MATH4100: Matrices and Calculus

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

Course Description

This module aims to introduce fundamental subjects of linear algebra and calculus, building on material from A-level, and providing essential pre-requisites for the rest of the undergraduate programme in mathematics. The linear algebra part will introduce vectors and matrices, key methods such as Gaussian elimination, and key concepts such as determinants and linear transformations. The module will study invertibility of matrices and introduce characteristic equations, eigenvalues and eigenvectors. Applications include using matrices to solve systems of linear equations, linear transformations to describe symmetries of the plane and eigenvalues and eigenvectors to understand Google’s page ranking algorithm. In calculus, the aim is to study the behaviour and properties of sequences and functions, exploring ideas such as convergence, continuity, differentiation and integration. The emphasis is on practical calculations and encouraging students to think of functions in terms of graphs, such as understanding derivatives via the gradient of the tangent to a graph. The module develops intuitive ideas such as monotonicity, continuity, rate of change, maxima and minima, and the area under a curve in the context of graphs. Limits are introduced in the context of simple examples sequences which will appear as fundamental examples in subsequent courses in analysis.

Educational Aims

Upon successful completion of this module students will be able to:

1. work with matrices, in particular by means of elementary row and column operations, and how they can be used to solve systems of linear equations with or without parameters.
2. express linear transformations of the real Euclidean space using matrices, determine whether a matrix is singular or not and obtain its characteristic equation and eigenspaces.
3. understand the concepts of convergence and limits on the real line; compute limits using standard limit laws.
4. calculate derivatives using both the limit definition and differentiation rules; locate and classify stationary points.
5. distinguish between definite and indefinite integrals, and perform integral calculations using standard techniques, including integration by parts and integration by substitution.
6. interpret the results discussed in MLOs 3,4 and 5, in terms of graphs, and conversely.
7. learn the importance of precise terminology and use the standard language to describe problems in linear algebra and calculus.

Outline Syllabus

The module starts with some basic theory of polynomials and mathematical induction, which will be used throughout the module (and elsewhere). Linear Algebra begins with an introduction to vectors and matrices. Students will learn standard matrix operations, and how to perform row operations on matrices. Invertibility and determinants of matrices will be covered. These concepts will then be used to solve systems of linear equations. Matrices will then be related to linear transformations, which are certain geometric transformations of the Euclidean space. Eigenvalues and eigenvectors, which characterise these transformations, are introduced. The student will learn how to calculate eigenvalues, via the characteristic polynomial, and eigenspaces, special examples of subspaces of the Euclidean space. Students will also see applications of linear algebra, for example in population growth and Google’s page rank algorithm. In calculus, we begin with convergence, which is introduced in the context of real sequences and then real series. The module then explores functions of a single real variable and their graphs, starting with polynomials and extending to rational and exponential functions. Trigonometric and hyperbolic functions are introduced in parallel with analogous power series, so that students understand the role of functional identities. The notion of a limit is crucial, serving as the main tool in the study of key concepts of calculus, such as continuity, differentiation and integration. The derivative measures the rate of change of a function and the integral measures the area under the graph of a function. The study of these concepts leads to the Fundamental Theorems of calculus, and applications to differential equations. Taylor series are calculated for trigonometric and hyperbolic functions. Finally, we combine the theory of vectors with calculus to study maxima and minima of functions of two variables.

Assessment Proportions

This module introduces fundamental techniques of linear and calculus, emphasising practical calculations and the translation of intuitive ideas into precise terminology. It is designed to provide the computational tools required for the subsequent module MATH4105 and supports modules MATH4115 and MATH4125. The theoretical foundation introduced in this module will be further developed and formalized in modules MATH5210 and MATH5220. The learning material will be delivered through four 1-hour lectures per week. These lectures will underpin the development of mathematical structures from basic concepts to advanced theories. Detailed proofs and worked examples will be presented, providing sufficient time for students to reflect and develop their self-understanding strategies. Students will have weekly 1-hour workshops, led by academics or GTAs. These workshops will take place in smaller classes of 20 students per tutor, and students will work on worksheets with exercises of varying difficulty, either individually or in small groups. This setup provides a great opportunity for students to receive formative feedback on their understanding of the module and to enhance their oral and writing skills. Summative coursework consists of written assessment and Moodle quizzes. Each week, students will submit written coursework to their tutor and receive detailed feedback. The Moodle quizzes will test the students’ understanding of the key concepts via multiple choice questions. The mid-module test will cover material from linear algebra, providing students with an opportunity to experience university-level exams and receive feedback on their progress. At the end of the module, the entire module will be assessed through a final exam.

MATH4105: Probability and Statistics

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

Course Description

This module introduces the mathematical and computational toolsets for modelling the randomness of the world we find ourselves in. Probability is the language used to describe random fluctuations, and statistics provides techniques to make inferences about the world. Computing is an essential tool for solving the most pressing problems in scientific research, artificial intelligence, machine learning and data science. The module aims to develop the axiomatic theory of probability and discover the theory and uses of random variables. It will give the basis of statistical inference, and introduces how to select appropriate probability models to describe simple univariate discrete and continuous distributions. Additionally, throughout the entirety of the module, the basics of R or Python will be introduced, and their use within probability and statistics. This will equip the students with the skills to deploy statistical methods in practice.

