Use mathematics to solve real world problems, such as in finance, energy, engineering or scientific research.
Overview
The focus of this course is using mathematics to solve real world problems, such as in finance, energy, engineering or scientific research. The combination of the applied nature of the mathematics that is taught, with the masters level of this course, makes this qualification highly attractive to employers.
Many of the topics taught are directly linked to the research that we do, so you will be learning at the cutting edge of applied mathematics.
We are a relatively small division and operate with an excellent staff/student ratio. One advantage of this is that we can get to know each student personally, and so can offer a friendly and supportive learning experience. Staff are ready and willing to help at all levels, and in addition, our StudentStaff Committee meets regularly to discuss matters of importance to our students.
We also offer students the chance to choose a selection of modules from other subject areas such as economics and finance.
Specialist software
We have a wide selection of mathematical software packages such as MATLAB, Maple and COMSOL, which are used throughout the course.
Weekly seminar programme
We have a weekly seminar programme in the mathematics division, which features talks in the areas of research strength in the division, Mathematical Biology, Applied Analysis, Magnetohydrodynamics and Numerical Analysis & Scientific Computing.
Who should study this course?
This course suits graduates with a degree in mathematics or in a subject with strong mathematical components such as physics, who wish to deepen their mathematical knowledge and related skills.
Teaching Excellence Framework (TEF)
The University of Dundee has been given a Gold award – the highest possible rating – in the 2017 Teaching Excellence Framework (TEF).
Teaching & Assessment
How you will be taught
You will learn by traditional methods such as lectures, tutorials, and workshops as well as via computer assisted learning. We teach the use of professional mathematical software packages in order to allow you to explore mathematics far beyond the limits of traditional teaching.
Individual reading and study takes a particularly important role in the Summer project. For the project, you will be guided to prepare your research project plan and to develop skills and competence in research including project management, critical thinking and problem solving, project reporting and presentation.
How you will be assessed
Assessment is via a mix of open book continual assessment and closed book examinations, with a substantial project completed over the Summer.
What you will study
This one year course involves taking four taught modules in semester 1 (SeptemberDecember), followed by a further 4 taught modules in semester 2 (JanuaryMay), and undertaking a project over the Summer (MayAugust).
A typical selection of taught modules would be eight of the following:
 Dynamical Systems
 Computational Modelling
 Stochastic Processes
 Inverse Problems
 Mathematical Oncology
 Mathematical Ecology & Epidemiology
 Mathematical Physiology
 Fluid Dynamics
 Optimization in Finance and Energy
 Measure Theory
 Functional Analysis
We also offer the option of relacing one or two mathematics modules with modules from subjects such as Global Risk Analysis, Energy Economics, Quantitative Methods and Econometrics for Finance.
Modules
About the module
This module, aimed at the Level 5 student, takes an advanced look at dynamical systems. The time evolution of many biological, chemical, or physical processes, as well as systems considered in engineering or economics, can be described by difference or differential equations. Dynamical systems theory allows us to study these systems of equations and inver information about the behaviour of the corresponding biological, chemical or physical systems. It addresses questions like the existence and stability of solutions, how the behaviour of solutions changes depending on the system parameters, or determines the existence of strange attractors or chaos in the system.
This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or Mathematics and Physics degrees. If you have questions about this module, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in each of the modules MA31002 and MA32001, or equivalent.
Indicative Content

OneDimensional Maps
Definition, Cobweb Plot: Graphical Representation of an Orbit, Stability of Fixed Points, Periodic Points, Chaos: Lyapunov Exponents.

Ordinary Differential Equations
Background, Examples of main Physical and Biological Processes described by Ordinary Differential Equations (ODEs), Existence and uniqueness of solutions of ODEs, Linearised Stability Analysis, Twodimensional Systems: Hamiltonian and Gradient systems, Periodic solutions: Floquet theory, Poincare Map and Stability of Periodic Orbits, Bifurcation and Chaos.

