This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here, students will select a small number of properties which these and other examples have in common, and use them to define a group.

They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure', a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same' will be discovered.

Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials; several less familiar examples will also be explored.

Complex Analysis has its origins in differential calculus and the study of polynomial equations. In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.

The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every non-constant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Students will gain a solid understanding of computation and computer programming within the context of maths and statistics. This module expands on five key areas:

• Programming and R
• Numerical solutions of equations
• Numerical differentiation and integration
• Monte Carlo methods
• Numerical solutions to ODEs

Under these headings, students will study a range of complex mathematical concepts, such as: data structures, fixed-point iteration, higher dimensions, first and second derivatives, non-parametric bootstraps, and modified Euler methods.

Throughout the module, students will gain an understanding of general programming and algorithms. They will develop a good level of IT skills and familiarity with computer tools that support mathematical computation.

Over the course of this module, students will have the opportunity to put their knowledge and skills into practice. Workshops, based in dedicated computing labs, allow them to gain relatable, practical experience of computational mathematics.

Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structure-preserving maps and their relationship to matrices.

They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces.

Probability provides the theoretical basis for statistics and is of interest in its own right.

Basic concepts from the first year probability module will be revisited and extended to these to encompass continuous random variables, with students investigating several important continuous probability distributions. Commonly used distributions are introduced and key properties proved, and examples from a variety of applications will be used to illustrate theoretical ideas.

Students will then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

Project Skills is a module designed to support and develop a range of key technical and professional skills that will be valuable for all career paths. Covering five major components, this module will guide students through and explore:

• Mathematical programmes
• Scientific writing
• Communication and presentation skills
• Individual projects
• Group projects

Students will gain an excellent grasp of LaTeX, learning to prepare mathematical documents; display mathematical symbols and formulae; create environments; and present tables and figures.

Scientific writing, communication and presentations skills will also be developed. Students will work on short and group projects to investigate mathematical or statistical topics, and present these in written reports and verbal presentations.

A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at A-level.

Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.

Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in A-level, such as estimations of discrete sums of series.

Further possible topics include Stirling's Formula, infinite products and Fourier series.

Statistics is the science of understanding patterns of population behaviour from data. In the module, this topic will be approached by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.

The focus will be on likelihood-based parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.

Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.

Students on the Study Abroad scheme spend their third year studying at one of our partner universities. They are matched to a suitable institution so they can make the most out of this exciting opportunity. During the year abroad students will study all the compulsory topics that must be covered in the third year of the degree, but they will also have the benefit of choosing topics that are not offered at Lancaster. A personal tutor will help them develop their study programme and will keep in touch whilst they are studying abroad.

At the end of Year 3 students will fill in a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This depends on the student's degree scheme and choice.

During the first term students will meet their supervisor on a weekly basis and will be guided into an in-depth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in the first week after the Easter recess. The grade is based 70% on your final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student demonstrates during the entire two terms of the module.

Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

This module is an introduction to elliptic curves, and hence students will have the opportunity to learn the basics of algebraic geometry. It also presents applications and results of the theory of elliptic curves and provides a useful link between concepts from algebra and geometry.

Students will look at how curves can be described by algebraic equations, and will develop an understanding of abstract groups, learning how to use them to deal with geometrical objects (curves). They will also investigate the way that algebra and geometry are linked via polynomial equations, performing algebraic computations with elliptic curves.

Galois Theory is, in essence, the systematic study of properties of roots of polynomials. Starting with such a polynomial f over a field k (e.g. the rational numbers), one associates a ‘smallest possible’ field L containing k and the roots of f; and a finite group G which describes certain ‘allowed’ permutations of the roots of f. The Fundamental Theorem of Galois Theory says that under the right conditions, the fields which lie between k and L are in 1-to-1 correspondence with the subgroups of G.

In this module students will see two applications of the Fundamental Theorem. The first is the proof that in general a polynomial of degree 5 or higher cannot be solved via a formula in the way that quadratic polynomials can; the second is the fact that an angle cannot be trisected using only a ruler and compasses. These two applications are among the most celebrated results in the history of mathematics.

Students will have the opportunity to learn about Hilbert space, consolidating their understanding of linear algebra and enabling them to study applications of Hilbert space such as quantum mechanics and stochastic processes.

The module will teach students how to use inner products in analytical calculations, to use the concept of an operator on an infinite dimensional Hilbert space, to recognise situations in which Hilbert space methods are applicable and to understand concepts of linear algebra and analysis that apply in infinite dimensional vector spaces.