Educational Aims

Upon successful completion of this module students will be able to:

1. Understand, derive, interpret and make use of axioms and theorems on events, independence and conditional probability
2. Derive, interpret and manipulate concepts of univariate random variables (probability mass function, probability density function, cumulative distribution function, expectation, variance) including specific random variable families (uniform, Bernoulli, binomial, geometric, Poisson, exponential, gamma, normal, beta) and their uses
3. Derive properties of transformations of univariate random variables via the cumulative distribution function method
4. Derive, understand and use Chebychev’s inequality, and understand it’s implication in a special case of the weak law of large numbers
5. Understand the key concepts underpinning the frequentist approach to statistics, including sampling distributions, confidence intervals, hypothesis testing and p values.
6. Derive simple single parameter likelihood functions, based on the distributions introduced in the module, or other clearly specified probability models
7. Implement basic statistical inference based on the likelihood principle, including maximum likelihood inference and asymptotic properties of estimators, such as Wald intervals
8. Write effective programs in R or Python to solve statistical problems, and compare and contrast good and bad programming practices.
9. Understand and implement the process that maps mathematical ideas to algorithms to computer code.

Outline Syllabus

This module will be split into three parts: Probability, Statistics and Scientific Programming. The students will be introduced to probability concepts first, then to statistics, with programming in R or Python interwoven throughout. The probability section will introduce the key mathematical tools for considering simple random quantities. We will build up from an axiomatisation, before introducing the key concepts of random variables, and learn how to work with these objects. We then move on to statistics, in which we learn how to use random variables and associated concepts to discover things about the world around us and quantify uncertainty. We will introduce the most important principles in statistical inference, and the fundamental object in mathematical statistics: the likelihood function. Whilst an understanding of classical probability and statistics requires little more than a pen and paper, in the modern workplace both classical methods and their contemporary counterparts are deployed using bespoke computer programmes and software. Over the course of this module we will also learn how to use computers to effectively implement and apply these methods; this will be starting from scratch and will not require any prior knowledge of programming. The module will allow you to choose between using Python or R according to your career aspirations.

Assessment Proportions

Formative assessment

• Problem-solving exercises where students are encouraged to collaborate, to be solved during the workshops, with peer-assessment and support from the GTAs.
• Programming exercises, to solve during the computer-lab sessions, followed by an automated programming quiz to check understanding of key concepts.

Summative assessment

• Weekly online quizzes following the workshops, confirming learning and exposing gaps in understanding. These will be worth a total of 10% of the module mark.
• Two programming courseworks, each worth 10% of the module mark.
• An end of module exam, covering the probability and statistics components, worth 70% of the module mark

MATH4110: Logic and Discrete Mathematics

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

Course Description

This module aims to introduce you to university mathematics where emphasis is placed far more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is more powerful than a calculation that deals only with a specific case. The language and structure of mathematical proofs will be studied, emphasising how logic can be used to express mathematical arguments in a concise and rigorous manner. Concepts from Number Theory will be used to illustrate these abstract ideas, including congruence of integers, and how equivalence relations can be used to construct the rationals from the integers. Discrete Mathematics is the study of discrete structures, including counting problems and mathematical graphs (or networks). The module introduces Set Theory, which is the language which underpins such discrete structures and mathematics in general. Counting problems considered will be both finite combinatorial problems as well as counting infinite sets: the rationals and integers are the same size, but are smaller than the set of real numbers. The language of Graph Theory will be introduced, including a study of Trees and colouring problems. Throughout the module there will be an emphasis on writing logically sound, concise and rigorous mathematical arguments.

Educational Aims

Upon successful completion of this module students will be able to…

1. Use the language of mathematics, including set theoretic notation, logical symbols and connectives, and truth tables, and apply these to write proofs and study mathematical functions and relations.
2. Understand the role of prime numbers in elementary number theory, be able to perform calculations using number theory concepts, and to write proofs about statements involving the integers.
3. Formulate and solve counting problems, both using standard formulae and by giving rigorous combinatorial arguments.
4. Use basic notions of graph theory to identify structural properties of graphs and to distinguish non-isomorphic graphs.

Outline Syllabus

Logic allows us to be precise about notions like True and False. We introduce truth tables and logical connectives like “and”, “or” and “implies”. The three main methods of mathematical proof are direct, contraposition and contradiction: it is surprising to see results which appear impossible to prove directly, yet have a simple proof by contradiction. A set is a collection of objects, and a function a way to associate elements of one set with those of another. Injectivity, surjectivity and bijectivity are fundamental properties of functions that you will encounter. Informally, the Fundamental Theorem of Arithmetic states that the prime numbers are the “building blocks” of the integers. The Euclidean Algorithm provides a fast way to find the common factors of two integers. Considerations of divisibility naturally lead to the notion of integer congruences. We solve linear congruences, and pairs thereof using Sun Zi’s Remainder Theorem. Such ideas extend from numbers to polynomials. “Equivalence relations” generalise equality and congruence, and enable us to rigorously construct number systems such as the integers and the rational numbers. Combinatorial counting problems address questions such as “how many ways can 4 people be seated around a circular table?”, or counting the number of choices from a finite set, with and without replacement. You will meet the famous Pigeonhole Principle. Counting infinite sets leads to surprising results, such as the sets of integers and rational numbers being the same size, and Cantor’s Diagonal argument. A graph is a collection of vertices and edges linking some vertices: a network. These model myriad real-world situations, and are the archetypical discrete mathematical objects. We introduce the language of graphs, including paths, connectedness, and vertex colourings. These ideas provide methods of telling if two graphs are essentially the same (isomorphic) or not. A tree is a connected graph without cycles, and Kruskal’s Algorithm provides a simple way to construct spanning trees.

