# Mathematics MMath, BSc

## Year of entry 2024

2025 course information- UCAS code
- G101
- Start date
- September 2024
- Delivery type
- On campus
- Duration
- 4 years full time
- Work placement
- Optional
- Study abroad
- Optional
- Typical A-level offer
- AAA/A*AB (specific subject requirements)
- Typical Access to Leeds offer
- ABB

Full entry requirements - Contact
- maths.admiss@leeds.ac.uk

## Course overview

Mathematics is key to the sciences and to a cross-section of business disciplines. The ongoing drive for economic efficiency, the increasing importance of technology and big data, and new emerging areas such as climate science all mean that mathematics continues to have a significant impact on the world. Demand for mathematics skills comes from all sectors – from business and technology to science, research and development and IT – and the career options available are varied and rewarding.

Studying a mathematics degree at Leeds will provide you with a range of core mathematical skills whilst enhancing your abilities in logical thinking, problem solving and decision making – all of which are highly valued by employers. You can also opt to explore topics as diverse as fluid mechanics, cosmology, topology, and numerical analysis specialising in a particular area of interest or delving into several different areas. Choosing this integrated Masters degree (MMath, BSc) is particularly suitable if you wish to work closer to the frontiers of research, or to use mathematics at a higher level in your career.

Here at Leeds, we understand the importance mathematics has in everyday life, which is why we have one of the largest mathematics research departments in the UK which shapes the curriculum. This will equip you with the relevant knowledge, skills and experience you need to begin your career in this highly valued specialism.

### Why study at Leeds:

- Our School’s globally-renowned research feeds into the course, shaping your learning with the latest thinking in areas such as pure and applied mathematics, statistics and financial mathematics.
- Learn from expert academics and researchers who specialise in a variety of mathematical areas.
- Small tutorial groups support the teaching, providing you with regular feedback and advice from the academic staff throughout your degree.
- Access excellent facilities and computing equipment, complemented by social areas, communal problem-solving spaces and quiet study rooms.
- Broaden your experience and enhance your career prospects with our industrial placement opportunities or study abroad programmes.
- Make the most of your time at Leeds by joining our student society MathSoc where you can meet more of your peers, enjoy social events and join the MathSoc football or netball team.

### Benefits of an integrated Masters

Learn more about what an integrated Masters is and how it can benefit your studies and boost your career.

**Accreditation**

Accreditation is the assurance that a university course meets the quality standards established by the profession for which it prepares its students.

The School of Mathematics at Leeds has a successful history of delivering courses accredited by the Royal Statistical Society (RSS). This means our mathematics courses have consistently met the quality standards set by the RSS.

As we are reviewing our curriculum, we are currently seeking reaccreditation from the RSS.

## Course details

Our core mathematics degree offers opportunities to study a broad range of topics within the discipline, spanning pure, applied and statistical mathematics. Our academic staff have extensive research interests, which is why we're able to offer a wide choice of industry-relevant modules. You’ll graduate as a multi-skilled mathematician, perhaps with particular expertise in an area of interest or with the training necessary to work in a particular industry.

Discovery modules are available in years 2 and 3 of your degree, as long as you’re taking enough credits of your own subject for that year.

Each academic year, you'll take a total of 120 credits.

The course structure shown below represents typical modules/components studied and may change from time to time. Read more in our terms and conditions.

**Year 1 **

The first year of your course will introduce you to core mathematical topics, including calculus, linear algebra, differential equations, probability and statistics. You’ll develop a solid understanding of these core areas, which will provide you with the necessary background knowledge you require to explore more advanced topics later in your programme. The wide range of subjects you’ll explore at this stage allows you to define what areas of mathematics really interest you.

**Modules**

**Core Mathematics – 40 credits **

This module introduces you to the fundamental topics in mathematics. You’ll learn the foundational concepts of function, number and proof, equipping you with the language and skills to tackle your mathematical studies. The module also consolidates basic calculus, extending it to more advanced techniques, such as functions of several variables. These techniques lead to methods for solving simple ordinary differential equations. Linear algebra provides a basis for wide areas of mathematics and this module provides the essential foundation.

**Real Analysis – 20 credits **

Calculus is arguably the most significant and useful mathematical idea ever invented, with applications throughout the natural sciences and beyond. This module develops the theory of differential and integral calculus of real functions in a precise and mathematically rigorous way.

**Probability and Statistics – 20 credits **

'Probability is basically common sense reduced to calculation; it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct.' So said Laplace. In the modern scientific and technological world, it is even more important to understand probabilistic and statistical arguments. This module will introduce you to key ideas in both areas, with probability forming the theoretical basis for statistical tests and inference.

**Introduction to Group Theory – 10 credits **

Group theory is a fundamental branch of mathematics, central also in theoretical physics. The concept of a group may be regarded as an abstract way to describe symmetry and structure. In this module, we will introduce group theory, with motivation from, and application to, specific examples of familiar mathematical structures such as permutations of lists and symmetries of shapes.

**Dynamics and Motion – 10 credits **

In its broadest sense, dynamics refers to the mathematical modelling of things which change with time. The main focus of this module is that of Newtonian mechanics, where forces cause accelerations which govern the motion of objects (their dynamics), but the module will also explore other examples and applications. You’ll build on the methods of calculus (especially solution of ordinary differential equations) from the “Core Mathematics” module. You’ll also be introduced to a simple numerical method which allows equations for dynamics to be solved approximately on computers.

**Computational Mathematics and Modelling – 20 credits **

You'll be introduced to computational techniques, algorithms and numerical solutions, as well as the mathematics of discrete systems. You'll learn basic programming using the language Python and apply computational techniques to the solution of mathematical problems.

**Year 2 **

In your second year, you’ll deepen your knowledge of topics such as advanced calculus, abstract algebra, differential equations, and computational mathematics. You will also have the freedom to explore the topics that interest you most by choosing from a diverse range of optional modules. Additional topics may include discrete mathematics, finance, geometry, logic, modelling, statistics, or topics of a similar mathematical depth.

**Compulsory modules **

**Investigations in Mathematics – 20 credits **

You’ll be introduced to ideas and methods of mathematical research. Examples and applications will be drawn from across the spectrum of pure mathematics, applied mathematics and statistics. You’ll investigate a mathematical theory or concept and produce a report.

