BSc Mathematics with Economics

BEE1036: Economics I

This module provides the introduction to economics for undergraduates in the Department of Economics. It puts the student at the centre of pedagogy using learning materials and experiences attuned both to the societal problems that students care about and to how students acquire facility and confidence in using and communicating economics. Digital technology and interactive teaching methods will introduce students to an empirical discipline. Students will learn to use evidence from history, experiments and other data sources to test competing explanations and policies. It introduces the characteristics of economies using historical and cross-country comparisons across the major dimensions of economic performance (growth, inequality, stability). By taking the main economic actors and showing how they make decisions, the module covers behaviour in goods, labour and credit markets, highlighting the role of the rules of the game (institutions), and showing the sources of market successes and market failures. Behaviour of households and firms is analysed in the economy, along with that of fiscal and monetary policy makers.

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BEE1037: Economics II

This module continues to provide the introduction to economics for undergraduates started in the module Economics I. It introduces the characteristics of economies using historical and cross-country comparisons across the major dimensions of economic performance (growth, inequality, stability).

By taking the main economic actors and showing how they make decisions, the course covers behaviour in goods, labour and credit markets, highlighting the role of the rules of the game (institutions), and showing the sources of market successes and market failures. Behaviour of households and firms is analysed in the economy as a whole, along with that of fiscal and monetary policy makers .

This module aims to provide students with a basic understanding of economics, and to apply this way of thinking to real world problems. It aims to help students understand the world around them, become more astute participants in the Economy and Society and help them understand Economic Policy so that they can better judge the decisions affecting the allocation of their society's resources.

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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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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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BEE2025: Microeconomics II

The module is designed to equip you with the key microeconomic principles necessary for the analysis of a range of basic economic problems and policies. The module builds on first-year microeconomics and aims to both deepen and widen your formal knowledge of economic theory and its application. It seeks, in particular, to increase your abilities to, independently, pose and solve economic questions, especially those relating to policy issues. It emphasizes the fundamental conceptual foundations in microeconomics and provides concrete examples of their applications.

Internationalisation

Microeconomics is relevant across countries as it is based on mathematical models,

Sustainability

All of the resources for this module are available on the ELE (Exeter learning Environment).

Employability

This module equips you with logical thinking, numeracy and writing skills, as well as an understanding and theoretical knowledge of economic issues. These help you think like economists, a quality highly valued by employers.

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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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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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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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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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MTH3026: Cryptography

Cryptography is the mathematical art and science of maintaining information security. In this module, you will learn practical algorithms for encrypting plain messages into secret messages. These algorithms ranges from the simple ciphers used in ancient Rome to the sophisticated modern ciphers that secure worldwide banking transactions and diplomatic communications.

You will study two broad classes of cryptosystems: symmetric and asymmetric. Modern symmetric ciphers such as AES (the Advanced Encryption Standard) have their mathematical basis in linear algebra and field theory, whereas asymmetric ciphers such as RSA and ElGamal are founded on number theory and group theory.

Knowing how to crack asymmetric cryptosystems requires developing methods for factorising large numbers and testing numbers for primality. These methods are at the heart of modern attempts to break secret codes.

The module concludes with an introduction to elliptic curve cryptography, a topic with connections to Algebraic Curves (MTHM029).

You will be given a brief introduction to the computer programming language Python in the particular context of solving cryptographic problems.

Prerequisite module: MTH3004.

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MTH3030: Mathematics of Climate Change

This module will provide a background in the mathematics underlying human-induced climate change. It will provide you with a good general understanding of the climate system, against which to assess the likely role of anthropogenic forcing factors. You will learn to apply a range of mathematical methods, including differential equations, calculus and the use of small parameters to approximate and simplify climate system problems. Topics of study will include observations of climate change, the greenhouse effect, regimes of atmospheric absorption, climate feedbacks, climate tipping points and geoengineering.