Assessment Proportions

This module is designed to introduce students to university mathematics, focussing on number-theoretic and discrete mathematical problems, and to communicate their solutions in a rigorous and concise way. Material will be taught using lectures, supported by comprehensive written materials. Lectures provide an opportunity for the mathematical thinking process to be displayed in real-time. A mathematical proof, or a counting argument, might be short when written down, yet each line of the argument may contain multiple steps of reasoning. A key role of lectures is to explain these steps and to describe how the arguments are arrived at. Lectures will also contain numerous worked, carefully motivated, problems. Abstract mathematical ideas are hard to communicate and require the formulation of internal “mental models” of the relevant concepts. This is greatly aided by students working through appropriately selected exercises. Some will be given in lectures, while weekly workshops – in small groups with a dedicated workshop tutor – will allow students to work on more substantial exercises, either individually or in small groups, with help at hand. The purely formative assessment of workshop exercises is complemented by a blend of formative and summative work submitted weekly, alternating between Moodle quizzes and written solutions marked by the workshop tutor. This allows students to practise problem-solving and proof-writing on their own, and to obtain prompt feedback. The iterative process of learning to write concisely yet with rigour depends critically on this continual feedback cycle. The mid-module test is a more formal closed-book exam, allowing students to experience what university-level exams are like. By situating this test in the middle of the course, we make use of the consolidation week and ensure feedback is provided well ahead of the final exam, which will assess all the learning outcomes of the module.

MATH4115: Symmetry and Sequences

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

Course Description

This module aims to provide foundations in abstract algebra and mathematical analysis, which are the two core disciplines in pure mathematics. The module will build on the development of language and structure of mathematical proofs by applying them in the context of the study of symmetry, convergence and continuity. The module will provide foundational knowledge and reasoning skills for all Year 2, 3 and 4 modules in mathematics. Group theory is the study of symmetry. The module will introduce the concept of a group using examples from geometry, linear algebra and discrete mathematics. The module will then introduce the ideas of subgroup and isomorphism, giving further examples of equivalence relations and bijective functions from MATH4110, developing students’ example space. The module will then pivot to the second fundamental stream of pure mathematics: mathematical analysis. It will introduce the notions of convergence and continuity and provide grounding in epsilon-delta formalism, which is essential for making the concept of limit rigorous and forms the basis of calculus.

Educational Aims

Upon successful completion of this module students will be able to:

1. State the definition of a group and check when a set with a binary operation satisfies the definition.
2. Give examples of groups and decide when groups of small order are isomorphic.
3. Decide and prove when a subset of a group is a subgroup.
4. Understand the structure of the real number system and the notions of supremum and infimum for sets of real numbers.
5. Define the mathematical notion of sequences, subsequences, boundedness, limit points, and convergence.
6. Provide examples and counterexamples to mathematical definitions and statements regarding the above topics.
7. Understand mathematical notation and how to read and write proofs related to the above topics

Outline Syllabus

An indicative syllabus is as follows:

1. Examples of groups: symmetry groups of regular polygons (dihedral groups), permutation groups, matrix groups, abelian groups arising from modular arithmetic, leading to the formal definition of a group.
2. Group isomorphisms and examples of isomorphic groups.
3. Subgroups and Lagrange’s theorem.
4. Maximum and minimum, supremum and infimum.
5. Least upper bound principle for Real Numbers.
6. Convergence, monotonicity, boundedness.
7. Cauchy sequences and the completeness of Real Numbers.
8. Subsequences and the Bolzano-Weierstrass theorem.
9. Real functions.
10. Epsilon-delta definition of continuity.

Assessment Proportions

Teaching will consist of lectures and examples classes with lecture notes being provided. Lectures will be used to define key concepts, develop the theory and illustrate the theory and definitions through examples. Examples classes will provide a forum for students to construct their own examples, practice relevant skills and methods, and receive feedback on their progress with reference to the learning outcomes. Assessment will be through:

• summative coursework (written and online) submitted on a weekly basis;
• an end-of-module closed book examination.

The summative coursework will not carry much credit to be used as vehicle for feedback and for students to monitor their own progress. The final assessment will be a closed-book examination in line with practice across mathematical sciences and to ensure academic integrity of the assessment.

MATH4120: Mathematical Modelling and Programming

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: high-school level maths required

Course Description

This module aims to introduce students to mathematical modelling and the mathematical modelling cycle, primarily through scenarios that lead to ordinary differential equation models. It will equip students with a variety of fundamental modelling techniques, as well as standard methods for solving differential equations, enabling them to make quantitative statements in the context of the original scenario. In parallel with the above, the module will equip students with the scientific programming and computing skills that will be used repeatedly throughout the programme.

Educational Aims

Upon successful completion of this module students will be able to…

1. Reduce a simplified real-world problem involving the evolution of a single variable to an appropriate mathematical model involving a differential equation, criticise the model and relate the model properties and solution back to the original problem.
2. Solve a range of single-variable ordinary differential equation models using standard techniques.
3. Write a structured computer programme involving functions and control flow to correctly perform a mathematical task or investigate a mathematical phenomenon.
4. Use a form of markdown to write a short report incorporating text, equations, computer code and output.

Outline Syllabus

A mathematical model is a representation, in the language of mathematics, of a real-world phenomenon such as a building vibrating during an earthquake or the spread of a disease within a population. In this module you will investigate mathematical models that lead to ordinary differential equations and will study a variety of core analytical methods for solving them. You will learn to extract the most important aspects from a real-world problem or scenario to develop a mathematical model. You will analyse the model and relate your findings to the original problem or scenario and subsequently refine the model, if necessary. Starting from scratch, you will also learn the fundamental programming skills and concepts that will be used in subsequent modules. Many mathematical models, including those used in artificial intelligence, are intractable analytically and, hence, require a computational approach. The skills obtained in this module will enhance your understanding of later material when you implement the computational techniques for yourself. Outline syllabus:

• Mathematical models: definition, examples, uses and limitations; the mathematical modelling cycle.
• Solution methods for differential equations, including separation of variables, the integrating factor and substitutions; second-order, linear equations with constant coefficients. A first look at numerical solution of differential equations.
• Programming: variable types, flow control, functions, and good programming practice.
• Markdown basics including headings, equations and incorporating code and output.