**Further Linear Algebra and Discrete Mathematics – 20 credits **

Explore the more abstract ideas of vector spaces and linear transformations, together with introducing the area of discrete mathematics.

**Vector Calculus and Partial Differential Equations – 20 credits **

Vector calculus is the extension of ordinary one-dimensional differential and integral calculus to higher dimensions, and provides the mathematical framework for the study of a wide variety of physical systems, such as fluid mechanics and electromagnetism.

These systems give rise to partial differential equations (PDEs), which can be solved and analysed. Students will learn to use, among others, techniques introduced in earlier modules as well as being introduced to Fourier methods for PDEs.

**Optional modules **

You’ll study optional modules within one of the following pathways:

**Pure and Applied Mathematics****Pure Mathematics and Statistics****Applied Mathematics and Statistics**

Please note: The modules listed below are indicative of typical options and some of these options may not be available, depending on other modules you have selected already.

**Pure and Applied Mathematics **

**Calculus, Curves and Complex Analysis – 20 credits **

The ideas of differential geometry and complex analysis are powerful products of nineteenth- and twentieth-century pure mathematics: interesting and beautiful in their own right, and with application to theoretical physics. Differential geometry is concerned with describing, understanding and quantifying properties of curved objects, while complex analysis extends and, in some ways, simplifies the differential calculus by considering functions of a complex variable. This module offers an introduction to each of these subjects.

**Mathematical Modelling – 20 credits **

Learn analytical and computational techniques for the solution of ordinary and partial differential equations, which describe particle motion in fields, fluids, waves, diffusion and many other phenomena.

**Optional modules: **

**Introduction to Logic – 10 credits **

This module is an introduction to mathematical logic introducing formal languages that can be used to express mathematical ideas and arguments. It throws light on mathematics itself, because it can be applied to problems in philosophy, linguistics, computer science and other areas.

**Optimisation – 10 credits **

Optimisation, “the quest for the best”, plays a major role in financial and economic theory, such as maximising a company's profits or minimising its production costs. This module develops the theory and practice of maximising or minimising a function of many variables, and thus lays a solid foundation for progression onto more advanced topics, such as dynamic optimisation, which are central to the understanding of realistic economic and financial scenarios.

**Rings and Polynomials – 10 credits **

Rings are one of the fundamental concepts of mathematics, and they play a key role in many areas, including algebraic geometry, number theory, Galois theory and representation theory. The aim of this module is to give an introduction to rings. The emphasis will be on interesting examples of rings and their properties.

**Calculus of Variations – 10 credits **

The calculus of variations concerns problems in which one wishes to find the extrema of some quantity over a system that has functional degrees of freedom. Many important problems arise in this way across pure and applied mathematics. In this module, you’ll meet the system of differential equations arising from such variational problems: the Euler-Lagrange equations. These equations and the techniques for their solution, will be studied in detail.

**Pure Mathematics and Statistics **

**Calculus, Curves and Complex Analysis – 20 credits **

The ideas of differential geometry and complex analysis are powerful products of nineteenth- and twentieth-century pure mathematics: interesting and beautiful in their own right, and with application to theoretical physics. Differential geometry is concerned with describing, understanding and quantifying properties of curved objects, while complex analysis extends and, in some ways, simplifies the differential calculus by considering functions of a complex variable. This module offers an introduction to each of these subjects.

**Statistical Methods – 20 credits **

Statistical models are important in many applications. They contain two main elements: a set of parameters with information of scientific interest and an "error distribution" representing random variation. This module lays the foundations for the analysis of such models. We’ll use practical examples from a variety of statistical applications to illustrate the ideas.

**Optional modules: **

**Introduction to Logic – 10 credits **

This module is an introduction to mathematical logic introducing formal languages that can be used to express mathematical ideas and arguments. It throws light on mathematics itself, because it can be applied to problems in philosophy, linguistics, computer science and other areas.

**Stochastic Processes – 10 credits **

A stochastic process refers to any quantity which changes randomly in time. The capacity of a reservoir, an individual’s level of no claims discount and the size of a population are all examples from the real world. The linking model for all these examples is the Markov process. With appropriate modifications, the Markov process can be extended to model stochastic processes which change over continuous time, not just at regularly spaced time points. You’ll explore the key features of stochastic processes and develop your understanding in areas like state, space and time, the Poisson process and the Markov property.

**Rings and Polynomials – 10 credits **

Rings are one of the fundamental concepts of mathematics, and they play a key role in many areas, including algebraic geometry, number theory, Galois theory and representation theory. The aim of this module is to give an introduction to rings. The emphasis will be on interesting examples of rings and their properties.

**Time Series – 10 credits **

In time series, measurements are made at a succession of times, and it is the dependence between measurements taken at different times which is important. This module will concentrate on techniques for model identification, parameter estimation, diagnostic checking and forecasting within the autoregressive moving average family of models and their extensions.

**Applied Mathematics and Statistics **

**Statistical Methods – 20 credits **

Statistical models are important in many applications. They contain two main elements: a set of parameters with information of scientific interest and an "error distribution" representing random variation. This module lays the foundations for the analysis of such models. We’ll use practical examples from a variety of statistical applications to illustrate the ideas.

**Mathematical Modelling – 20 credits **

Learn analytical and computational techniques for the solution of ordinary and partial differential equations, which describe particle motion in fields, fluids, waves, diffusion and many other phenomena.

**Optional modules: **

**Stochastic Processes – 10 credits **

A stochastic process refers to any quantity which changes randomly in time. The capacity of a reservoir, an individual’s level of no claims discount and the size of a population are all examples from the real world. The linking model for all these examples is the Markov process. With appropriate modifications, the Markov process can be extended to model stochastic processes which change over continuous time, not just at regularly spaced time points. You’ll explore the key features of stochastic processes and develop your understanding in areas like state, space and time, the Poisson process and the Markov property.

**Optimisation – 10 credits **

Optimisation, “the quest for the best”, plays a major role in financial and economic theory, such as maximising a company's profits or minimising its production costs. This module develops the theory and practice of maximising or minimising a function of many variables, and thus lays a solid foundation for progression onto more advanced topics, such as dynamic optimisation, which are central to the understanding of realistic economic and financial scenarios.

**Time Series – 10 credits **

In time series, measurements are made at a succession of times, and it is the dependence between measurements taken at different times which is important. This module will concentrate on techniques for model identification, parameter estimation, diagnostic checking and forecasting within the autoregressive moving average family of models and their extensions.

