Level 3 Modules
- AMA3011 Applied Mathematics Project (1st and 2nd semester)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of BSc Mathematics, Applied Mathematics and Physics or Theoretical Physics programmes, and a mathematical knowledge and ability commensurate with this stage is assumed.
Module Co-ordinators: Prof J Kohanoff (1st semester), Dr C Ramsbottom (2nd semester)
This module (or alternatively, PMA3013) is compulsory for all students on the BSc pathway and constitutes a self-study project on an advanced mathematical topic under the supervision of a member of staff. Students will be offered a choice of topic subjects which can span the entire range of applied mathematics and statistics, including some theoretical physics.
Self-study of an advanced mathematical topic under the supervision of a member of staff. Students will be offered a choice of subjects, which can span the entire range of applied mathematics, including theoretical physics. The study concludes with a written report and a poster presentation.
- AMA3020 Investigations (Sem 2)
Pre-Requisites: While there are no specific pre-requisites, this course is intended for MSci students in Mathematics, Mathematics and Statistics & Operational Research, Applied Mathematics and Physics, Mathematics and Computer Science, and Theoretical Physics, and a mathematical knowledge and ability commensurate with this stage is assumed.
Lecturers: Prof M Paternostro and Dr C Ballance
Problem-solving is a key skill in many domains, from finance to academia, from software development to data analysis. Being able to address accurately a previously unseen problem, finding creative manners to solve it, and presenting complex concepts in a simple and effective way are invaluable assets. This module will facilitate the development of problems-solving skills that are expendable in a broad domain of fields and areas’
This module provides an introduction to project development and management in topics of Applied Mathematics. Students are trained in research methods by working on a range of projects.
In the first part of the module the students conduct a short practice investigation, followed by two short investigations (one in small groups [typically in pairs] and one individually) in a range of problems in Applied Mathematics and Theoretical Physics. The results are presented in the form of typed reports.
This is followed by a long investigation, which is a literature study each a topic in Mathematical or Theoretical Physics not covered in the offered (or chosen) modules. The results of the investigation are presented in the form of a set of notes and a presentation that takes the form of a short lecture and is delivered by the students individually.
The two short investigations are typed up in reports and submitted for assessment. The set of notes on which the presentation for the long investigation is based, is typed up and submitted for assessment.
Report 35% (x 1)
Report 45% (x 1)
- AMA3022 Team Project: Mathematics with Finance (Sem 2)
Pre-requisite: Only available to students on the BSc Mathematics with Finance programme.
Lecturers: Dr D Dundas and Dr S Moutari
As a result of taking this module, students will learn to respond to a briefing on a problem by a client. They will be able to work successfully as part of a team to address the problem. They will also be able to make a final presentation on the outcome of the work.
Business skills workshop. Presentation skills. Negotiation skills. Customer relationship. Project management/team building. Teams required to negotiate, plan, develop and deliver a completed task working as a group, commissioned by the `client' company. The project will require software development skills.
Business Proposals 30%
Report & Presentation 40%
Business Plan 10%
Peer Evaluation 20%
- MTH3012 Rings and Modules (Sem 1)
Pre-requisite: MTH2011 Linear Algebra and MTH2014 Group Theory, or
MTH2001 Linear Algebra and Complex Variables and PMA2008 Group Theory [for 2021/22]
Lecturer: Dr Y-F Lin
Rings, subrings, prime and maximal ideals, quotient rings, homomorphisms, isomorphism theorems, integral domains, principal ideal domains, modules, submodules and quotient modules, module maps, isomorphism theorems, chain conditions (Noetherian and Artinian), direct sums and products of modules, simple and semisimple modules.
- MTH3021 Dynamical Systems (Sem 2)
Pre-requisite: MTH2011 Linear Algebra, MTH2013 Metric Spaces
Lecturer: Dr G Kiss
Continuous dynamical systems
- Fundamental theory: existence, uniqueness and parameter dependence of solutions;
- Linear systems: constant coefficient systems and the matrix exponential; nonautonomous linear systems; periodic linear systems.
- Topological dynamics: invariant sets; limit sets; Lyapunov stability.
- Grobman-Hartman theorem.
- Stable, unstable and centre manifolds.
- Periodic orbits: Poincare-Bendixson theorem.
- Applications: the Van der Pol oscillator; the SIR compartmental model; the Lorenz system.
Discrete dynamical systems
- One-dimensional dynamics: the discrete logistic model; chaos; the Cantor middle-third set.
