MMath Mathematics

MTH1000: Foundations

University level mathematics differs from that taught in schools not only in the difficulty of the topics and higher abstraction, but also in the style of teaching. This module aims to ease the transition to university level mathematics by bridging the gap between mathematics taught prior to university level, and the material covered in the first year of our mathematics degree, including the programming languages which will be taught in depth in other modules. The module eases you into a university teaching and learning environment and helps revise material from A-level. You will revisit essential skills in algebra, coordinate geometry, vectors, series and sequences, as well as some topics which are covered in Further Mathematics A-level such as complex numbers, matrix algebra, differential equations, and Maclaurin series. In this module, you will go over the theory and see many solved examples, as well as practice many examples to master these essential topics. Attending the lectures of this module is highly recommended to those students who do not have an A-level in Further Mathematics or equivalent, but those who do can also utilise these sessions to review the material and gain more practise experience. This module will also provide the skills needed to communicate mathematics which is a vital skill in all modules to be taken throughout a mathematics course.

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MTH1001: Mathematical Structures

A key aspect of mathematics is its ability to unify and generalise disparate situations exhibiting similar properties by developing the concepts and language to describe the common features abstractly and reason about them rigorously. In this module, you will be introduced to the language of logic, sets, and functions which underpins of all modern pure mathematics, and will learn how to use it to construct clear and logically correct mathematical proofs. The content goes beyond mathematics taught at A-level: you will learn and use methods to prove rigorous general results about the convergence of sequences and series, justifying the techniques developed in MTH1002 and laying the foundations for a deeper study of Analysis in MTH2008. You will also learn the definitions and properties of abstract algebraic structures such as groups and vector spaces. These ideas are developed further in MTH2010 and MTH2011. The material in this module is fundamental to many other modules in the mathematics degree programmes. It underpins the topics you will see in more advanced modules in fundemental mathematics and enables a deeper understanding and rigorous justification of the mathematical tools you will meet in more applied mathematics modules and which are widely used in physics, economics, and many other disciplines.

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MTH1002: Mathematical Methods

During your mathematics degree, you will be solving problems and proving theories in several branches of mathematics such as in pure mathematics, in applications to science and engineering, and in statistics. Inevitably you need to be able to calculate. That is what gives the mathematics its great power. This module covers developed bodies of useful techniques as a toolkit of common knowledge. It brings emphasis on the techniques rather than the applications of the techniques. Such techniques will enable you to deepen your familiarity with, and generalise, methods that you have seen at school level mathematics. This module will study topics that include the geometry of conic sections, properties of functions such as continuity and differentiability, differential and integral calculus, limits and convergence of sequences and series including Power Series and Taylor Series. The module also develops the fundamentals of vector and matrix theory, multivariate calculus, and the classification of various types of differential equations as well as analytical methods for solving them. The material in this module provide intuition for, and examples of, many of the mathematical structures that will be discussed in the module MTH1001 Mathematical Structures, and supply a firm understanding of methods required in future modules in the mathematics degree. In particular, it develops methods that underpin the modules MTH2003 Differential Equations and MTH2004 Vector Calculus and Applications.

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MTH1003: Mathematical Modelling

This module will introduce you to the theory and tools for analysing real physical systems, such as pendulums, planetary motion, and predator-prey models. You will also develop programming and coding skills using a language such as Python, and learn how mathematical theory and computer-based modelling can complement each other to help us understand and predict the world around us.

This module will also introduce you to the process of mathematical research and help you to understand the nature of the mathematical research community that you will be joining at the University of Exeter. You will work individually or as part of a team to carry out three short projects that will develop a range of individual and group research and communication skills. The ideas and skills in the module are developed further in MTH2005 Modelling: Theory and Practise.

The module aims to introduce you to Newtonian dynamics and its applications; to show you the use of calculus and vectors in the modelling of physical systems; to introduce you to applied mathematics as a tool for investigating natural phenomena. As examples, you will explore the consequences of physical laws, as well as the behaviour of physical and natural systems from projectiles to predator-prey systems and planetary motion.