Partial Differential Equations
Definitions, Background, Wellposedness, Maximum Principles, Spectral Theorem for Laplace Equation, Semigroups for Evolution Equations in Banach Spaces, Nonlinear Evolution Equations: Linearised Stability Analysis for ReactionDiffusion Equations.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/presentation classes and assessed via an exam (60%) and coursework (40%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
In this module the Level 5 student will learn to write their own code and to apply builtin "black box" solvers in MATLAB and COMSOL to mathematical modelling problems. This module is mandatory for Level 5 students taking the MMath in Mathematics or the MSci in Mathematical Biology. This module may be taken in combination with another at Level 5 by students taking the MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in each of the modules MA32005 and MA42003, or equivalent.
Indicative Content

MATLAB fundamentals
Students will learn basic operations in MATLAB, and implement various finite difference schemes to solve ODEs (primarily initial value problems) originating in celestial mechanics, population dynamics, and cell biomechanics.

MATLAB ODE solvers for initial value problems
Students will learn to use standard builtin solvers with MATLAB, particularly ode45 and ode23s, and possibly dde23. We will apply these solvers to initial value problems (and possibly delay differential equations) stemming from celestial mechanics, cell biomechanics, and population dynamics.

MATLAB random variables, stochastic processes, and SDEs
After a brief introduction to stochastic differential equations (SDEs), students will learn MATLAB solution techniques, with applications to Brownian motion and related physical processes. We will also learn to simulate discrete and continuous stochastic processes, and generate samples from random variables with arbitrary distributions.

MATLAB ODE solvers for boundary value problems
Students will implement a standard "shooting" method to solve a BVP from heat transfer. We will learn to use the standard builtin solvers, particularly bvp4c. We explore alternate solution techniques, such as by formulating the discretised equation as a linear algebraic system, and as the steady state solution to a PDE; these approaches help drive us towards PDE solution methods. The class will apply these solvers to boundary value problems stemming from heat transfer and fluid mechanics.

MATLAB for PDEs
Students will implement explicit finite difference methods in MATLAB, with a focus on reactiondiffusion problems. The overall goal will be to solve coupled reactiondiffusion problems (with heterogeneous coefficients) and cell growth.

Weak formulations for partial differential equations; introduction to FEMs
We repose PDEs using a weak formulation, using the context of function spaces. Using this framework, we develop an understanding of finite element methods (FEMs).

FEMs and COMSOL fundamentals
Students will learn to solve reactiondiffusion equations using the builtin FEMs in COMSOL.
Delivery and Assessment
Delivery of this module will take a handson, interactive approach, where lectures are integrated with guided computer lab time. Assessment will be based on computational coursework (100%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module, aimed at Level 5 students, gives a nonmeasure theoretic introduction to stochastic processes, considering the theory and some applications and going on to introduce stochastic differential equations and their solutions. This module may optionally be taken in combination with others by Level 5 students taking the MMath in Mathematics or the MSci in Mathematical Biology or Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module would find it beneficial to have taken each of the modules MA32001 and MA51007, or equivalent.
Indicative Content

Probability fundamentals
Elementary probability concepts such as random variables, expected value, moment generating and characteristic functions, conditional exceptions, probability inequalities and limit theorems, etc.

The Poisson process
(Homogeneous) Poisson process and related examples such as interarrival and waiting time distributions and conditional distribution of the arrival times. Some practical examples such as the busy period of the M/G/1 queueing system. Introduction to the nonhomogeneous Poisson process.

Markov chains
(Discretetime) Markov chains and some related examples. ChapmanKolmogorov equations and classification of states.

Continuoustime Markov chains
Continuoustime Markov chains, birth and death processes, and the Kolmogorov differential equations.

Brownian motion and stochastic differential equations
Basics of Brownian motion, Ito^ integral and Ito^ formula, and then stochastic differential equations (SDEs). A number of commonly used SDEs and their solutions will be discussed.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/presentation classes and assessed via coursework (100%) consisting of homeworks and a presentation.
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This is a Level 5 course that offers a robust understanding of the inverse problems theoretical framework and methods suitable for medical and financial applications. The aim is to achieve comprehensive knowledge in the theoretical fundaments and general methodology for inverse problems in various heterogeneous media, including medical applications and finance. This module may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA32001, or equivalent.
Indicative Content

Examples of Inverse Problems
Examples from medical applications and finance.