Climate change is a high-profile subject that is often covered in the media. However, debate about climate change is often presented in a polarized way, divided along political or ideological lines. In contrast, there is now an urgent need to develop a new generation of thinkers capable of objectively analyzing the evidence for climate change and its causes, and the options for dealing with it (including mitigation, adaptation and geoengineering). Mathematically-minded people are especially sort after by organizations such as the Met Office-Hadley Centre in Exeter. This module aims to develop the skills required to meet these needs, by providing a strong-background in the science surrounding the climate change issue to mathematically-minded undergraduates.

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MTH3038: Galois Theory

Drawing on key ideas in the theory of groups and fields, you will learn core elements of the theory of field extensions. You are already familiar with the idea that the real numbers can be extended to the complex numbers by introducing a new number as the square root of -1; Galois theory formalises such constructions and explores the intriguing relationship between groups and field extensions.

As an important application of Galois Theory, you will understand why there can be no algebraic solution to the general quintic polynomial with rational coefficients.

Prerequisite module: MTH2010 Groups, Rings, and Fields and MTH2011 Linear Algebra, or equivalent.

The aim of this module is to motivate and develop Galois Theory both as an abstract theory and through the study of important applications.

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MTH3039: Computational Nonlinear Dynamics

Nonlinear dynamical systems are used in almost all disciplines: from applied mathematics to physics of any kind, to biology, chemistry, sociology, ecology, economics, engineering, and computer science. This is also why this module is welcoming students from all disciplines, from mathematics to sociology. The nonlinearity of systems makes them widespread, but it comes at a price: almost nothing about nonlinear systems can be estimated analytically.

Computational nonlinear dynamics is the process of studying nonlinear dynamical systems by devising and running numerical algorithms. Throughout this module we will be discussing many interesting aspects of nonlinear dynamical systems, such as multistability, deterministic chaos, critical transitions, … (see Topics Covered). For each aspect, we will be devising algorithms that can identify it for arbitrary dynamical systems. In the coursework we will be creating computer programs that apply these algorithms to dynamical systems. Sometimes we may have data obtained directly from some real-world source instead of a dynamical system, but the process will be the same. As such, this module will not only teach you how nonlinear dynamical systems behave, and how to understand them, but also how to design computer algorithms that fulfil a certain goal. This is an invaluable experience for your future employability in a world increasingly reliant on programming.

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MTH3040: Topology and Metric Spaces

Description – Summary of the module content

Topology and metric spaces provide a set of powerful tools that are used in many branches of mathematics (from algebraic topology and numerical analysis to dynamical systems and ergodic theory over to topological data analysis). Fundamental to these topics is the idea of generalising the concept of “closeness” of two objects in a set to a very general setting. These techniques are key to the understanding of more advanced topics in mathematics such as measure theory, functional analysis, algebraic topology and algebraic geometry. This course aims to give an introduction to point set topology and metric spaces. In every section covered in this course we will start by studying the core definitions and then present some examples and discuss some basic properties. Some important theorems will be stated and proved. With this module you will have the opportunity to further refine your skills in problem-solving, axiomatic reasoning and the formulation of mathematical proofs.

Pre-requisite modules

MTH2008

Aims – Intentions of the module

The objective of this module is to provide you an introduction to point set topology and metric spaces. Our main objective will be to define the foundational concepts and to provide proofs of important results.

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MTH3042: Integral Equations

Similarly to differential equations, integral equations provide an effective way to model real life situations, particularly those that arise in physics and engineering. Using certain techniques, many initial and boundary value problems can be converted to integral equations where the unknown function lies in the integrand. This module will introduce you to the mathematics of integral equations, techniques of analysing such equations, and methods of solving them, analytically or numerically.

Following the introduction integral equations, you will be introduced to a large class of integral equations. Volterra integral equations and Fredholm integral equations will be explained and methods on how to solve these equations will be described for cases of having integral equations of the first kind and the second kind, as well as looking at homogeneous and nonhomogeneous equations. Examples, that model real life problems, will be given and solutions will be interpreted.

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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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