- MTH3022 Discrete Mathematics (Sem 2)
Pre-requisite: MTH1011 Introduction to Algebra and Analysis, or MTH1001 Analysis and Calculus and MTH1002 Numbers, Vectors and Matrices [for 2021/22]
Lecturer: Dr F Pausinger
1. Intro Enumerative Combinatorics: basic counting, pigeonhole principle, inclusion-exclusion, recurrence relations, generating functions.
2. Intro Elementary Number Theory: Divisibility and primes, Euclidean algorithm, linear congruences, Chinese Remainder Theorem.
3. Intro Graph Theory: basic notions, trees, connectivity, matchings, graph colouring, planarity, basic Ramsey theory.
4. Intro Algorithmics: analysis of algorithms, sorting, greedy and divide-and-conquer algorithms, basic graph and NT algorithms.
- MTH3023 Numerical Analysis (Sem 1)
Pre-requisite: MTH2011 Linear Algebra, or MTH2001 Linear Algebra and Complex Variables [for 2021/22]
Lecturer: Dr D Dundas
Introduction and basic properties of errors: Introduction; Review of basic calculus; Taylor's theorem and truncation error; Storage of non-integers; Round-off error; Machine accuracy; Absolute and relative errors; Richardson's extrapolation.
• Solution of equations in one variable: Bisection method; False-position method; Secant method; Newton-Raphson method; Fixed point and one-point iteration; Aitken's "delta-squared" process; Roots of polynomials.
• Solution of linear equations: LU decomposition; Pivoting strategies; Calculating the inverse; Norms; Condition number; Ill-conditioned linear equations; Iterative refinement; Iterative methods.
• Interpolation and polynomial approximation: Why use polynomials? Lagrangian interpolation; Neville's algorithm; Other methods.
• Approximation theory: Norms; Least-squares approximation; Linear least-squares; Orthogonal polynomials; Error term; Discrete least-squares; Generating orthogonal polynomials.
• Numerical quadrature: Newton-Cotes formulae; Composite quadrature; Romberg integration; Adaptive quadrature; Gaussian quadrature (Gauss-Legendre, Gauss-Laguerre, Gauss-Hermite, Gauss-Chebyshev).
• Numerical solution of ordinary differential equations: Boundary-value problems; Finite-difference formulae for first and second derivatives; Initial-value problems; Errors; Taylor-series methods; Runge-Kutta methods.
Project 1 20%
Project 2 30%
- MTH3024 Modelling and Simulation (Sem 2)
Pre-requisite: MTH2021 Mathematical Methods 2, or
MTH2002 Introduction to Partial Differential Equations [for 2021/22]
Lecturers: Dr M Gruening and Prof H van der Hart
In this module, students will analyse real-life situations, build a mathematical model, solve it using analytical and/or numerical techniques, and analyse and interpret the results and the validity of the model by comparing to actual data. The emphasis will be on the construction and analysis of the model rather than on solution methods. Two group projects will fix the key ideas and incorporate the methodology. This will take 7-8 weeks of term and will be supported with seminars and workshops on the modelling process. Then students will focus on a solo project (relevant to their pathways) with real-life application and work individually on this for the remaining weeks of term. They will present their results in seminars with open discussion, and on a Webpage.
The starting group project will be focused, and offer a limited number of specific modelling problems. For the other projects, students will build on these initial problems by addressing a wider problem taken from, but not exclusively, the following areas: classical mechanics, biological models, finance, quantum mechanics, traffic flow, fluid dynamics, and agent-based models, including modelling linked to problems of relevance to the UN sustainable development goals. A pool of options will be offered, but students will also have the opportunity to propose a problem of their own choice.
- MTH3025 Financial Mathematics (Sem 2)
Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 3 of either an MSci or a BSc Mathematics pathway, and a mathematical knowledge and ability commensurate with this stage is assumed.
Lecturer: Dr A Ferraro
Introduction to financial derivatives: forwards, futures, swaps and options; Future markets and prices; Option markets; Binomial methods and risk-free portfolio; Stochastic calculus and random walks; Ito's lemma; the Black-Scholes equation; Pricing models for European Options; Greeks; Credit Risk.
- MTH3031 Classical Fields (Sem 1)
Pre-requisite: MTH2031 Classical Mechanics or PHY2004 Electricity, Magnetism & Optics, or AMA2001 Classical Mechanics or PHY2004 Electricity, Magnetism & Optics [for 2021/22]
Lecturer: Dr G Gribakin
• Recapping of the least action principle in Classical Mechanics. Lagrangians for continuous systems (e.g., a string), and derivation of the wave equation from Lagrange’s equation.