Assessment Proportions

Exam 50%, Test 20%, Coursework 30% The summative coursework (worth 30%) will comprise a mixture of handwritten coursework and Moodle quizzes (20%), submitted regularly over the teaching part of the semester, and a small group-based coursework submitted towards the end of the teaching period (10%). The regular coursework and quizzes cover both analytical aspects and computational techniques, allowing us to assess the theoretical, applied and computing parts of the module. On the theoretical and applied side, the coursework will include activities such as deriving a mathematical model of a given real-world system or finding the analytical solution of a differential equation. On the programming side, the coursework will include assessments such as short Markdown write-ups of code and output, for example, investigating the behaviour of a numerical approximation to an integral or a derivative, and automated assessment via CodeRunner. The group coursework, due towards the end of the teaching period will be assessed through a pdf of several pages created using Markdown. The document will contain text, equations, code from a task related to differential equations and modelling and output in the form of figures, all sensibly formatted.

MATH4125: Multivariate Calculus

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic 1st - year mathematics at univeristy level required

Course Description

This module aims to expand students’ knowledge and understanding of calculus from the univariate material covered at A-level and in MATH4100 to calculus of several variables, fundamental in modern pure and applied mathematics, in the natural sciences, and in engineering. Students will develop core skills in the techniques of multivariate calculus, crucial in the formulation, analysis and solution of differential equation models; in optimisation and machine learning; and in high-dimensional data analysis. They will also learn to work with their peers to solve a problem and explain the solution to a small audience.

Educational Aims

Upon successful completion of this module students will be able to…

1. Understand scalar and vector functions of several variables and apply techniques involving differentiation and integration of such functions. Solve problems involving line, surface, and volume integrals of scalar and vector fields.
2. Recognise and manipulate the main operators of vector calculus and use the key identities relating these operators. Understand and apply the divergence theorem and related results.
3. Apply the techniques of vector calculus appropriately and correctly to formulate and analyse problems arising across the mathematical and natural sciences, including problems involving differential equations, and problems arising in engineering and machine learning.
4. Collaborate with colleagues to solve a problem and, together, explain the solution to a small audience.

Outline Syllabus

Many real-world problems involve functions with vector inputs and/or outputs, where the vectors could describe, for example, a position in space, the state of a biological system, or the weights of an artificial neural network. In this module students will explore the world of functions with multiple inputs and/or outputs using the techniques of multivariate calculus. They will deepen their understanding of the geometry of curves, surfaces and volumes in two, three, and higher dimensions, and learn how to use different co-ordinate systems to simplify the description and analysis of models with different underlying geometries. Students will encounter multidimensional derivatives and integrals, and practice multidimensional analogues of techniques such as the chain rule and integration by substitution. They will learn and apply the theorems of vector calculus, fundamental in modern geometry and in the study of differential equations. The connection to real-world problems will be emphasised throughout, and the new understanding and skills will enlarge the set of mathematical models students can analyse, while also being foundational for more advanced study in later years of the course. Outline syllabus:

• Vectors and angles, scalar and vector products, and key identities of vector algebra.
• Functions with multidimensional inputs and/or outputs: parameterised curves and surfaces; scalar and vector fields.
• The use of different coordinate systems to transform problems into simpler forms.
• Applications of differentiation of multivariate functions: finding extrema, Taylor expansion and local approximation, the chain rule, Jacobian matrices and the Hessian. Applications to root finding and optimisation.
• Multiple integrals and integration over curves, surfaces and volumes, with applications in the natural sciences, probability, and engineering.
• Differential operators of vector calculus: gradient of a scalar field; divergence and curl of vector fields, with an emphasis on physical intuition and applications. The divergence theorem, Green's theorem, and Stokes' theorem, with applications to formulating and analysing differential equation models.

Assessment Proportions

The core content will be covered in the lectures. Multivariate calculus is a very practical and hands-on subject, made more exciting by the wide range of applications. Consequently, teaching will be focussed on providing maximum geometrical insight, with numerous examples. More challenging proofs will not be lectured (but students will be provided references to follow up if they choose). Problem solving is key to a deepened understanding of multivariate calculus. A selection of problems will be set each week, with help and guidance given during whole-cohort problem classes and smaller-group workshops. The problems will develop students' ability to formulate questions in a form amenable to analysis using multivariate calculus techniques, and to apply these techniques. They will include problems drawn from the natural sciences, engineering and machine learning. Assessment will consist of: coursework worth 20%, comprising four randomised moodle quizzes, and four written courseworks; a group project, assessed via a group presentation (10%); and a final exam (70%). Students will work on a small group project, with colleagues from their workshop block, over the final three teaching weeks of the module, presenting their findings to the rest of the workshop block during their final workshop. Support will be available during the penultimate workshop and from the lecturer through office hours and during the final problems class, which will be dedicated to this.