**Calculus of Variations – 10 credits **

The calculus of variations concerns problems in which one wishes to find the extrema of some quantity over a system that has functional degrees of freedom. Many important problems arise in this way across pure and applied mathematics. In this module, you’ll meet the system of differential equations arising from such variational problems: the Euler-Lagrange equations. These equations and the techniques for their solution, will be studied in detail.

**Year 3 **

In your third year, you’ll focus on higher-level optional modules. You’ll have complete control over which modules you study, provided you meet each module’s prerequisites. You could specialise in pure mathematics, applied mathematics or statistics, or choose modules across the spectrum of topics.

You’ll also have a final-year project where you’ll be grouped into similar topic areas. You’ll develop vital skills in carrying out independent research, producing your own final project report and giving a group presentation on that topic area. There is a long list of topics from a wide range of areas that your project can relate to and you’ll be able to choose a topic which interests you.

**Compulsory modules **

**Project in Mathematics – 40 credits **

This project is a chance for you to build invaluable research skills and develop and implement a personal training plan by conducting your own independent research project in a topic in mathematics. You’ll meet in groups to discuss the project topic, with each group member researching a specific aspect of the topic and producing an individual project report. You’ll then come together as a group to present your results, with each person contributing their own findings.

**Optional modules **

Please note: The modules listed below are indicative of typical options and some of these options may not be available, depending on other modules you have selected already.

**Pure and Applied Mathematics **

**Optional modules: **

**Groups and Symmetry – 20 credits **

Group theory is the mathematical theory of symmetry. Groups arise naturally in pure and applied mathematics, for example in the study of permutations of sets, rotations and reflections of geometric objects, symmetries of physical systems and the description of molecules, crystals and materials. Groups have beautiful applications to counting problems, answering questions like: "How many ways are there to colour the faces of a cube with m colours, up to rotation of the cube?"

**Methods of Applied Mathematics – 20 credits **

This module develops techniques to solve ordinary and partial differential equations arising in mathematical physics. For the important case of second-order PDEs, we distinguish between elliptic equations (e.g., Laplace's equation), parabolic equations (e.g., heat equation) and hyperbolic equations (e.g., wave equation), and physically interpret the solutions. When there is not an exact solution in closed form, approximate solutions (so-called perturbation expansions) can be constructed if there is a small or large parameter.

**Metric Spaces and Measure Theory – 20 credits **

If you would like to undertake a rigorous study of a physical, geometrical or statistical law, it is likely you’ll need to use both of these concepts. Metric spaces have a notion of distance between points and measure theory generalises familiar ideas of volume, underpinning integration. We will study these exciting topics, proving results fundamental to pure and applied mathematics; Picard-Lindelöff's theorem from ODEs; the inverse and implicit function theorems; Lebesgue’s dominated convergence theorem.

**Computational Applied Mathematics – 20 credits **

The equations that model real-world problems can only rarely be solved exactly. The basic idea employed in this module is that of discretising the original continuous problem to obtain a discrete problem, or system of equations, that may be solved with the aid of a computer. This course introduces and applies the techniques of finite differences, numerical linear algebra and stochastic simulation.

**Numbers and Codes – 20 credits **

Number theory explores the natural numbers. Central themes include primes, arithmetic modulo n, and Diophantine equations as in Fermat's Last Theorem. It is a wide-ranging current field with many applications, e.g. in cryptography.

Error-correcting codes tackle the problem of reliably transmitting digital data through a noisy channel. Applications include transmitting satellite pictures, designing registration numbers and storing data. The theory uses methods from algebra and combinatorics.

This module introduces both subjects. It emphasises common features, such as algebraic underpinnings, and applications to information theory, both in cryptography (involving secrecy) and in error-correcting codes (involving errors in transmission).

**Proof and Computation – 20 credits **

The main goal of this module is to prove Gödel's First Incompleteness Theorem (1931) which shows that, if any reasonable formal theory has strong enough axioms, there are statements which it can neither prove nor refute. This module will also provide background to the impact of Gödel's Theorem on the modern world and the way it sets an agenda for further research.

**Entropy and Quantum Mechanics – 20 credits **

The material world is composed of countless microscopic particles. When three or more particles interact, their dynamics is chaotic, and impossible to predict in detail. Further, at the microscopic atomic-scale particles behave like waves, with dynamics that is known only statistically. So, why is it that the materials around us behave in predictable and regular ways? One reason is that random behaviour on the microscopic scale gives rise to collective behaviour that can be predicted with practical certainty, guided by the principle that the total disorder (or entropy) of the universe never decreases. A second reason is that the mathematics of quantum mechanics provides incredibly accurate predictions at the atomic scale. This module studies calculations involving both entropy and quantum mechanics, as applied to the matter that makes up our world.

**Fluid Dynamics – 20 credits **

Fluid dynamics is the science that describes the motion of materials that flow. It constitutes a significant mathematical challenge with important implications in an enormous range of fields in science and engineering, including aerodynamics, astrophysics, climate modelling, and physiology. This module sets out the fundamental concepts of fluid dynamics, for both inviscid and viscous flows. It includes a formal mathematical description of fluid flow and the derivation of the governing equations, using techniques from vector calculus. Solutions of the governing equations are derived for a range of simple flows, giving you a feel for how fluids behave and experience in modelling everyday phenomena.

**Mathematics in Social Context C – 20 credits **

Mathematics is possessed of what Bertrand Russell called a cold and austere beauty; and yet it has roots in deeply human concerns. In this module, you’ll gain insight into ways in which mathematicians can bridge the ‘two cultures’ and see how mathematics shapes our world and our cultures.

**Graph Theory and Combinatorics – 20 credits **

Graph theory is one of the primary subjects in discrete mathematics. It arises wherever networks as seen in computers or transportation are found, and it has applications to fields diverse as chemistry, computing, linguistics, navigation and more. More generally, combinatorics concerns finding patterns in discrete mathematical structures, often with the goal of counting the occurrences of such patterns. This module provides a foundation in graph theory and combinatorics.

**Differential Geometry – 20 credits **

Differential geometry is the application of calculus to describe, analyse and discover facts about geometric objects. It provides the language in which almost all modern physics is understood. This module develops the geometry of curves and surfaces embedded in Euclidean space. A recurring fundamental theme is curvature (in its many guises) and its interplay with topology.