Inverse Methodology Preliminary Foundation
Necessary Basic Definitions and Theorems in Measure Theory and Function Spaces

General Regularisation Theory
Tikhonov′s regularization method. Landweber Iteration. The Discrepancy Principle of Morozov. Conjugate gradient method.

Galerkin Methods
Galerkin General formulation. The Least Squares Method. The Dual Least Squares Method.

The Truncated Singular Value Decomposition Method

Stable inversion via the Mollification Method

Inverse Problems in General Heterogeneous Media and Medical Applications
Backward heat conduction problem. Inverse problems in reactiondiffusion equations.

Inverse problems in finance
Formulation of forward model: BlackScholes and Dupire's Formula. Inverse Problem formulation of market volatility. Reconstruction of time and pricedependent volatilities.
Delivery and Assessment
The module is delivered in the form of lectures and assessed via coursework (100%) consisting of tests and homeworks.
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module, aimed at the Level 5 student, covers the fundamentals of Measure Theory including Lebesgue Measure and Integration and Lebesgue Spaces as well as their pivotal implications in the modern analysis of partial differential equation and probability theory. This module may optionally be taken in combination with other modules at this level by Level 5 Students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA32001 or equivalent.
Indicative Content

Construction of a general measure space
Definition and properties of a σalgebra. The concept of measures as a nonnegative σadditive real valued set functions defined on a σalgebra. Measurable sets, σfinite measurable sets, the notion of algebra of generators of a σ algebra of parts. Construction of the Lebesgue Measure. Charatheodory Extension Theorem: characterising the unique extension of a finitely additive nonnegative real valued function on an algebra of generators to a measure on the generated σalgebra.

Measurable functions
Definition. Approximation by measurable simple/step functions as well as by continuous functions.

Construction of the Lebesgue Integral
Construction of the integral for measurable indicators (characteristic) functions as well as for measurable simple/step functions. Construction of the integral for a general measurable function. Properties of Lebesgue Integrable functions (additivity, multiplication by scalars, positivity, monotonicity). Definition and basic vectorial properties of the space of Lebesgue Integrable Functions L^{1}.

Limit theorems (concerning pointwise convergence, almost everywhere convergence, and convergence in measure)
Fatou Lemma. Monotone Convergence Theorem. Dominated Convergence Theorem.

Product Measures and the Fubini Theorem

Absolutely continuous measures and the LebesgueRadonNikodym Theorem

Definition and properties of general Lebesgue Spaces L^{p} and their embedding relations
If time permits then the following will also be covered:

The Sobolev Spaces H^{1} and their implications for the analysis of elliptic equations partial differential equations

Connection with probability theory, including random variables, cumulative distribution functions and martingales
Delivery and Assessment
The module is delivered in the form of lectures and seminars and assessed via an exam (60%) and coursework (40%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This Level 5 module will undertake analysis and computational simulation of mathematical models of cancer growth and treatment. In addition to the mathematical formulation and analysis of such systems which will take place during the formal lectures, through tutorial work students will undertake computational analyses using tools such as MATLAB and COMSOL introduced in MA51004. This module is mandatory for students on the MSci in Mathematical Biology degree and may optionally be taken in combination with other modules by students on the MMath in Mathematics or MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA31002, or equivalent.
Indicative Content

Overview of Cancer Growth and Spread
The biology of cancer.

Avascular Solid Tumour Growth
ODE models of solid tumour growth Greenspan′s model. Reactiondiffusion moving boundary models.

Tumourinduced Angiogenesis
Continuum PDE models. Hybrid discretecontinuum models.

Cancer Invasion
Continuum PDE models. Hybrid discretecontinuum models.

Immune Response to Cancer
ODE models of the immune response to cancer. PDE models of the immune response to cancer.