MATH6310: Metric Spaces and Topology

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic real analysis

Course Description

The module aims to extend the scope of the students’ understanding of the notions of convergence and continuity gained in Real Analysis. This is done in two stages. In the first stage the distance between real numbers, as modulus of their difference, is replaced by a notion of distance between the points of a set, governed by a few simple rules. The set may consist of real-valued functions, matrices, points of a sphere, probability distributions, binary sequences, or even subsets of the plane. In the last and first of these, the theory has the power to respectively deliver fractal sets, and establish existence and uniqueness for solutions of differential equations. In the second stage, the very notions of convergence and continuity are abstracted. A topological space equips each of its points with so-called neighbourhoods, in terms of which one finds natural notions of convergence and continuity. The topology may or may not be derivable from a metric and, when it is, the metric is typically far from unique.

The theory and application of metric spaces and topology are vast in scope and pervade the mathematical sciences and theoretical physics. This module will be useful for many later modules including Hilbert Spaces, Knots and Geometry; Measure and Integration; Lie Groups and Lie Algebras; and Operators and Spectral Theory.

Educational Aims

Upon successful completion of this module students will be able to:

1. Employ a range of arguments to settle countability/uncountability questions.
2. Describe a variety of metrics and topologies.
3. Distinguish the different metric/topological types of convergence and continuity.
4. Complete a metric space.
5. Work with (total) boundedness, (sequential) compactness, initial, relative and metrisable topologies.
6. Identify separable, first countable and second countable topological spaces.
7. Form product topologies and Cauchy products of metric spaces.
8. Understand and explain the basic concepts of metric spaces and topology.
9. Be able to formulate questions and solve problems involving all of the above.
10. Marshal and apply key theorems listed in the module syllabus.

Outline Syllabus

Metrics appear in familiar guises, such as the Euclidean distance between points in the plane – given by the Pythagorean rule, and geodesic distance between points on the surface of a sphere. The following list illustrates the breadth of examples. The Hausdorff metric gives an effective `distance’ between two nonempty closed and bounded subsets of the line/plane/space, and the theory then delivers fractal sets. The Hamming metric gives a distance between binary sequences, and plays a key role in information theory. The p-adic metric on the field of rationals is a basic tool in number theory. The Wasserstein metric gives a distance between probability distributions and is central to the modern theory of optimal transport.

Metrics deliver sound notions of continuity for functions, and neighbourhoods for points. They deliver much besides however, and, concentrating on and abstracting from just these, one arrives at topology. The module will focus on the key topological properties of compactness (rooted in the Bolzano-Weierstrass property of closed and bounded subintervals of the real line), and `Hausdorffness’ (a basic criterion for separating points), which work very effectively in tandem; the property of `normality’ which delivers a plentiful supply of continuous real-valued functions; and the question of metrisability of topological spaces.

The module begins with a review of real analysis, linear algebra and sets-and-functions basics. The theory of countable/uncountable sets is developed, and Cantor’s power set theorem proved and applied. This all serves as the basis for the rest of the module, which divides into two parts: metric space theory and general (point-set) topology; with the latter abstracted from, and illuminated by, the former. The syllabus will cover the main metric space concepts including: types of convergence and continuity, equivalences of metrics, Cauchy products of metric spaces, total boundedness, completeness and sequential compactness. In topology we begin with the concepts of neighbourhood and continuity, and cover interior and closure, density and separability, first and second countability, initial, relative and product topologies, metrisability, normality, compactness and the Hausdorff property.

Key theorems include Cantor’s intersection theorem; Banach’s fixed point theorem; the completability of metric spaces; the preservation of total boundedness under Cauchy continuous functions, and compactness under continuous functions; the normality of compact Hausdorff spaces; the Baire category theorem; the sequential compactness of complete totally bounded metric spaces; and Tychonoff’s theorem.

The module ends with a study of the `Cantor space’, and a discussion of Urysohn’s theorem, which characterises the metrisable spaces each of whose points is approximable from a single countable subset.

Assessment Proportions

This module will be taught using live interactive lectures and comprehensive lecture notes, accompanied by regular workshops and coursework, both formative and assessed. Model solutions will be supplied for all of the coursework exercises. Feedback on coursework will also play an essential formative role. Plenty of purely formative exercises will be provided, and students will be strongly encouraged to regularly work on these in order to gain command of the material. The lecturer will go over a selection of the exercises in the workshops.

Students will be encouraged to actively participate in the lectures, both by asking questions and by responding to questions posed by the lecturer. Workshops will involve working on problems, either alone or in small groups according to the preference of the student, with advice and guidance offered throughout.

This module is the natural successor to Real Analysis. It also builds on the abstract notion of vector length from Linear Algebra. The module will be helpful for a good many subsequent modules too, such as those listed in the Module Aims. This reflects the central role of metric spaces and topology in pure mathematics and its applications.

Students will be assessed through regular fortnightly coursework, a short mid-term test, and a final written exam. The mid-term test will take place immediately after the consolidation week, and will cover the first part of the module (countability and metric space theory).

MATH6315: Hilbert Spaces

• Terms Taught: Michaelmas term
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

Course Description

This module introduces students to the theory of Hilbert spaces, which is a powerful and elegant synthesis of techniques from linear algebra and analysis, and which provides a fundamental toolkit for many modern applications of analysis to engineering, physics and statistics. Students will see how the general theory is built up logically from a small set of axioms/conditions, so that the main results are presented as part of a cohesive whole rather than isolated claims to be taken on faith.

The module provides a rigorous underpinning for results that students may encounter in topics such as signal processing, probability theory and statistical learning. Examples and applications are chosen from a range of settings, to emphasise connections between Hilbert space theory and other areas of mathematics.

The module is a natural development of techniques and concepts seen in Y2 linear algebra and analysis. Together with MATH6310 (Metric Spaces and Topology), this module provides the main foundation for Level 7 studies (modules or dissertations) in functional analysis.