• 4-dimensional space time, interval, 4-vectors, tensors, Lorentz covariance.
• Action and Lagrangian for a particle, energy and momentum.
• 4-potential and the Lagrangian for a charged particle in an electromagnetic field, relativistic equation of motion and Lorentz’s force, electric and magnetic fields.
• Lagrangian of the electromagnetic field, Maxwell’s equations in covariant form, charge density and current density, continuity equation, Maxwell’s equations in conventional (3+1) form, and in integral form.
• Electrostatics (general ideas, Coulomb’s law, fields of various charge distributions, electric dipole moment).
• Magnetostatics (general ideas, Biot-Savart-Laplace law, fields of systems of currents, magnetic dipole moment).
• Electromagnetic waves, plane wave, polarisation, monochromatic wave.
• Electromagnetic radiation: retarded potentials, dipole radiation (electric, magnetic), Larmor formula.
Assignments 15% (x 2)
- MTH3032 Quantum Theory (Sem 1)
Pre-requisite: MTH2031 Classical Mechanics or PHY2001 Quantum and Statistical Physics, and MTH2021 Mathematical Methods 2, or None [for 2021/22]
Lecturer: Prof M Paternostro
• Overview of classical physics and the need for new theory.
• Basic principles: states and the superposition principle, amplitude and probability, linear operators, observables, commutators, uncertainty principle, time evolution (Schrödinger equation), wavefunctions and coordinate representation.
• Elementary applications: harmonic oscillator, angular momentum, spin.
• Motion in one dimension: free particle, square well, square barrier.
• Approximate methods: semiclassical approximation (Bohr-Sommerfeld quantisation), variational method, time-independent perturbation theory, perturbation theory for degenerate states (example: spin-spin interaction, singlet and triplet states).
• Motion in three dimensions: Schrödinger equation, orbital angular momentum, spherical harmonics, motion in a central field, hydrogen atom.
• Atoms: hydrogen-like systems, Pauli principle, structure of many-electron atoms and the Periodic Table.
Assignments 15% (x 2)
- PMA3013 Mathematical Investigations (Sem 2)
Lecturers: Dr S Shkarin and Dr A Zhigun
This module is concerned with the investigation processes of mathematics, including the construction of conjectures based on simple examples and the testing of these with further examples, aided by computers where appropriate. A variety of case studies will be used to illustrate these processes. A series of group and individual investigations will be made by students under supervision, an oral presentation will be made on one of these investigations. While some of the investigations require little more than GCSE as a background, students will be required to undertake at least one investigation which needs knowledge of Mathematics at Level 2 or Level 3 standard and/or some background reading.
Solo Project 50%
Group Project 40%
- PMA3014 Set Theory (Sem 1)
Pre-requisite: MTH2011 Linear Algebra. This module is a compulsory component of the MSci course in Pure Mathematics.
Lecturer: Dr M Mathieu
Set Theory is a rich and beautiful subject of Pure Mathematics whose fundamental concepts permeate every branch of mathematics. No undergraduate mathematics education is complete without a thorough study of this discipline. This module will teach fluency in the language of elementary axiomatic set theory, facility in the use of key techniques such as transfinite induction and maximality principles, and basic arithmetic of cardinal and ordinal numbers (the ‘arithmetic of infinity’).
Naïve vs. Axiomatic Set Theory, Zermelo-Fraenkel axioms, Axiom of Choice, Well-Ordering Principle, Zorn’s Lemma, Transfinite Induction, Construction of the number systems, Ordinal and cardinal numbers and their arithmetic.
70% of the module mark is provided by a written 3-hour exam
30% of the module mark are made up by two marked homework assignments due in week 4 and in week 9.
D. W. Cunningham, Set theory: A first course, Cambridge Math. Textbooks, Cambridge Univ. Press, New York, 2017.
I. Kaplansky, Set theory and metric spaces, Chelsea Publ. Company, New York, 1977.
- PMA3017 Metric and Normed Spaces (Sem 2)
In 2021-22 this module will be co-taught with MTH2013 Metric Spaces, but will be examined separately.