**Mathematical Biology – 20 credits **

Mathematics is increasingly important in biological and medical research. This module aims to introduce you to some areas of mathematical biology and medicine, using tools from applied mathematics.

**Nonlinear Dynamical Systems and Chaos – 20 credits **

Many applications, ranging from biology to physics and engineering, are described by nonlinear dynamical systems, in which a change in output is not proportional to a change in input. Nonlinear dynamical systems can exhibit sudden changes in behaviour as parameters are varied and even unpredictable, chaotic dynamics. This module will provide you with the mathematical tools to analyse nonlinear dynamical systems, including identifying bifurcations and chaotic dynamics.

**Pure Mathematics and Statistics **

**Optional modules: **

**Groups and Symmetry – 20 credits **

Group theory is the mathematical theory of symmetry. Groups arise naturally in pure and applied mathematics, for example in the study of permutations of sets, rotations and reflections of geometric objects, symmetries of physical systems and the description of molecules, crystals and materials. Groups have beautiful applications to counting problems, answering questions like: "How many ways are there to colour the faces of a cube with m colours, up to rotation of the cube?"

**Statistical Modelling – 20 credits **

The standard linear statistical model is powerful but has limitations. In this module, we study several extensions to the linear model which overcome some of these limitations. Generalised linear models allow for different error distributions; additive models allow for nonlinear relationships between predictors and the response variable; and survival models are needed to study data where the response variable is the time taken for an event to occur.

**Metric Spaces and Measure Theory – 20 credits **

If you would like to undertake a rigorous study of a physical, geometrical or statistical law, it is likely you’ll need to use both of these concepts. Metric spaces have a notion of distance between points and measure theory generalises familiar ideas of volume, underpinning integration. We will study these exciting topics, proving results fundamental to pure and applied mathematics; Picard-Lindelöff's theorem from ODEs; the inverse and implicit function theorems; Lebesgue’s dominated convergence theorem.

**Actuarial Mathematics 1 – 20 credits **

The module introduces the theory of interest rates and the time value of money in the context of financial transactions such as loans, mortgages, bonds and insurance. The module also introduces the basic theory of life insurance where policy payments are subject to mortality probabilities.

**Numbers and Codes – 20 credits **

Number theory explores the natural numbers. Central themes include primes, arithmetic modulo n, and Diophantine equations as in Fermat's Last Theorem. It is a wide-ranging current field with many applications, e.g. in cryptography.

Error-correcting codes tackle the problem of reliably transmitting digital data through a noisy channel. Applications include transmitting satellite pictures, designing registration numbers and storing data. The theory uses methods from algebra and combinatorics.

This module introduces both subjects. It emphasises common features, such as algebraic underpinnings, and applications to information theory, both in cryptography (involving secrecy) and in error-correcting codes (involving errors in transmission).

**Proof and Computation – 20 credits **

The main goal of this module is to prove Gödel's First Incompleteness Theorem (1931) which shows that, if any reasonable formal theory has strong enough axioms, there are statements which it can neither prove nor refute. This module will also provide background to the impact of Gödel's Theorem on the modern world and the way it sets an agenda for further research.

**Stochastic Calculus and Derivative Pricing – 20 credits **

Stochastic calculus is one of the main mathematical tools to model physical, biological and financial phenomena (among other things). This module provides a rigorous introduction to this topic. You’ll develop a solid mathematical background in stochastic calculus that will allow you to understand key results from modern mathematical finance. This knowledge will be used to derive expressions for prices of derivatives in financial markets under uncertainty.

**Mathematics in Social Context C – 20 credits **

Mathematics is possessed of what Bertrand Russell called a cold and austere beauty; and yet it has roots in deeply human concerns. In this module, you’ll gain insight into ways in which mathematicians can bridge the ‘two cultures’ and see how mathematics shapes our world and our cultures.

**Graph Theory and Combinatorics – 20 credits **

Graph theory is one of the primary subjects in discrete mathematics. It arises wherever networks as seen in computers or transportation are found, and it has applications to fields diverse as chemistry, computing, linguistics, navigation and more. More generally, combinatorics concerns finding patterns in discrete mathematical structures, often with the goal of counting the occurrences of such patterns. This module provides a foundation in graph theory and combinatorics.

**Differential Geometry – 20 credits **

Differential geometry is the application of calculus to describe, analyse and discover facts about geometric objects. It provides the language in which almost all modern physics is understood. This module develops the geometry of curves and surfaces embedded in Euclidean space. A recurring fundamental theme is curvature (in its many guises) and its interplay with topology.

**Multivariate Analysis and Classification – 20 credits **

Multivariate datasets are common: it is typical that experimental units are measured for more than one variable at a time. This module extends univariate statistical techniques for continuous data to a multivariate setting and introduces methods designed specifically for multivariate data analysis (cluster analysis, principal component analysis, multidimensional scaling and factor analysis). A particular problem of classification arises when the multivariate observations need to be used to divide the data into groups or “classes”.

**Actuarial Mathematics 2 – 20 credits **

The module expands on the theory of life insurance introduced in Actuarial Mathematics 1. Instead of considering a single life and single decrement, we will consider policies with multiple lives and multiple decrements. In addition, the module includes profit testing for different types of insurance policies.

**Applied Mathematics and Statistics **

**Optional modules: **

**Methods of Applied Mathematics – 20 credits **

This module develops techniques to solve ordinary and partial differential equations arising in mathematical physics. For the important case of second-order PDEs, we distinguish between elliptic equations (e.g., Laplace's equation), parabolic equations (e.g., heat equation) and hyperbolic equations (e.g., wave equation), and physically interpret the solutions. When there is not an exact solution in closed form, approximate solutions (so-called perturbation expansions) can be constructed if there is a small or large parameter.

**Statistical Modelling – 20 credits **

The standard linear statistical model is powerful but has limitations. In this module, we study several extensions to the linear model which overcome some of these limitations. Generalised linear models allow for different error distributions; additive models allow for nonlinear relationships between predictors and the response variable; and survival models are needed to study data where the response variable is the time taken for an event to occur.

**Computational Applied Mathematics – 20 credits **

The equations that model real-world problems can only rarely be solved exactly. The basic idea employed in this module is that of discretising the original continuous problem to obtain a discrete problem, or system of equations, that may be solved with the aid of a computer. This course introduces and applies the techniques of finite differences, numerical linear algebra and stochastic simulation.