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MTH1004: Probability, Statistics and Data

Our ability to collect and analyse data is increasingly driving our world. Statistics is concerned with both the practice of analysing data to learn about the world, and the theory that underpins the methods and models used for data collection and analysis. This theory is itself based on probability, the mathematics of chance and uncertainty. In this module, you will learn about the mathematics of combinatorics and probability, and the key ideas of statistical modelling and inference, in which probability is used to quantify uncertainty. You will also gain experience of employing these ideas to analyse data using statistical software such as the R programming environment. The module develops key ideas and techniques that form the foundation of modules such as MTH2006 Statistical Modelling and Inference.

The aim of this module is to introduce you to basic topics in probability, statistics and data analysis. This module provides the foundation for the second-year stream in Statistical Modelling and Inference, and subsequent modules in statistics in years 3 and 4.

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MTH2003: Differential Equations

Differential equations are at the heart of nearly all modern applications of mathematics to natural and man-made phenomena. Mathematically, all rates of change and acceleration can be described by derivative functions. These include the growth of populations, the spread of diseases, movement of physical objects in response to forces acting on them, or even the fluctuations of the stock market. This course will enable you to demonstrate an understanding of, and competence in, a range of analytical tools for posing and solving differential equations, and their application to situations in science and technology.

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MTH2004: Vector Calculus and Applications

This module introduces  vector calculus and its applications in particular fluid dynamics and electromagnetism. The module consists of two parts, which are closely linked. In the first part of the module, you will learn about the mathematical theory and techniques of vector calculus. You will develop your competence in using vector calculus in both differential and integral forms. The second part of the module gives an introduction to fluid dynamics and electromagnetism as two applications of vector calculus. It lays down some basic principles using a number of simplifying assumptions.

This introductory vector calculus course aims to increase your understanding of fluid dynamics and electromagnetism. It examines how one can use vector formalism and calculus together to describe and solve many problems in two and three dimensions. For example, the rules that govern the flow of fluids can be described using vector calculus, with resulting laws of motion described by partial differential equations rather than ordinary differential equations.

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MTH2008: Real Analysis

Description – summary of the module content

Infinite processes appear naturally in many contexts, from science and engineering to economics. From solving the equation that finds the wave function of a quantum system in physics, processing sensor data in engineering, to calculating prices for options in economics, at the foundation of all of these are infinite processes and the pure mathematics developed to rigorously and correctly handle these processes. That field of pure mathematics is called analysis, and the central object of study in analysis is that if a limit, which further extends to the notions of convergence, continuity, differentiation and integrability.

In this module, you will be introduced to the pioneering work of Cauchy, Riemann and many other notable mathematicians. By building on material from MTH1001 and MTH1002, we will carefully and rigorously develop how to handle real-variable differentiation, Riemann integration, power series, and basic notions of point set topology.

The material in this module is a prerequisite for the study of Complex Analysis (MTH2009) Topology and Metric Spaces (MTH3040) and Fractal Geometry (MTHM004). It is also recommended for those studying Dynamical Systems and Chaos (MTHM018), and is the basis for applications in economics, engineering and physics.

Pre-requisite modules

MTH1001 and MTH1002 (or equivalent)

Aims – intentions of the module

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MTH2009: Complex Analysis

The central object of study in analysis is the notion of a limit and related concepts of convergence, continuity, differentiation, and integration.

The objective of this module is to provide you with a rigorous introduction to complex analysis. We will carefully develop an understanding of the analysis of functions of a complex variable, and prove the central theorems governing the differentiation and integration of such functions. You will learn how to handle power series, singularities and contour integration, and see how to apply these to solve a wide range of problems. Quite surprisingly, complex analysis turns out to be a great deal more rigid, and more algebraic, than real analysis, and has many practical applications.

The material in this module has close links with Vector Calculus MTH2004 (although these modules are logically independent), and provides the foundations for further study in a range of subjects, most notably in number theory and geometry.

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MTH2010: Groups, Rings and Fields

In this module, you will explore some of the key techniques of modern algebra, including groups, rings, and fields. These topics have their roots in the desire to solve certain equations that arise from arithmetic and geometry.

