MMath Mathematics

You'll study the basic concepts and standard methods of mathematical notation and proof, polynomial equations and inequalities, sequences and series, functions, limits and continuity, differentiation and integration.

The fundamental concepts of calculus (differentiation and integration) presented in Applications of Calculus will be examined in more detail, extended to a larger class of functions by means of more sophisticated methods, including an introduction to complex numbers and variables. These will all be demonstrated in application to practical problems including solving basic first and second-order differential equations.  

This module will introduce you to vectors and matrices, along with the idea of mathematical modelling through their application to real-world problems.

Some basic ideas and techniques of statistics will be presented while introducing some essential study skills, allowing you to develop and practice personal and technical skills, for example, self-study, teamwork, analysing data, writing reports, and making presentations.

This module will introduce you to some elementary number theory with interesting modern applications, including:

• mathematical background: natural numbers and integers, factorisation, proof by induction
• highest common factor, lowest common multiple, the Euclidean Algorithm
• prime numbers and the Fundamental Theorem of Arithmetic
• Diophantine equations, modular arithmetic, congruence mod n, solving equations in Z_n
• Euler's phi-function and Euler’s product formula
• affine and exponential ciphers, and the RSA cryptosystem

This module will introduce you to an area of mathematics (graph theory) not usually met in school or college courses, including: 

• graphs and multigraphs
• isomorphic graphs and subgraphs
• paths, trails, circuits and cycles,
• Eulerian and Hamiltonian graphs
• weighted graphs: the shortest path problem, the travelling salesman problem and minimal spanning trees
• planar graphs and Euler’s formula
• colouring of graphs and matchings on graphs
• digraphs, critical path analysis and network flows

This module will introduce you to the basic ideas of linear algebra, such as matrices and determinants, vector spaces, bases, eigenvalues and eigenvectors. You'll study various standard methods for solving ordinary differential equations and understand their relevance.

This module will present basic ideas, techniques and results for calculus of two and three variables, along with differentiation and integration over curves, surfaces and volumes of both scalar and vector fields.

This module will give a rigorous treatment of convergence of sequences and infinite series of real numbers and of continuity, differentiability and integrability of functions of a real variable. It will illustrate the importance of these concepts in the analysis of problems arising in applications.

This module will present the basic concepts of probability theory and statistical inference and provide you with the tools to appropriately analyse a given data set and effectively communicate the results of such analysis.

This module will develop your appreciation of the basic concepts of force, momentum and energy, and of Newton’s laws of motion. The module will equip you to apply these concepts to model physical systems, in particular the orbital motion of bodies.

This module will introduce you to the R computing environment. It'll enable you to use R to import data and perform statistical tests, allow you to understand the concept of an algorithm and what makes a good algorithm and will equip you for implementing simple algorithms in R.

This module will introduce functions of a complex variable, define concepts such as continuity, differentiability, analyticity, line integration, singular points, etc. You will examine some important properties of such functions and consider some applications of them, for example, conformal mappings and the evaluation of real integrals using the Residue Theorem. You will also be introduced to Fourier and Laplace transform methods for solving linear ordinary differential equations and convolution type integral equations.

In this module we'll introduce basic algebraic structures, with particular emphasis on those pertaining to finite dimensional linear spaces and deepen your understanding of linear mappings. We'll also provide an introduction to inner product spaces and bilinear forms.

In this module we’ll introduce you to analytical methods for solving ordinary and partial differential equations, so you'll develop an understanding along with technical skills in this area.

In this module you will be introduced to the basic theory and applications of:

• metric spaces
• normed vector spaces and Banach spaces
• inner product spaces and Hilbert spaces
• bounded linear operators on normed linear spaces

This module will:

• review the concepts of probability distributions and how to work with these
• present approaches to parameter estimation, focusing on maximum likelihood estimation, bootstrap estimation, and properties of estimators
• present hypothesis testing procedures, including classical likelihood ratio tests and computer-based methods for testing parameter values, and goodness-of-fit tests
• introduce and provide understanding of the least squares multiple regression model, general linear model, transformations and variable selection procedures
• present use of R functions for regression and interpretation of R output

This module will:

• convey the generalisation of the mechanics of single-particle systems to many-particle systems
• convey the central ideas of a continuum description of material behaviour and to understand relevant constraints
• ground students in the basic principles governing three-dimensional motions of rigid bodies
• convey how the ideas of continuum theory are applied to static and inviscid fluids

This module will motivate the need for numerical algorithms to approximate the solution of problems that can’t be solved with pen and paper. You’ll develop your skills in performing detailed analysis of the performance of numerical methods and will continue to develop your skills in the implementation of numerical algorithms using R.

You'll be introduced to the basic concepts of random phenomena evolving in time, from two complementary points of view: probabilistic modelling and data-driven analysis. Presentation of underlying ideas of simple stochastic processes, time series models, and the associated probability theory and statistical techniques will be covered. In addition to applications of the methods to financial and economic systems, including modelling, data analysis, and forecasting.

This module provides you with experience of the skills required to undertake project work, and to communicate the findings in written and oral form using a variety of sources, such as books, journals and the internet. You will undertake an individual research project, researching a mathematical or statistical topic and writing a short report on it.

In this module you'll get an introduction to ideas in mathematics and statistics that can be used to model real systems, with an emphasis on the valuation of financial derivatives. This module places equal emphasis on deterministic analysis (calculus, differential equations) and stochastic analysis (Brownian motion, birth and death processes). In both cases, in addition to theoretical analysis, appropriate computational algorithms are introduced.