Educational Aims

Upon successful completion of this module students will be able to…

1. Recognize concrete examples of inner products on function spaces, and accurately perform calculations with them.
2. Rigorously derive properties of abstract inner product spaces from the defining axioms.
3. Formulate appropriate classes of optimization problems in terms of orthogonal projections, and apply the general theory correctly to obtain solutions for specific examples.
4. Recognize and construct orthonormal systems in inner product spaces, and accurately perform calculations with them.
5. Extend results and proofs from analysis on the real line to the setting of Hilbert spaces.
6. Write logically coherent proofs that distinguish between premises and conclusions, and clearly present a chain of reasoning.

Outline Syllabus

Key topics in this module include:

• Real and complex inner products. Examples of infinite-dimensional inner product spaces.
• Orthogonality and orthogonal complements. Invariant subspaces and reducing subspaces.
• The abstract Cauchy-Schwarz inequality. The norm and distance induced by an inner product.
• Characterising the closest point in a linear subspace via orthogonality. Finding best approximations by solving linear systems.
• Gram matrices and positive-semi-definite kernels. Feature spaces and the representer theorem.
• Orthonormal sequences and formulas for orthogonal projections. Examples of orthogonal polynomials.
• Closure points, density, and separability. Examples of closed and non-closed subspaces.
• Bessel’s inequality and Parseval’s identity. Application to Fourier series.
• Convergence and completeness. Completions of inner product spaces.
• The Fourier transform, revisited. Isometry of all separable infinite-dimensional Hilbert spaces.
• The theorem of the closest point for Hilbert spaces. Bounded linear functionals and the Riesz-Frechet theorem. The reconstruction theorem for reproducing kernel Hilbert spaces.
• Bounded linear operators. Adjoints and duality.

Assessment Proportions

The module will encourage students to consolidate and synthesise prior knowledge from the 2nd year, by presenting proofs and results that draw on this knowledge and extend it to new settings. In particular, the module builds on students’ experience in Year 2 with linear algebra, and their geometric intuition about distance and angles. At the same time, through the lectures and workshops/tutorials, students will develop skills in digesting and writing rigorous formal arguments.

The module is taught through a combination of course notes that are provided to students in advance, with lectures that go over selected parts of the notes. The notes are structured pedagogically, providing motivation and scaffolding. The lectures provide commentary on particular examples, discussion of any common difficulties or misunderstandings, and explanation of selected proofs. Emphasis is placed on demonstrating how proofs can be reconstructed from underlying ideas and principles, rather than rote memorization.

Fortnightly tutorial/workshop sessions will ask students to work on exercises related to the notes, including certain proofs that are deliberately omitted. Through this work, students will gain practice in applying general theoretical results to make concrete calculations, and also in writing proofs with an appropriate level of mathematical rigour and logical structure. Students will be able to review model solutions to these exercises and compare with their own attempts.

Students will be assessed through a combination of assessed coursework exercises (fortnightly) and a final exam. The coursework is aimed at testing and consolidating understanding of the basic elements of the course. The exam will assess more fully the students’ summative knowledge, their ability to apply general principles to specific problems, and their ability to communicate logical reasoning.

MATH6320: Commutative Algebra

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

Course Description

This module aims to further develop students’ learning in abstract algebra, building on MATH5225 Abstract Algebra. As well as deepening their knowledge and understanding of this topic, this module also stands as a gateway module to further advanced algebra modules, notably MATH6325 Representation Theory and MATH7420 Galois Theory (the module is recommended but not a prerequisite for the former, but is a prerequisite for the latter).

Specifically, the module will examine in detail questions about factorizing and divisibility in a variety of contexts, through the abstract frameworks of unique factorization domains and related objects. The particular case of polynomials leads in two directions: one is an algebraic approach to geometry and the other is Galois’ renowned theory answered the question of the solvability of polynomials through the addition of suitable square and higher roots.

Educational Aims

Upon successful completion of this module students will be able to...

1. Recall and apply the key definitions relating to factorizability and divisibility in a variety of contexts.
2. Explain the relationships between the different classes of rings introduced in the module (PID, UFD, Euclidean).
3. Solve polynomials via field extensions and give both algebraic and geometric interpretations of their sets of roots where appropriate.
4. Evaluate the validity of statements in commutative algebra based on their experience from the module and either formulate rigorous proofs or find counterexamples to justify these assessments.

Outline Syllabus

Commutative rings play very important roles in a wide variety of areas of mathematics. As well as being of central importance in algebra, they sit at the heart of algebraic approaches to geometry and number theory, in part because rings of functions occur so naturally there, as they do in analysis too.

Basic but crucial questions one often wants to answer about commutative rings include their factorizability and divisibility properties. For example, what is the analogue of the set of prime integers, or which are the invertible elements? This module sets out the general theory that enables us to ask and answer these questions. It begins by looking at rings with certain properties and finding the key examples of these. It continues by describing several constructions that allow us to produce rings with properties we would like, and concludes with some discussion of the applications to the areas mentioned above.

An indicative syllabus is as follows:

• Principal ideal domains (PIDs) and unique factorization domains (UFDs): motivation, definition, examples.
• Invertible and associated elements; greatest common divisors; Bézout's Theorem, Euclidean rings and the Euclidean algorithm.
• Polynomial algebras over fields are PIDs and therefore UFDs.
• Localisation, with the field of fractions as the main example.
• Gauss' lemma and Eisenstein's criterion, at the generality of UFDs and their fields of fractions.
• Solving polynomials by taking field extensions. Cyclotomic polynomials and their roots. Finite fields.
• Advanced applications-oriented topics, taken from: (i) Introduction to chain conditions and zero sets of polynomials (via low-dimensional examples), (ii) Contrasts with the noncommutative situation: 1-sided versus 2-sided ideals, matrix rings, division rings, and (iii) Implications for algebraic number theory and discriminants.