**Actuarial Mathematics 1 – 20 credits **

The module introduces the theory of interest rates and the time value of money in the context of financial transactions such as loans, mortgages, bonds and insurance. The module also introduces the basic theory of life insurance where policy payments are subject to mortality probabilities.

**Stochastic Calculus and Derivative Pricing – 20 credits **

Stochastic calculus is one of the main mathematical tools to model physical, biological and financial phenomena (among other things). This module provides a rigorous introduction to this topic. You’ll develop a solid mathematical background in stochastic calculus that will allow you to understand key results from modern mathematical finance. This knowledge will be used to derive expressions for prices of derivatives in financial markets under uncertainty.

**Entropy and Quantum Mechanics – 20 credits **

The material world is composed of countless microscopic particles. When three or more particles interact, their dynamics is chaotic, and impossible to predict in detail. Further, at the microscopic atomic-scale particles behave like waves, with dynamics that is known only statistically. So, why is it that the materials around us behave in predictable and regular ways? One reason is that random behaviour on the microscopic scale gives rise to collective behaviour that can be predicted with practical certainty, guided by the principle that the total disorder (or entropy) of the universe never decreases. A second reason is that the mathematics of quantum mechanics provides incredibly accurate predictions at the atomic scale. This module studies calculations involving both entropy and quantum mechanics, as applied to the matter that makes up our world.

**Fluid Dynamics – 20 credits **

Fluid dynamics is the science that describes the motion of materials that flow. It constitutes a significant mathematical challenge with important implications in an enormous range of fields in science and engineering, including aerodynamics, astrophysics, climate modelling, and physiology. This module sets out the fundamental concepts of fluid dynamics, for both inviscid and viscous flows. It includes a formal mathematical description of fluid flow and the derivation of the governing equations, using techniques from vector calculus. Solutions of the governing equations are derived for a range of simple flows, giving you a feel for how fluids behave and experience in modelling everyday phenomena.

**Mathematics in Social Context C – 20 credits **

Mathematics is possessed of what Bertrand Russell called a cold and austere beauty; and yet it has roots in deeply human concerns. In this module, you’ll gain insight into ways in which mathematicians can bridge the ‘two cultures’ and see how mathematics shapes our world and our cultures.

**Multivariate Analysis and Classification – 20 credits **

Multivariate datasets are common: it is typical that experimental units are measured for more than one variable at a time. This module extends univariate statistical techniques for continuous data to a multivariate setting and introduces methods designed specifically for multivariate data analysis (cluster analysis, principal component analysis, multidimensional scaling and factor analysis). A particular problem of classification arises when the multivariate observations need to be used to divide the data into groups or “classes”.

**Mathematical Biology – 20 credits **

Mathematics is increasingly important in biological and medical research. This module aims to introduce you to some areas of mathematical biology and medicine, using tools from applied mathematics.

**Nonlinear Dynamical Systems and Chaos – 20 credits **

Many applications, ranging from biology to physics and engineering, are described by nonlinear dynamical systems, in which a change in output is not proportional to a change in input. Nonlinear dynamical systems can exhibit sudden changes in behaviour as parameters are varied and even unpredictable, chaotic dynamics. This module will provide you with the mathematical tools to analyse nonlinear dynamical systems, including identifying bifurcations and chaotic dynamics.

**Actuarial Mathematics 2 – 20 credits **

The module expands on the theory of life insurance introduced in Actuarial Mathematics 1. Instead of considering a single life and single decrement, we will consider policies with multiple lives and multiple decrements. In addition, the module includes profit testing for different types of insurance policies.

**Year 4 **

**Compulsory modules **

**Assignment in Mathematics – 45 credits **

You'll engage in independent research on an individual basis, on a title negotiated with an academic supervisor. This will include training in the skills necessary to plan, execute and report on a project in advanced mathematics. Although this is an independent project, our academic staff will be there to supervise and support you throughout.

**Optional modules **

Please note: The modules listed below are indicative of typical options and some of these options may not be available, depending on other modules you have selected already.

**Pure and Applied Mathematics **

**Optional modules: **

**Topology – 15 credits **

A topological space is a set with the minimal added structure that makes it possible to define continuity of functions. In this module, we will define topological spaces and explain what it means for them to be connected, path-connected and compact. In the second half of this module, we study algebraic topology and show some applications to Euclidean space: e.g any continuous map from a disk to itself must have a fixed point.

**Models and Sets – 15 credits **

Set Theory is generally accepted as a foundation for mathematics, in an informal sense, but is also a formal axiomatic system. Model Theory is the study of formal axiomatic systems and depends on Set Theory for many of its basic definitions and results. Model Theory and Set Theory constitute two of the basic strands of mathematical logic. In this module, we explain the basic notions of these interrelated subjects.

**Functional Analysis and its Applications – 15 credits **

Solving problems in infinite dimensional space is fundamental to our understanding of the world; finding the optimal depth for a wine cellar, or the natural frequencies of an object, fall within the same mathematical playground of Functional Analysis. Many such problems admit solutions in the form of an infinite sum or integral expression, but how, why, and are these expressions genuine? We will develop the mathematical theory to rigorously ask and answer these fundamental questions.

**Evolutionary Modelling – 15 credits **

Darwin’s natural selection theory is a cornerstone of modern science. Recently, mathematical and computational modelling has led to significant advances in our understanding of evolutionary puzzles, such as what determines biodiversity or the origin of cooperative behaviour. On this module, you will be exposed to fundamental ideas of evolutionary modelling, and to the mathematical tools needed to pursue their study. These will be illustrated by numerous examples motivated by exciting developments in mathematical biology.

**Advanced Mathematical Methods – 15 credits **

Many real-world problems can be modelled by ordinary or partial differential equations or formulated as a complicated integral. In this module, advanced techniques are developed to solve such problems and interpret their solutions, motivated by examples from mathematical physics, continuum mechanics, and mathematical biology. These techniques include so-called asymptotic methods, which yield approximate solutions if the problem contains a small or large parameter.

**Astrophysical and Geophysical Fluids – 15 credits **

This module concerns mathematical modelling of various phenomena observed in astrophysical and geophysical flows, meaning those in planetary and stellar atmospheres and interiors. The focus is on understanding key dynamical processes in such flows, including those due to rotation and density stratification and, in many astrophysical flows, the electrical conductivity of the fluid (which can thus support a magnetic field). These effects lead to various interesting waves and instabilities, with physical and observational significance.