The most familiar example of a ring is the set of all integers Z=...,-3,-2,-1,0,1,2,3... equipped with the usual operations of addition and multiplication. The familiar properties of these operations serve as a model for the axioms for rings. We can consider whether certain equations have solutions in rings such as the integers. For example, Fermat's Last Theorem famously asserts that if n is a fixed integer that is at least 3, then the equation x^n + y^n = z^n has no solutions for which x, y and z are non-zero integers. Though this problem is easy to state, its solution is extremely difficult: it was first stated in 1637 but the first complete and correct proof was given in 1994. Ring theory is essential for the fourth year module MTHM028 Algebraic Number Theory, which in turn lays the foundations for solving problems such as Fermat's Last Theorem.

Fields are special types of ring in which every non-zero element has a multiplicative inverse. Examples include the rational numbers Q, the real numbers R and the complex numbers C.

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MTH2011: Linear Algebra

Abstract vector spaces are important objects in linear algebra, which has its origins in solving linear equations over a field such as the rational, real or complex numbers. The elements of a vector space can be somewhat abstract: for example, they can be functions. However, it is precisely this abstraction that makes the theory of vector spaces such a powerful tool. They arise in almost every area of (pure and applied) mathematics and statistics. For example, PDEs (partial differential equations) of some types are just ODEs (ordinary differential equations) in vector spaces of functions, and numerical and data analysis methods consider vector spaces of increasing dimension to approximate function spaces.

Prerequisite modules: MTH1001and MTH1002 (or equivalent).

This module aims to develop the theories and techniques of modern algebra, particularly in relation to vector spaces and inner product spaces.

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MTH2006: Statistical Modelling and Inference

Statistical modelling lies at the heart of modern data analysis, helping us to describe and predict the real world. Statistical inference is the way that we use data and other information to learn about and apply statistical models. In this module, you will learn the theory underpinning modern statistical methods such as fitting normal linear models, evaluating how well they fit the data and taking inferences from it. You will apply the theory using statistical software such as R to analyse and draw conclusions from a range of real-world data sets. Topics covered in the module range from estimators, confidence intervals, design of experiments and hypothesis testing to statistical modelling, regression, inference and comparison of models. Skills developed in the module are taken further in modules such as MTH3012 Advanced Statistical Modelling.

This module aims to develop understanding and competence in statistical modelling by introducing you to the Normal linear model from a modern perspective. It will provide you with the ability to formulate and apply these models in a range of practical settings, to carry out associated inference appreciating how this relates to the general likelihood inferential framework, and to perform appropriate model selection and model checking procedures. Use will be made of a suitable statistical computer language for practical work.

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MTH2014: Numerical Modelling

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MTH2015: Mathematics of Machine Learning and AI

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MTH3100: Industrial Placement

This module will provide you with extensive practical work experience in a business or commercial setting that is of direct relevance to your development as an experienced mathematician. You will apply the knowledge and skills from taught modules to scientific, business or industrial problems at a professional level. You will be encouraged to use imagination and creativity in problem solving and to develop communication skills, planning and time management and team-working skills.

Placements will involve a substantial role in the host organisation. Individual placements are subject to availability and approval by the module leader.

Placements are normally for one year, and must be at least 6 months. International placements are acceptable. It is not required that you are paid a salary for the placement.

This module aims to provide you with the experience of working in science, business or industry in order for you to apply the knowledge and skills acquired in an academic environment to a professional work setting.

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MTHM036: Research in Mathematical Sciences

This is a unique module that runs in several research themes and provides you with a taste for and experience in research in mathematical sciences. The themes are broadly aligned with Fourth Year options and would cover subjects from pure maths, applied maths and statistics, although the number and range of topics may change from year to year. Each theme consists of lectures, student-led discussion, reading, and practical sessions, getting deeper experience in research approaches and skills. Students will learn to write a short essay, make an oral presentation and write a longer scientific report.

Set in at least three distinct research themes, this module will introduce you to scientific thinking and abstraction and give you knowledge/experience in core research skills: reading (and understanding), writing and presenting for the mathematical sciences.