Assessment Proportions

Teaching will be centred on lecturer-led teaching through lectures, written notes and examples classes. The necessarily technical definitions will be illustrated through frequent use of examples and students’ understanding of them developed via calculations, both by hand and with the support of computer algebra software.

Assessment will be through summative coursework on a fortnightly basis, supported by prior formative work, and a closed book examination. This is in line with the other level 6 modules on the programmes to which this module contributes.

MATH6321: Mathematical Cryptography

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

Course Description

This module provides an introduction to Mathematical Cryptography, beginning with a range of classical methods of encryption and discussing the advantages, disadvantages and efficiency. In particular, we will see significant statistical attacks on these methods of encryption, motivating the need for better methods. After this, we move towards more modern methods of encryption that are used in the real world and rely heavily on the robustness of modular arithmetic. Security is now provided by the fact that certain mathematical problems are very hard to solve efficiently (i.e. integer factorization and the discrete log problem). While most of these encryption methods are still considered secure, we consider examples of potential attacks on these hard problems (e.g. factorization algorithms) and explore attacks in situations where bad key generation or implementation has occurred. The production of a big enough quantum computer renders the above schemes useless! We finish the course with a short introduction to Post-Quantum Cryptography, i.e. the production of next-gen cryptographic schemes that are considered to be impenetrable to both classical and quantum computers. We will introduce the theory of lattices and see how these can be used to produce new schemes that are considered to be quantum secure (e.g. NTRU).

This module will both build on and explore applications of some of the material covered in MATH4110 and MATH5225 (e.g. modular arithmetic, polynomials and basic group theory).

Educational Aims

Upon successful completion of this module students will be able to...

1. Work with discrete mathematical objects;
2. Formulate real-world problems in mathematical formality;
3. Explain basic cryptographic concepts, methods and proofs;
4. Compare and contrast basic cryptographic schemes;
5. Critique the security of classical cryptographic schemes;
6. Investigate and understand the implications of bad implementation, key generation, key management;
7. Recognise the vast applications of cryptography to other mathematical fields, such as Algebra and Number Theory;
8. Explain basic cryptographic concepts, methods and proofs.

Outline Syllabus

Ket topics in this module include:

• Classical Ciphers and Statistical Attacks: Caesar, Substitution, Frequency Analysis, Vigenère, (Mutual) Index of Coincidence, One Time Pad.
• Mechanical ciphers: Enigma, Turing’s attack (the gist of it, non-examinable), LFSR’s and pseudorandom sequence generation.
• Public Key Cryptography: Diffie-Hellman, RSA, El Gamal, Digital signatures.
• Elliptic curve Cryptography
• Basic attacks on factorization and Discrete Logs: Fermat, Pollard Rho, Dixon, Baby-step Giant-step, Pollard Rho, Index Calculus.
• Intro to Post Quantum Crypt: Simple noisy schemes, relation to rank 2 lattice problems, break using Gauss reduction, Knapsack scheme, relation to higher rank lattice problems, NTRU (non-examinable).

Assessment Proportions

Two lectures will be given per week to the full class, allowing the subject specific knowledge to be imparted efficiently. Weekly tutorials will then supplement the lectures, allowing students to practice the relevant skills and learning outcomes, getting useful feedback as they do so.

Students will be assessed through a mixture of exam and assessed coursework. The exam will test overall knowledge of the theory behind cryptographic schemes and why/how they work, whereas the assessed coursework will test more practical skills in cryptography (e.g. real world weaknesses of cryptographic schemes, implementation/coding, numerical examples).

MATH6325: Representation Theory

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

Course Description

This module aims to introduce students to modern representation theory. Representation theory is the study of mathematical objects through linear approximations, which is such a fundamental idea that it crops up in mathematical sciences from quantum physics to algebraic geometry to topological data analysis. The course will consolidate and develop students’ learning in abstract and linear algebra and provide tools for further study in algebra, geometry and physics at postgraduate level. The interplay between abstraction on one hand and concrete examples from linear algebra and combinatorics on the other will provide students insight into how examples-based reasoning facilitates mathematical creativity and proof construction.

Specifically, the module will show how the specialisation of the concept of group action to linear spaces allows more concrete understanding of mathematical structure by using linear algebra to represent finite groups and finite-dimensional algebras. It will examine in detail questions about the representation theory of a group or algebra can be decomposed into its simplest building blocks, and how linear and combinatorial data can be used to make sense of often complex mathematical structures. The module will provide examples and indications of the widespread application of representation theory throughout mathematical and physical sciences.

Educational Aims

Upon successful completion of this module students will be able to:

1. Recall and apply the definitions of associative algebra, group algebra and a module over an algebra.
2. Compute examples of modules, representations and homomorphisms.
3. Expand their example space of noncommutative rings beyond matrix rings.
4. Explain complete reducibility of the representation theory of group algebras (in characteristic zero) and how this fails more generally.
5. Evaluate the validity of statements in representation theory by formulating rigorous proofs or explicit counterexamples.
6. Use example-based reasoning to test conjectures, understand structure and to help the construction of mathematical proofs.