**Riemannian Geometry – 15 credits **

Riemannian geometry is the study of length, angle, volume and curvature. It is a far-reaching generalisation of the theory of curves and surfaces to higher dimensions. Famously, it formed the basis of Einstein's theory of general relativity and it remains a primary language of modern theoretical physics. In this module, you’ll learn the basic concepts of Riemannian geometry and study some fascinating theorems relating the geometry of a manifold to its topological properties.

**Algebras and Representations – 15 credits **

An algebra is a vector space with a compatible binary operation, or multiplication. A natural example is the set of all square matrices of a fixed size with complex entries, under matrix multiplication. Semisimple algebras form an important class of algebras and one of the highlights of the course is the structure theorem classifying the semisimple algebras in terms of matrix algebras. The module will also study representations of general algebras via matrices.

**Environmental and Industrial Flows – 15 credits **

Many flows found in nature such as avalanches and glaciers or in industrial applications such as 3D printing and coatings involve complex fluids, whose properties can be very different from those of simple Newtonian fluids like air and water. This course gives an introduction into the often-surprising behaviour of these fluids and how they can be modelled mathematically using differential equations.

**Classical and Quantum Hamiltonian Systems – 15 credits **

The Hamiltonian formulation of dynamics is the most mathematically beautiful form of mechanics and a stepping stone to quantum mechanics. Hamiltonian systems are conservative dynamical systems with a very interesting algebraic and geometric structure: the Poisson bracket. Hamilton's equations are invariant under a very wide class of transformations and this leads to a number of powerful solution techniques.

**Pure Mathematics and Statistics **

**Optional modules: **

**Topology – 15 credits **

A topological space is a set with the minimal added structure that makes it possible to define continuity of functions. In this module, we will define topological spaces and explain what it means for them to be connected, path-connected and compact. In the second half of this module, we study algebraic topology and show some applications to Euclidean space: e.g any continuous map from a disk to itself must have a fixed point.

**Models and Sets – 15 credits **

Set Theory is generally accepted as a foundation for mathematics, in an informal sense, but is also a formal axiomatic system. Model Theory is the study of formal axiomatic systems and depends on Set Theory for many of its basic definitions and results. Model Theory and Set Theory constitute two of the basic strands of mathematical logic. In this module, we explain the basic notions of these interrelated subjects.

**Functional Analysis and its Applications – 15 credits **

Solving problems in infinite dimensional space is fundamental to our understanding of the world; finding the optimal depth for a wine cellar, or the natural frequencies of an object, fall within the same mathematical playground of Functional Analysis. Many such problems admit solutions in the form of an infinite sum or integral expression, but how, why, and are these expressions genuine? We will develop the mathematical theory to rigorously ask and answer these fundamental questions.

**Statistical Theory – 15 credits**

This module gives a general unified theory of the problems of estimation and hypotheses testing. It covers Bayesian inference, making comparisons with classical inference.

**Statistical Computing – 15 credits**

Statistical computing is the branch of mathematics that concerns the use of computational techniques for situations that either directly involve randomness, or where randomness is used as part of a mathematical model. This module gives an overview of the foundations and basic methods in statistical computing.

**Riemannian Geometry – 15 credits**

Riemannian geometry is the study of length, angle, volume and curvature. It is a far-reaching generalisation of the theory of curves and surfaces to higher dimensions. Famously, it formed the basis of Einstein's theory of general relativity and it remains a primary language of modern theoretical physics. In this module, you’ll learn the basic concepts of Riemannian geometry and study some fascinating theorems relating the geometry of a manifold to its topological properties.

**Algebras and Representations – 15 credits**

An algebra is a vector space with a compatible binary operation, or multiplication. A natural example is the set of all square matrices of a fixed size with complex entries, under matrix multiplication. Semisimple algebras form an important class of algebras and one of the highlights of the course is the structure theorem classifying the semisimple algebras in terms of matrix algebras. The module will also study representations of general algebras via matrices.

**Advanced Statistical Modelling – 15 credits**

This module builds on statistical models introduced in earlier studies, developing advanced techniques for analysis of datasets. These include methods for estimating the probability density function from a data set and approaches to constraining the number of variables that contribute to a linear model.

**Applied Mathematics and Statistics**

**Optional modules:**

**Statistical Theory – 15 credits**

This module gives a general unified theory of the problems of estimation and hypotheses testing. It covers Bayesian inference, making comparisons with classical inference.

**Statistical Computing – 15 credits**

Statistical computing is the branch of mathematics that concerns the use of computational techniques for situations that either directly involve randomness, or where randomness is used as part of a mathematical model. This module gives an overview of the foundations and basic methods in statistical computing.

**Evolutionary Modelling – 15 credits**

Darwin’s natural selection theory is a cornerstone of modern science. Recently, mathematical and computational modelling has led to significant advances in our understanding of evolutionary puzzles, such as what determines biodiversity or the origin of cooperative behaviour. On this module, you will be exposed to fundamental ideas of evolutionary modelling, and to the mathematical tools needed to pursue their study. These will be illustrated by numerous examples motivated by exciting developments in mathematical biology.

**Advanced Mathematical Methods m– 15 credits**

Many real-world problems can be modelled by ordinary or partial differential equations or formulated as a complicated integral. In this module, advanced techniques are developed to solve such problems and interpret their solutions, motivated by examples from mathematical physics, continuum mechanics, and mathematical biology. These techniques include so-called asymptotic methods, which yield approximate solutions if the problem contains a small or large parameter.

**Astrophysical and Geophysical Fluids – 15 credits**

This module concerns mathematical modelling of various phenomena observed in astrophysical and geophysical flows, meaning those in planetary and stellar atmospheres and interiors. The focus is on understanding key dynamical processes in such flows, including those due to rotation and density stratification and, in many astrophysical flows, the electrical conductivity of the fluid (which can thus support a magnetic field). These effects lead to various interesting waves and instabilities, with physical and observational significance.

**Advanced Statistical Modelling – 15 credits**

This module builds on statistical models introduced in earlier studies, developing advanced techniques for analysis of datasets. These include methods for estimating the probability density function from a data set and approaches to constraining the number of variables that contribute to a linear model.