Pre-requisite: MTH2012 Analysis
Lecturers: Dr Y-F Lin and Dr G Kiss
Analysis is the study of convergence and continuity, it is fundamentaly linked to the structure of the real numbers. The aim of this course is to move gradually away from real analysis to the more general setting of metric and normed spaces. The fundamental question is: what aspects of real analysis survive to the setting of metric and normed spaces?A metric space is a set with a notion of distance, called a metric. Them most familiar example is the real line, with the distance from x to y given by | x - y | . This notion of distance allows us to define convergence and continuity in a much more abstract setting.Often, we want the set underlying our metric space to have an addition operation. To get interesting structures, we must then add some extra compatibility conditions to our metric. This leads us to the concept of a normed vector space, this is a vector space as we have encountered before, but now it comes with a measure of the size of a vector, called the norm. As every normed vector space is a metric space, we will begin with the study of metric spaces.
Definition and examples of metric spaces; open sets, closed sets, closure points, sequential convergence, compactness; completeness; continuous mappings between metric spaces; Banach's fixed-point theorem and applications; Baire category theorem and applications. Normed spaces; Banach spaces; finite dimensional normed spaces; subspaces and quotient spaces; linear operators; boundedness; compact operators; dual spaces.
Continuous Assessment 10%
Lecturers: Prof J Kohanoff, Dr M Gruening
Operational Research is the application of quantitative analysis to problems outside of the physical sciences and in particular to the problems of business, industry and administration. However, all the techniques can be used outside of Operational Research and emphasis is given to formulation of problems and recognition of the appropriate technique. Blackboard examples are given for each technique and the homework problems are drawn from a wide field. Most techniques are in the form of algorithms and for hand calculations the problems have to be fairly simple, but indications are given how computers could be applied to bigger problems in the project and presentation components.
The aim of this course is to develop competence in two of the most important mathematical techniques used in Operational Research, in formulating problems and expressing answers clearly.
This course covers the two main general-purpose mathematical techniques used in Operational Research together with some specialized applications. These are all optimisation techniques, although they also give insight into the problems and the interpretation of answers is stressed. Calculus cannot be used because either (i) the objective function and the constraints cannot be expressed simply, or (ii) the constraints dominate the problem and the optimal solution will be at the edge of the feasible region, or (iii) the variables are not continuous (e.g., they are integer).
Some students may have met some of the techniques in courses at other levels, but no previous knowledge is assumed, and more mathematical rigour and understanding is required than at those levels. However, the mathematical knowledge assumed does not extend beyond that in Level 1. The main knowledge assumed is elementary linear algebra (bases, linear dependence of vectors, matrix notation, partitioning of matrices). No knowledge of economics is required but the economic interpretation of some results is explored. Apart from a passing reference to stochastic Dynamic Programming this course is purely deterministic, and no knowledge of statistics is required.
The scope of Operational Research, formulating a problem from a verbal description.
Dynamic programming: formulation, principle of optimality, value iteration, applications including equipment replacement, allocation, production planning and optimal routes. Special algorithms for optimal routes.
Linear programming: formulation, theory, Primal Simplex Method, interpretation of the final tableau, Revised Simplex Method, duality theory including economic interpretation, Dual Simplex Method, Post-optimal Analysis, Transportation and Assignment problems. A wide variety of practical problems and applications is discussed.
Group work 20%
- SOR3004 Linear Models (1st semester) NOT TO BE TAKEN AFTER SOR2004
Pre-requisite: SOR2002. Make sure you are enrolled for this module in the 1st semester.
Lecturer: Dr H Mitchell
The aim of this module is to cover linear models encompassing multiple linear regression and analysis of variance (ANOVA). These models are the workhorses of statistical data analysis and are found in virtually all branches of the sciences as well as in the industrial and financial sectors.
Multiple linear regression is concerned with modelling a measured response as a function of explanatory variables. For example, a pharmaceutical company might use a a regression model to relate the effectiveness of a new cancer drug to the patients age, gender, weight, diet, tumour size, etc. ANOVA is concerned with the analysis of data from designed experiments. A materials manufacturer for example, may wish to analyse the results from an experiment to compare the heat resisting properties of four different polymers.
Regression and ANOVA will be initially developed using a classic least squares approach and later the correspondence between least squares and the method of maximum likelihood will be examined. After a thorough development of linear models the groundwork will have been laid to allow an extension to the broader class of Genealized Linear Models (GLM). These permit regression models to be applied to situations where the recorded response is not normally distributed. One famous example of the use of GLM was the analysis of O-ring failures on the space shuttle Challenger.
An important element of this module will be a weekly practical data analysis class using the SAS software package. SAS is probably the leading statistical package used in industry. These classes, lasting up to three hours, will introduce the student to elementary data entry in SAS, elementary matrix manipulation using the SAS Interactive Matrix Language (IML) and analysis of data using linear and generalized linear models. Each week the student will complete a data analysis task using SAS and is required to submit a report the following week.