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EMP3001: Commercial and Industrial Experience

This module will provide you with an opportunity to undertake practical work experience in a business, commercial or public sector setting that is of direct relevance to your development as an experienced professional. You will apply the knowledge and skills from taught modules to authentic problem solving in the workplace, which will give you important insights into your potential job role once you graduate from university. You will be encouraged to use imagination and creativity in problem solving and to develop communication skills, planning and time management and team-working skills. Placements will involve taking responsibility for a substantial project, which may be a problem to be solved in the host organisation, in line with your degree programme. Placements are subject to availability, approval by the module convener and full compliance with important Health and Safety procedures and requirements. Placements are normally three months some time during May-September, finishing before autumn classes start. Placements must be a minimum of six weeks full time. It is understood that this will entail around 210 hours of supervised work in order to generate the depth of experience equivalent to the 125 hours of self‑directed study on a focused topic specified under the regulations, as workplace activity is not counted directly as academic study International placements are allowed. Placements can be paid or volunteer.

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NSC3009: Aerosols, Clouds and Climate

Climate change is arguably one of the most urgent issues over the next two decades as humanity struggles to meet the 1.5C above pre-industrial target set by the Paris COP21. Concentrations of both greenhouse gases (GHG) and aerosols (particulate matter suspended in the atmosphere) have increased considerably since pre-industrial time. Whilst anthropogenic emissions of GHG warm the planet, aerosol emissions exert a significant, yet poorly quantified cooling that acts to offset a fraction of global warming from GHG.

Reducing current uncertainties associated with estimates of climate change sensitivity to GHG emissions is hampered by our understanding of the strength of the cooling effect aerosol particles have on the climate via their interactions with clouds and sunlight. Despite decades of research the Intergovernmental Panel on Climate Change Assessment Report continue to highlight our low understanding of aerosol-cloud-interactions (ACI) as the key uncertainty hampering our understanding of climate change.

This module is designed to explore the atmospheric physical processes determining the role of aerosols and their interaction with clouds on the climate to provide insight on the importance in reducing current uncertainties associated with aerosol - cloud - interactions (ACI) for adoption of more robust adaptation and mitigation strategies.

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MTH3025: Semester of Mathematical Studies Abroad

This module provides a full semester of studies in the Mathematics sciences and related subjects at an agreed partner university in a foreign country. It provides you with the excellent opportunity to gain knowledge of mathematical sciences material that is not covered at Exeter, or that is studied from a new perspective. In addition it allows you to experience how other cultures learn, perceive and use the mathematical sciences. Study at another agreed partner university where the language of instruction is not English may be approved subject to the requirement that you can prove fluency in that language to A level equivalence. This module is available to students on a four-year integrated masters programme in term one of stage three. To register on this module, you must achieve 55% at stage two and 60% average in term one of stage two.

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MTHM005: Mathematical Sciences Project

This independent research project gives you the chance to showcase the skills you have developed throughout your degree programme. It consists of a piece of work in an area of the mathematical sciences that you will choose yourself from a provided list. Using your independent learning skills, you will undertake the project individually, supervised by a member of staff. There will be a range of projects for you to choose from: some involve working on a practical problem, some involve developing or using computer programmes and packages, and others involve reviewing a theoretical area or tackling a theoretical problem.

The aim is to extend the knowledge you have acquired so far on your degree programme and to strengthen your skills in research and report writing. You will be able to exercise more autonomy on this research project than in more formal modules.

You will receive guidance on the responsible use of AI-assisted tools to support aspects of your research, report writing and presentation preparation, in line with University policy.

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MTHM044: MMath Project in Statistics

In this module you will become part of the research group in Statistics and Data Science and will write an independent project with a substantial research element. You will undertake the project individually, under the supervision of a member of faculty in statistical science and as a part of that member of staff’s wider research group, potentially including post doctoral researchers and PhD students tackling problems in the same or similar areas to yours. There will be a range of potential projects to choose from, aligning with the research interests of the participating members of the statistical science faculty. Projects will typically require mastering some theory beyond that given in undergraduate statistics courses and modelling/data analysis using statistical software, with the weighting of these elements varying according to the research problem. Each student will leave a lasting mark on the department. They will produce a poster, which will be displayed in the department for 1 year, and presented at a formative poster session.

Note: projects offered will build on material in level 3 Statistics modules and as a result each project will have one or more further prerequisites and/or co-requisites. These will be specified in the list of projects offered to students and available on the module ELE page.

Prerequisites: MTH2006, Project dependent 3rd year modules

Corequisites: Project dependent 3rd/4th year modules

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