**Environmental and Industrial Flows – 15 credits**

Many flows found in nature such as avalanches and glaciers or in industrial applications such as 3D printing and coatings involve complex fluids, whose properties can be very different from those of simple Newtonian fluids like air and water. This course gives an introduction into the often-surprising behaviour of these fluids and how they can be modelled mathematically using differential equations.

**Classical and Quantum Hamiltonian Systems – 15 credits**

The Hamiltonian formulation of dynamics is the most mathematically beautiful form of mechanics and a stepping stone to quantum mechanics. Hamiltonian systems are conservative dynamical systems with a very interesting algebraic and geometric structure: the Poisson bracket. Hamilton's equations are invariant under a very wide class of transformations and this leads to a number of powerful solution techniques.

**One-year optional work placement or study abroad **

During your course, you’ll be given the opportunity to advance your skill set and experience further. You can apply to either undertake a one-year work placement or study abroad for a year, choosing from a selection of universities we’re in partnership with worldwide.

**Learning outcomes **

- Demonstrate a good understanding of the basic body of knowledge in a range of mathematical areas;
- Demonstrate a good level of skill in calculation and manipulation within specific areas of mathematics;
- Apply concepts and principles in well-defined contexts, showing judgement in the selection and application of tools and techniques;
- Understand complex logical arguments, identifying the assumptions and conclusions made;
- Demonstrate a reasonable level of skill in comprehending problems, formulating them mathematically and obtaining solutions by appropriate methods;
- Present straightforward arguments and conclusions accurately and clearly;
- Demonstrate by the writing of a project appropriate transferable skills and the ability to work under guidance.
- One-year optional work placement or study abroad
- During your course, you’ll be given the opportunity to advance your skill set and experience further. You can apply to either undertake a one-year work placement or study abroad for a year, choosing from a selection of universities we’re in partnership with worldwide.

### Learning and teaching

You’ll be taught through lectures, tutorials, workshops and practical classes. You’ll enjoy extensive tutorial support and have freedom in your workload and options.

We offer a variety of welcoming spaces to study and socialise with your fellow students. There are social and group study areas, a library with a café and a seminar room, as well as a Research Visitors Centre and a Mathematics Active Learning Lab.

**Taster lectures**

Watch our taster lectures to get a flavour of what it’s like to study at Leeds:

- Playing with Infinity ∞ Two Famous Infinite Series
- What Does it Mean to be Round?
- Fractals – What, How, Why?

On this course you’ll be taught by our expert academics, from lecturers through to professors. You may also be taught by industry professionals with years of experience, as well as trained postgraduate researchers, connecting you to some of the brightest minds on campus.

### Assessment

You’re assessed through a range of methods, including formal exams and in-course assessment.

## Entry requirements

A-level: AAA/A*AB including a minimum of grade A in Mathematics.

AAA/A*AB including a minimum of grade A in Mathematics, AAB/A*BB including a minimum of grade A in Mathematics plus Further Mathematics, or AAB/A*BB including a minimum of grade A in Mathematics, plus A in AS Further Mathematics.

Where an A-Level Science subject is taken, we require a pass in the practical science element, alongside the achievement of the A-Level at the stated grade.

Excludes A-Level General Studies or Critical Thinking.

GCSE: GCSE: English Language at grade C (4) or above, or an appropriate English language qualification. We will accept Level 2 Functional Skills English in lieu of GCSE English.

Other course specific tests:

Extended Project Qualification (EPQ), International Project Qualification (IPQ) and Welsh Baccalaureate Advanced Skills Challenge Certificate (ASCC): We recognise the value of these qualifications and the effort and enthusiasm that applicants put into them, and where an applicant offers the EPQ, IPQ or ASCC we may make an offer of AAB/A*BB including a minimum of grade A in Mathematics, plus A in EPQ/IPQ/Welsh Bacc ASCC.

### Alternative qualification

### Access to HE Diploma

Normally only accepted in combination with grade A in A Level Mathematics or equivalent.

### BTEC

### Cambridge Pre-U

### International Baccalaureate

17 at Higher Level including 6 in Higher Level Mathematics (Mathematics: Analytics and Approaches is preferred).

### Irish Leaving Certificate (higher Level)

H2 H2 H2 H2 H2 H2 including Mathematics.

### Scottish Highers / Advanced Highers

Suitable combinations of Scottish Higher and Advanced Highers are acceptable, though mathematics must be presented at Advanced Higher level.Typically AAAABB Including grade A in Advanced Higher Mathematics.

### Other Qualifications

We also welcome applications from students on the Northern Consortium UK International Foundation Year programme, the University of Leeds International Foundation Year, and other foundation years with a high mathematical content.

Read more about UK and Republic of Ireland accepted qualifications or contact the School’s Undergraduate Admissions Team.

### Alternative entry

We’re committed to identifying the best possible applicants, regardless of personal circumstances or background.

Access to Leeds is a contextual admissions scheme which accepts applications from individuals who might be from low income households, in the first generation of their immediate family to apply to higher education, or have had their studies disrupted.

Find out more about Access to Leeds and contextual admissions.

**Typical Access to Leeds offer:** ABB including A in Mathematics and pass Access to Leeds OR A in Mathematics, B in Further Mathematics and C in a 3rd subject and pass Access to Leeds.

**Foundation years**

If you do not have the formal qualifications for immediate entry to one of our degrees, you may be able to progress through a foundation year.

We offer a Studies in Science with Foundation Year BSc for students without science and mathematics qualifications.

You could also study our Interdisciplinary Science with Foundation Year BSc which is for applicants whose background is less represented at university.

On successful completion of your foundation year, you will be able to progress onto your chosen course.

### International Foundation Year

International students who do not meet the academic requirements for undergraduate study may be able to study the University of Leeds International Foundation Year. This gives you the opportunity to study on campus, be taught by University of Leeds academics and progress onto a wide range of Leeds undergraduate courses. Find out more about International Foundation Year programmes.

### English language requirements

IELTS 6.0 overall, with no less than 5.5 in any one component, or IELTS 6.5 overall, with no less than 6.0 in any one component, depending on other qualifications present. For other English qualifications, read English language equivalent qualifications.

**Improve your English**

If you're an international student and you don't meet the English language requirements for this programme, you may be able to study our undergraduate pre-sessional English course, to help improve your English language level.

## Fees

UK: £9,250 (per year)

International: £27,250 (per year)

**Tuition fees for UK undergraduate students starting in 2024/25**

Tuition fees for UK full-time undergraduate students are set by the UK Government and will be £9,250 for students starting in 2024/25.