- Multiple linear regression: ordinary least squares, model selection and diagnostics, weighted least squares.
- Analysis of variance: Non-singular and singular cases; extra sum of squares principle, analysis of residuals, generalized inverse solution, estimable functions, testable hypotheses.
- Experimental designs: completely randomized, randomized block, factorial, contrasts, analysis of covariance.
- Generalised linear model: maximum likelihood and least squares, exponential family, Poisson and logistic models, model selection for GLM.
- SOR3008 Statistical Data Mining (Sem 2)
Pre-requisite: SOR2004 or currently enrolled in SOR3004 in semester 1.
Lecturer: Dr K Cairns
In the 1990's there was an explosive growth in both the generation and collection of data due mainly to the advancement of computing technology in processing and storage of data and the ease of scientific data collection. As a result, overwhelming mountains of data are being generated and stored. For example, in the business world large supermarket chains such as Wal-Mart and Sainsbury's collect data amounting to millions of transactions per day. In the US all health-care transactions are stored on computers yielding terabyte databases which are constantly being analysed by insurance companies. There are huge scientific databases as well. Examples include the human genome database project and NASA's Earth Observatory System. This has brought about a need for vital techniques for the modelling and analysis of these large quantities of data: data mining.
Data Mining is the process of selection, exploration, and modelling of large quantities of data to discover previously unknown regularities or relations with the aim of obtaining clear and useful results for the owner of the database. The application of data mining includes many different areas, such as market research (customer preferences), medicine, epidemiology, risk analysis, fraud detection and more recently within bioinformatics for modelling DNA.
This module will focus on data mining techniques which have evolved from and are strongly based on statistical theory.
Introduction to Data Mining.
Exploratory data analysis: Principal Component Analysis; Multiple Imputation.
Cluster analysis: Hierarchical clustering; Partitioning algorithms.
Classification: Nearest neighbour algorithms; Classification trees; Naïve Bayes Classifier; Bayesian networks; Ordinal Regression; Multinomial Logit; Conditional Logit; Nested Logit;Techniques for comparing classifiers - including bagging and boosting in ensemble methods.
Prediction (continuous targets): Regression Trees; Random Forests; Neural Networks; Support Vector Machines.
Association Rule mining.
Coursework (Continuous assessments) 40%
SOR3012 Stochastic Processes and Risk (2nd semester)
Pre-requisite: SOR1020 and SOR1021
Lecturer: Dr A Munaro
Uncertainty or risk is a natural part of life. Continually faced with choices, we are required to make decisions based on the possible consequences or outcomes. We often use the phrase ‘a calculated risk’ to describe how we analyse a problem and come up with our decision. Now the question is: how do we calculate risk - mathematically.
Statistical analysis of a random process helps determine whether there are any underlying patterns. If there are, we can use this to our advantage in predicting the future - or at least make an educated guess! Not surprisingly, this topic is primarily recognised for its career opportunities where ‘risk’ assessment and planning matter: for example the provision of power stations, telecoms, transport planning, the spread of infectious diseases (swine flu, for example). A major application of this kind of mathematics is the financial services sector: insurance and investment, credit risk, capital market trading etc. Many students that have taken this course are currently employed as highly-paid actuaries.
Aside from the educational and commercial value of this subject, it leads into very interesting and fun topics such as measure theory, martingales and potential theory, noise, fractals, etc.
Logic and Boolean algebra, counting and combinatorics, set algebra, inclustion-exclusion theorem, mutually exclusive events, De Morgan Laws.
Axioms of probability, events and probability spaces, sigma-field, random variables, conditional probability, and expectation, Bayes’ theorem, discrete and continuous random variables, moments and moment generating function. Laws of large numbers and central limit theorem.
Pairs of random variables, marginal probabilities, Cauchy Schwartz Inequality in statistics, correlation and covariance.
Discrete time Markov chains, Chapman Kolmogorov relation, limiting behaviour, transient, recurrent states and periodic states, limiting stationary distribution, hitting times and hitting probabilities.
Continuous time Markov chains, Kolmogorov forward equations, stationary distribution for continuous time Markov chains, Poisson process, MM1 Queue, inhomogeneous Poisson process and compound Poisson process.
Practical Lab / Log Book 45%
- 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
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.
Continual Assessment 30%
- 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
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
Continual Assessment 25%
- 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
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.
Class Test 10%