Chemotherapy and Radiotherapy Treatment
Chemotherapy drug scheduling models. The linearquadratic radiobiological model.
Delivery and Assessment
The module is delivered in the form of lectures and tutorials/computer labs and assessed via exam (100%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
In this module we will undertake analysis and computational simulation of mathematical models of ecological and epidemiological systems including predatorprey systems, hostparasitoid systems, plantherbivore systems, spread and transmission of disease (e.g. AIDS, SARS, measles, rabies). In addition to the mathematical formulation and analysis of such systems which will take place during the formal lectures, through tutorial work students will undertake computational analyses using tools such as MATLAB and COMSOL introduced in MA51004. This module is mandatory for students on the MSci in Mathematical Biology degree and may optionally be taken in combination with other modules by students on the MMath in Mathematics or MSci in Mathematics and Physics. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA31002, or equivalent.
Indicative Content

Nonspatial models
Difference equation models. Ordinary differential equation models. Delay difference/differential equation models.

Spatial models
Integrodifference equation models. Partial differential equation models. Integrodifferential equation models.
Delivery and Assessment
The module is delivered in the form of lectures and tutorials/computer labs and assessed via exam (100%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module, aimed at Level 5 students, studies mathematical models of cellular physiology in metabolism, genetic networks, and electrophysiology. This module is mandatory for Level 5 students taking the MSci in Mathematical Biology and may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA32009, or MA41002, or equivalent.
Indicative Content

Biochemical Reactions
Enzyme kinetics. Glycolysis

Genetic Networks
Central Dogma of biology, review of bifurcation theory. Feedback loops and Oscillation. Circadian rhythms. Cell cycle model.

Electrophysiology
The HodgkinHuxley model. The FitzHughNagumo equations. Small network dynamics of coupled neurons.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/presentation classes and assessed via exam (70%) and coursework (30%).
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
Optimisation problems arise from modelling a wide variety of systems in science, technology, industry, business, economics as well as in many other fields. This module, aimed at Level 5 students, covers practical methods of optimisation that are supported by a growing body of mathematical theory. Students are expected to implement the methods and solve problems numerically. This module may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA32005, or equivalent.
Indicative Content

Introduction
Examples of optimization problems. Mathematical background.

Unconstrained Optimization
Line search and Descent methods. Newton′s method. Conjugate gradient method.

Linear Programming
Simplex method. Slack and artificial variables. Simplex tableau.

Constrained Optimization
Lagrange multipliers. Theory of constrained optimization.

Application in Finance and Energy
Application problems such as factory location problem, oil pipeline problem.
Delivery and Assessment
The module is delivered in the form of lectures and workshops/presentation classes and assessed via coursework (100%) consisting of homeworks, tests, presentations and project work.
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module covers advanced topics in Fluid Dynamics for the Level 5 student who has already undertaken an introductory course. This module may optionally be taken by students on Mathematics or Physics MSc programmes as well as MMath in Mathematics, MSci in Mathematical Biology or MSci Mathematics and Physics.
The course focuses on vortex dynamics and uses this as basis to understand and describe turbulence, one of the most intriguing phenomena of fluid dynamics. It also provides an introduction to nonNewtonian fluids and their fascinating properties. It takes the viewpoint of an applied mathematician/theoretical physicist and derives fundamental properties from the underlying system of equations.
If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in the module MA42007 (Mathematics of Fluids I), or a similar course.
Indicative Content

Introduction (recap of basic fluid dynamics)
NavierStokes equations and their approximations and modifications

Boundary Layers
Nondimensionalisation, Blasius boundary layer

Vortex Dynamics
2D dynamics, point vortices, vortex sheets, dynamics of vortex filaments, effects of viscosity, energy and enstrophy dissipation

Turbulence
Instabilities, scales, description in Fourier space, Kolmogorov′s theory, 3D vs. 2D turbulence