The fee may increase in future years of your course in line with inflation only, as a consequence of future changes in Government legislation and as permitted by law.

**Tuition fees for UK undergraduate students starting in 2025/26**

Tuition fees for UK full-time undergraduate students starting in 2025/26 have not yet been confirmed by the UK government. When the fee is available we will update individual course pages.

**Tuition fees for international undergraduate students starting in 2024/25 and 2025/26**

Tuition fees for international students for 2024/25 are available on individual course pages. Fees for students starting in 2025/26 will be available from September 2024.

**Tuition fees for a study abroad or work placement year**

If you take a study abroad or work placement year, you’ll pay a reduced tuition fee during this period. For more information, see Study abroad and work placement tuition fees and loans.

Read more about paying fees and charges.

There may be additional costs related to your course or programme of study, or related to being a student at the University of Leeds. Read more on our living costs and budgeting page.

### Scholarships and financial support

If you have the talent and drive, we want you to be able to study with us, whatever your financial circumstances. There is help for students in the form of loans and non-repayable grants from the University and from the government. Find out more in our Undergraduate funding overview.

## Applying

Apply to this course through UCAS. Check the deadline for applications on the UCAS website.

We may consider applications submitted after the deadline. Availability of courses in UCAS Extra will be detailed on UCAS at the appropriate stage in the cycle.

**Admissions guidance **

Read our admissions guidance about applying and writing your personal statement.

**What happens after you’ve applied **

You can keep up to date with the progress of your application through UCAS.

UCAS will notify you when we make a decision on your application. If you receive an offer, you can inform us of your decision to accept or decline your place through UCAS.

**How long will it take to receive a decision**

We typically receive a high number of applications to our courses. For applications submitted by the January UCAS deadline, UCAS asks universities to make decisions by mid-May at the latest.

**Offer holder events **

If you receive an offer from us, you’ll be invited to an offer holder event. This event is more in-depth than an open day. It gives you the chance to learn more about your course and get your questions answered by academic staff and students. Plus, you can explore our campus, facilities and accommodation.

**International applicants **

International students apply through UCAS in the same way as UK students.

We recommend that international students apply as early as possible to ensure that they have time to apply for their visa.

Read about visas, immigration and other information here.

If you’re unsure about the application process, contact the admissions team for help.

### Admissions policy

University of Leeds Taught Admissions Policy 2024

### This course is taught by

### Contact us

School of Mathematics Undergraduate Admissions

Email: maths.admiss@leeds.ac.uk

Telephone:

## Career opportunities

Mathematical skills are highly valued in virtually all walks of life, which means that the employment opportunities for mathematics graduates are far-reaching and have the potential to take you all over the world.

Plus, University of Leeds students are among the top 5 most targeted by top employers according to The Graduate Market 2024, High Fliers Research.

Qualifying with a degree in Mathematics from Leeds will give you the core foundations you need to pursue an exciting career across a wide range of industries and sectors, including:

- Accountancy
- Insurance
- Banking and finance
- Asset management and investment
- Engineering
- Teaching
- Data analysis
- Law
- Consultancy

The numerical, analytical and problem-solving skills you will develop, as well as your specialist subject knowledge and your ability to think logically, are highly valued by employers. This course also allows you to develop the transferable skills that employers seek.

Here’s an insight into the job roles some of our most recent graduates have obtained:

- Category Management Analyst, Accenture
- Business Intelligence Engineer, Amazon
- Financial Analyst, American Express
- Consultant Statistician, AstraZeneca
- Audit Associate, Deloitte
- Senior Credit Risk Analyst, HSBC
- Senior Actuary, KPMG
- Retail Analyst, Emma Bridgewater
- Statistician, Nestle
- Senior Actuarial Associate, PwC
- Risk Analyst, SkyBet
- Statistical Analyst, Office of National Statistics

### Careers support

At Leeds, we help you to prepare for your future from day one. Our Leeds for Life initiative is designed to help you develop and demonstrate the skills and experience you need for when you graduate. We will help you to access opportunities across the University and record your key achievements so you are able to articulate them clearly and confidently.

You will be supported throughout your studies by our dedicated Employability team, who will provide you with specialist support and advice to help you find relevant work experience, internships and industrial placements, as well as graduate positions. You’ll benefit from timetabled employability sessions, support during internships and placements, and presentations and workshops delivered by employers.

You will also have full access to the University’s Careers Centre, which is one of the largest in the country.

## Study abroad and work placements

### Study abroad

Studying abroad is a unique opportunity to explore the world, whilst gaining invaluable skills and experience that could enhance your future employability and career prospects too.

From Europe to Asia, the USA to Australasia, we have many University partners worldwide you can apply to, spanning across some of the most popular destinations for students.

This programme offers you the option to spend time abroad as part of your four-year MMath course.

Once you’ve successfully completed your year abroad, you'll be awarded the ‘international’ variant in your degree title upon completion which demonstrates your added experience to future employers.

Find out more at the Study Abroad website.

### Work placements

A placement year is a great way to help you decide on a career path when you graduate. You’ll develop your skills and gain a real insight into working life in a particular company or sector. It will also help you to stand out in a competitive graduate jobs market and improve your chances of securing the career you want.

Benefits of a work placement year:

- 100+ organisations to choose from, both in the UK and overseas
- Build industry contacts within your chosen field
- Our close industry links mean you’ll be in direct contact with potential employers
- Advance your experience and skills by putting the course teachings into practice
- Gain invaluable insight into working as a professional in this industry
- Improve your employability

If you decide to undertake a placement year, this will extend your period of study by 12 months and, on successful completion, you will be awarded the ‘industrial’ variant in your degree title to demonstrate your added experience to future employers.

With the help and support of our dedicated Employability team, you can find the right placement to suit you and your future career goals.

Here are some examples of placements our students have recently completed:

- Data Scientist, Department for Work & Pensions
- Cyber Crime Researcher, Department for Work & Pensions
- Risk Analyst - Infrastructure/ Strategy, Lloyds Banking Group
- Operations Analyst, Tracsis Rail Consultancy
- Risk analyst, Lloyds Banking Group

Find out more about Industrial placements.

### Rankings and awards

## Student profile: Amelie Davies

I found the staff extremely friendly, the course details aligned with my goals, and I could see myself really succeeding.Find out more about Amelie Davies's time at Leeds