shape and structure in data in order to answer questions from the data’s domain.

The aim of this module is to introduce the main mathematical tools and techniques needed to understand and apply modern methods from topological data analysis.  

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

Continuous assessment 25% 

Written examination 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

Statistical mechanics is a theoretical framework in physics that can be used to rationalise the behaviour of physical systems based on an understanding of the individual atoms of which they are composed.  This module provides you with a very practical introduction to the subject.  You will learn the relevant theory by writing computer programs to simulate systems of spins and atoms.  In the final assignment you will then engage with the cutting-edge research literature for this field. 

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)

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)

• 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

In the MSci Statistics Project, students will complete a substantial investigation of a statistical or operational research problem over the course of two semesters. This project will incorporate a review of relevant literature, use of statistical software packages and, when necessary, the construction of computer programs to solve specific stages of the problem. The research undertaken will be presented in the form of a technical report and a sequence of oral presentations culminating in a 30-minute assessed presentation.

Content

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

Assessment

Dissertation 80%

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 putting the theory into practice using statistical software packages. Computer practical sheets and online quizzes will give students the opportunity to analyse real world survival datasets and gain further insights into the kind of interpretations and knowledge that can be gained from this type of analysis.

Content

Survival data, survivor and hazard functions. Nonparametric method: estimating median and percentile survival and confidence intervals. Comparing two groups of survival data, the log-rank and Wilcoxon tests. Comparison of k-groups. The Cox proportional hazard model, baseline hazard, hazard ratio, including variates and factors, maximum likelihood, treatment of ties. 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.

Assessment

Exam 70%

Coursework 20%

Presentation 10%