NonNewtonian Fluids
Types of nonNewtonian fluids, Ostwald  de Waele model, Bingham plastic
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%) consisting of homeworks.
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module, aimed at the Level 5 student, specifically aims to develop students' knowledge, skills and understanding of the fundamentals of function spaces (infinite dimensional spaces) and linear operators and functionals defined on function spaces.
Functional analysis plays an important role in many areas of applied mathematics and is essential for the theory of Partial Differential Equations, Numerical Analysis, Probability theory and Theoretical Physics. Functional analysis originated from classical analysis and is formed by the study of infinite dimensional vector spaces and linear functions defined on these spaces. The theory of functional analysis was developed by some of the most famous mathematicians of the 20th century such as Hilbert, Schmidt, Riesz, Banach and von Neumann. Functional analysis can be characterised as a combination of infinitedimensional linear algebra and classical analysis. Methods of functional analysis will allow us to analyse the properties of function spaces and to characterise solutions of integral and differential equations, arising in modelling of many biological and physical systems.
This module may optionally be taken by students on the MMath in Mathematics, or the MSci in Mathematical Biology or Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Prerequisites
Students taking this module must usually have achieved a pass mark in MA32001, or equivalent.
Indicative Content

Function Spaces
 Normed spaces and inner product spaces (Banach and Hilbert spaces)
 Lebesgue integral and Lebesgue spaces
 Orthogonality in Hilbert spaces
 Weak derivatives and Sobolev spaces
 Notion of weak convergence in Function Spaces

Linear Operators and Linear Functionals
 Definition of linear operators and linear functionals
 Definition of a norm of a bounded linear operator
 Riesz and LaxMilgram theorems
 Definition, main properties and examples of compact operators
 Spectral theory for compact operators

Wellposedness results for Integral and Partial Differential Equations
 Application of LaxMilgram theorem and theory of compact operators to prove the wellposedness results
Delivery and Assessment
The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (60%) and coursework (40%) consisting of homeworks and tests.
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.
About the module
This module is a personal research project for students on the MSc in Applied Mathematics and Mathematical Biology, which runs over the summer. If you have questions about this module please contact your Advisor of Studies.
Prerequisites
Students taking this module must have achieved an average mark of at least 50% (C3) over the Mathematics MSc modules taken in Semesters 1 and 2 and have obtained at least 75 credits in these modules.
Indicative Content

Project
Carry out a substantial project in an area of mathematics and document the work in a project report.
Delivery and Assessment
The project will require substantial independent work as well as regular meetings with the project supervisor(s). Assessment will be based on coursework (100%) consisting of the project report.
Credit Rating
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 60 SCOTCAT credits or 30 ECTS credits.
Careers
Mathematics is central to the sciences, and to the development of a prosperous, modern society. The demand for people with mathematical qualifications is considerable, and a degree in mathematics is a highly marketable asset.
Mathematics graduates are consistently amongst those attracting the highest graduate salaries and can choose from an ever widening range of careers in research, industry, science, engineering, commerce, finance and education.
Entry Requirements
Students should have a 2.2 Honours BSc or above (or a suitable qualification) in a relevant mathematical discipline.
EU and International qualifications
English Language Requirement
IELTS Overall  6.0 

Listening  5.5 
Reading  5.5 
Writing  6.0 
Speaking  5.5 
Equivalent grades from other test providers
English Language Programmes
We offer PreSessional and Foundation Programme(s) throughout the year. These are designed to prepare you for university study in the UK when you have not yet met the language requirements for direct entry onto a degree programme.
Discover our English Language Programmes
Fees & Funding
The fees you pay will depend on your fee status. Your fee status is determined by us using the information you provide on your application.
Find out more about fee status
Fee status  Fees for students starting 201819 

Scottish and EU students  £6,950 per year of study See our scholarships for UK/EU applicants 
Rest of UK students  £6,950 per year of study See our scholarships for UK/EU applicants 
Overseas students (nonEU)  £16,450 per year of study See our scholarships for international applicants 
Your Application
You apply for this course via the UCAS Postgraduate (UKPASS) website which is free of charge. You can check the progress of your application online and you can also make multiple applications.
You'll need to upload relevant documents as part of your application. Please read the How to Apply page before you apply to find out about what you'll need.
Degree  Course Code  

Apply Now  Applied Mathematics MSc  P052171 
Course Contact
Dr Dumitru Trucu
Science and Engineering
pgtmaths@dundee.ac.uk
+44 (0)1382 384462