At the end of Year 3 students will complete a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics (MATH491), statistics (MATH492), or on an industrial project (MATH493), which is in cooperation with an external industrial partner. This depends on the degree scheme and choice.

In the first term there will be weekly supervisor meetings and students will be guided into their in-depth study of a specific topic. During the second term students will write a dissertation on what has been learnt and give an oral presentation. The dissertation will be submitted in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on oral presentation, and 20% on the initiative and effort that is demonstrated during the entire two terms of the module.

Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.

Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.

The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

The theory of Lie groups and Lie algebras will be introduced during this module. The relationship between the two will be explored, and students will develop an understanding of the way that this forms an important and enduring part of modern mathematics and a great number of fields including theoretical physics. They will learn to appreciate the subtle and pervasive interplay between algebra and geometry, and to appreciate the unified nature of mathematics.

The abstract nature of the module will give them a taste of modern research in pure mathematics. At the end of the module, students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras.

At the end of Year Three students will fill in a form stating their mathematical or statistical interests,and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This will depend on the student’s degree scheme and personal choice.

In the first term they will meet their supervisor weekly and will be guided into their in-depth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in on the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student has demonstrated during the entire two terms of the module.

Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.

This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.

Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and Black-Scholes models.

Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:

• Conditional expectation
• Filtrations
• Martingales
• Stopping times
• Brownian motion
• Black-Scholes formula

Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated risk-neutral probability; performing calculations with the Black-Scholes formula; and proving various steps in the derivation of the Black-Scholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.

In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the real-world; improve their written and oral communication skills; and develop their critical thinking.

In this module,students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. Looking at some basic results about the set of real numbers, students will explore properties of countable sets, open sets and algebraic numbers.T

They will also have the opportunity to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Operator theory is a modern mathematical topic in analysis which provides powerful general methods for the analysis of linear problems, and possibly even infinite dimensional problems.

Early successes were in the solution of differential and integral equations. Now operator theory is also an extensive subject in its own right in the general area of functional analysis.

First students will review Hilbert spaces, before spending some time studying infinite-dimensional operators, notably the unilateral shift and multiplication operators, as well as basic concepts. They will then consider the criteria for invertibility of self adjoint operators, leading to the spectral theory of such operators.

The aim of this module is to develop an analytical and axiomatic approach to the theory of probabilities.

Students will consider the notion of a probability space, illustrated by simple examples featuring both discrete and continuous sample spaces. They will then use random variables and the expectation to develop a probability calculus, which is applied to achieve laws of large numbers for sums of independent random variables. Finally the characteristic function will be used to study the distributions of sums of independent variables, applying the results to random walks and to statistical physics.

In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of R-module and group representations, and the main results pertaining to group representations, as well as learning to handle basic applications in the study of finite groups. They will also develop their skills in performing computations with representations and morphisms in a selection of finite groups.

The first part of the module is an introduction to the ordinary representation theory of finite groups. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between both is discussed and special examples and constructions are studied.

The second part of the module concerns the ordinary character theory of finite groups, intrinsic to representation theory. The main objectives are to prove the orthogonality relations of characters and construct the character table of a finite group.

Stochastic calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discrete-time stochastic processes, defining key concepts such as martingales and stopping times. This then leads on to the exploration of continuous-time processes, in particular, Brownian motion.

Students will learn to derive basic properties of Brownian motion and explore integration with respect to it. They will also examine the derivation of Ito's formula and how to apply this to Brownian motion.

Over the course of the module, students will also learn to justify and critique the use of stochastic models for real-life applications, and to use the stochastic calculus framework to formulate and solve problems involving uncertainty – a skill that underpins financial mathematics.

This module shows how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced.

In Stochastic Processes students will learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. They will also learn to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times, as well as how to determine whether a Markov process has an asymptotic distribution and how to calculate it. Finally, they will develop an understanding of how stochastic processes are used as models.

Fractals, roughly speaking, are strange and exotic sets in the plane (and in higher dimensions) which are often generated as limits of quite simple repeated procedures. The 'middle thirds Cantor set' in [0,1] is one such set. Another, the Sierpinski sieve, arises by repeated removal of diminishing internal triangles from a solid equilateral triangle.

This analysis module will explore a variety of fractals, partly for fun for their own sake but also to illustrate fundamental ideas of metric spaces, compactness, disconnectedness and fractal dimension. The discussion will be kept at a straightforward level and you’ll consider topological ideas of open and closed sets in the setting of R^2.