The first half of the module introduces general modelling and simulation tools, and the second half focuses on the specific application of valuing financial derivatives, including the celebrated Black-Scholes theory.

This module will present the main results in Functional Analysis. You will also be introduced to linear operators on Banach and Hilbert spaces and study applications to integral and differential equations.

You will be provided with a range of applied statistical techniques that can be used in professional life. This module provides you with the fundamental principles of statistical modelling through experimental design and multivariate analysis.

In this module you'll be introduced to the theory of Newtonian fluids and its application to flow problems and the dynamics of waves on water and in other contexts.

In this module you'll be presented with the basic theory and practice of finite element methods and polynomial and piecewise polynomial approximation theory.

In this module you'll be introduced to a range of modern statistical methods and practices used in industry, commerce and research, and you will develop skills in your application and presentation.

In this module, you'll learn the application of mathematical models to a variety of problems in biology, medicine, and ecology. The module will show:

• the application of ordinary differential equations to simple biological and medical problems
• the use of mathematical modelling in biochemical reactions
• the application of partial differential equations in describing spatial processes such as cancer growth and pattern formation in embryonic development
• the use of delay-differential equations in physiological processes.

The marine population modelling element will introduce the use of difference models to represent population processes through applications to fisheries, and the use of coupled ODE system to represent ecosystems. Practical work will include example class case studies that will explore a real-world application of an ecosystem model.

This module will demonstrate the central role network theory plays in mathematical modelling. It'll also show the intimate connection between linear algebra and graph theory and how to use this connection to develop a sound theoretical understanding of network theory. Finally, it'll apply this theory as a tool for revealing structure in networks.

This module will cover the application of classical statistical methods to data collected for health care research. There will be an emphasis on the use of real data and the interpretation of statistical analyses in the context of the research hypothesis under investigation. Topics covered will include:

• survival analysis
• experimental design and sampling
• categorical data analysis
• clinical measurement

The module will provide you with experience and expertise in skills required to undertake a sustained and significant individual project in a mathematical or statistical area, using a variety of sources, such as books, journals and the internet, and to communicate the findings in written and oral form.

This module provides an introduction to ideas in mathematics and statistics that can be used to model real systems, with an emphasis on the valuation of financial derivatives. It places equal emphasis on deterministic analysis (calculus, differential equations) and stochastic analysis (Brownian motion, birth and death processes). In both cases, in addition to theoretical analysis, appropriate computational algorithms are introduced.  The first half of the module introduces general modelling and simulation tools, and the second half focuses on the specific application of valuing financial derivatives, including the celebrated Black-Scholes theory.

This module will present the main results in Functional Analysis, give an introduction to linear operators on Banach and Hilbert spaces and study applications to integral and differential equations.

This module will introduce the theory of Newtonian fluids and its application to flow problems and the dynamics of waves on water and in other contexts.

This module aims to present you with the basic theory and practice of finite element methods and polynomial and piecewise polynomial approximation theory.

This module will introduce you to a range of modern statistical methods and practices used in industry, commerce and research, and to develop skills in their application and presentation.

This module will teach the application of mathematical models to a variety of problems in biology, medicine, and ecology. It will show:

• the application of ordinary differential equations to simple biological and medical problems
• the use of mathematical modelling in biochemical reactions
• the application of partial differential equations in describing spatial processes such as cancer growth and pattern formation in embryonic development
• the use of delay-differential equations in physiological processes

The marine population modelling element will introduce the use of difference models to represent population processes through applications to fisheries, and the use of coupled ODE system to represent ecosystems. Practical work will include example class case studies that'll explore a real-world application of an ecosystem model.

This module will demonstrate the central role network theory plays in mathematical modelling. It'll show the intimate connection between linear algebra and graph theory and how to use this connection to develop a sound theoretical understanding of network theory. Finally, application of the theory as a tool for revealing structure in networks.

This module will introduce you to general continuum theory, with applications to Newtonian and non-Newtonian fluids and elastic materials.

This module will provide a mathematical understanding of modern approaches to optimization and the calculus of variations.

This module will develop approaches to understanding complex or random systems in or out of equilibrium. These are based on ideas from statistical mechanics that incorporate familiar concepts and methods from neighbouring subjects such as classical mechanics and probability and statistics. This will explain, through various examples and techniques, how macroscopic properties of systems arise from the ensemble action of many microscopic ingredients. Specifically, you will see how deterministic “laws” may arise from the basic randomness of a system having many variables or degrees of freedom. Fundamental examples include Brownian motion and the ideal gas.

This module will introduce mathematical models which arise in epidemiology and population dynamics. You will develop techniques for analysing the qualitative behaviour of the associated dynamical systems.

This module will expose you to a number of diverse topics in modern applied analysis, explaining how key theoretical ideas underpin methods that play an important role in investigations into problems arising in the applied sciences.

This module will expose you to a number of diverse topics in modern applied statistics, explaining how key theoretical ideas underpin methods that play an important role in investigations into problems arising in the applied sciences.

This module will expose you to a number of diverse topics in a range of applied mathematics.

This module will expose you to a number of diverse topics in modern numerical analysis, explaining how key theoretical and practical ideas underpin computational methods in science, engineering and technology.

This module will make you aware of modern mathematical and statistical methods used to address real life research problems in the biological and ecological problems. In particular to bring together theoretical principles, computational practices and statistical data to construct models that describe the growth of populations and the spread of infectious diseases. You will learn that these models can form the basis for decision making. The module will be based on the research interests and expertise of the Department (please note not all of the syllabus topics will be taught in a particular year).