Outline Syllabus

An indicative syllabus is as follows:

• Motivation from group actions on why mathematicians like to consider linear actions.
• Definition and examples of associative algebras.
• Definition of a module over an algebra, module homomorphisms, submodules, quotient modules, (semi)simple modules and Schur's lemma.
• Group algebras and Maschke's theorem (complete reducibility for group algebras).
• Discussion of complete reducibility and its failure in general, generalisations to the Jordan--Hölder theorem and Krull—Remak--Schmidt theorem for arbitrary finite-dimensional algebras.
• Classification of finite dimensional representations of the polynomial ring in one variable.
• Advanced topics taken from: (i) Quivers and path algebras, finite representation type via persistence theory (from topological data analysis), (ii) Classification of modules over principal ideal domains, derivation of normal forms in linear algebra, first theorems of persistence theory, and (iii) Character theory of finite groups.

Assessment Proportions

Teaching will consist of lectures and examples classes with lecture notes being provided. Lectures will be used to define key concepts, develop the theory and illustrate the theory and definitions through examples. Examples classes will provide a forum for students to construct their own examples, practice relevant skills and methods, and receive feedback on their progress with reference to the learning outcomes.

Assessment will be through:

• summative coursework submitted on a fortnightly basis;
• a short mid-term test;
• an end-of-module closed book examination.

The summative assessment will not carry much credit in order to act as a learning incentive with lower cost for making mistakes allowing it to be used as vehicle for feedback. The mid-term test will serve to consolidate learning and ensure a solid foundation from the first half the course to prevent students falling behind in the second half of the course as the subject builds on the basic concepts from the first half. The final assessment will be a closed-book examination in line with practice across mathematical sciences and to ensure academic integrity of the assessment in the context AI.

MATH6326: Graph Theory and Algorithms

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic (Linear) algebra

Course Description

The study of graphs – mathematical objects used to model networks and pairwise relations between objects – is a cornerstone of discrete mathematics. Graphs can represent important real-world situations, and the study of algorithms for graph-theoretical problems has strong practical significance.

During the module you will learn about structural and topological properties of graphs, including graph minors, planarity and colouring. We will introduce several theoretical tools, including matrices relating to graphs and the Tutte polynomial. We will also study fundamental algorithms for network exploration, routing and flows, with applications to the theory of connectivity and trees, considering implementation, proofs of correctness, and efficiency of algorithms.

You will gain experience of following and constructing mathematical proofs, correctly and coherently using mathematical notation, and choosing and carrying out appropriate algorithms to solve problems. The module will enable you to develop an appreciation for a range of discrete mathematical techniques.

Educational Aims

Upon successful completion of this module students will be able to…

1. Define a variety of standard graph parameters, find their values in suitable cases and prove relationships between them.
2. Choose and implement appropriate algorithms to find routes, connected structures and flows in graphs and digraphs, prove correctness and understand complexity.
3. Understand the relationship between matrices associated with graphs and their structural properties.
4. Recognise graphs from fundamental minor-closed families, and deduce properties shared by graphs in each family.
5. Determine and interpret the Tutte and chromatic polynomials of small graphs.
6. Follow and construct mathematical proofs of appropriate complexity, using suitable notation.

Outline Syllabus

The most fundamental property of a graph or network is connectedness. Minimally connected graphs are trees. You will study theoretical properties of trees, and algorithms for graph exploration and for finding efficient spanning trees. For directed graphs there are different notions of connectedness, and an important algorithmic problem is to find the strongly connected components and relationships between them. Paths correspond to routes through a network, and you will study algorithms for finding optimal paths in different contexts. Network flow is another key algorithmic area of the course, with duality properties that give insight into matching problems and higher-order connectivity.

The course will highlight how matrices and tools from linear algebra can give insight into graph properties, yielding elegant proofs of theoretical results as well as a method for counting the spanning trees of a graph. The topology of graphs will also be explored. A starting point is the question of which graphs can be drawn in the plane; this led to the development of graph minor theory, and we will consider other types of graphs which can be classified in this way, such as series-parallel graphs. Graph colouring is an important research area which links to graph minor theory via the celebrated four-colour theorem. We shall study how to calculate and interpret the chromatic polynomial, and its generalisation the Tutte polynomial, algebraic objects that summarise colourability and connectedness information about the graph.

Assessment Proportions

Lectures will be the main means of teaching, supported by detailed written materials and workshops where students are split into smaller groups to work on sample problems and receive direct feedback. There will be three lectures in five of the eleven teaching weeks, and two lectures and a workshop in the other six. Rather than separate examples classes, many worked examples will be integrated into the lectures to better support learning in a timely manner. Additional sample problems will be provided for independent study outside of workshops.

Students will be assessed through a combination of an end of module exam and assessed coursework spread throughout the course. Students will receive individual feedback on each segment of written coursework, allowing them to develop their skills throughout the module. The written coursework will be divided into five short sections on different parts of the syllabus, to be handed in in alternate weeks. These should each take at most three hours. On other weeks there will be an online quiz, with simple problems which should take at most one hour to complete.

MATH6327: Knots and Geometry

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability

Course Description

Geometry and topology is one of the core areas of pure mathematics; it is useful for applications in engineering, physics and biology, but it also provides an enlightening perspective on other parts of pure mathematics. This module aims to develop your geometric and topological intuition and to help you to turn vague intuitive statements (this looks knotted, this looks curved, these look different, these look the same) into precise mathematical statements. We will introduce ideas from geometry like curvature which play a central role in physical applications (from wavefront propagation in optics to gravitation and black holes) and ideas from topology like knot invariants and the fundamental group.

Educational Aims

Upon successful completion of this module students will be able to…

1. Formulate mathematically precise hypotheses based on geometrical or topological intuition and either prove these hypotheses or find counterexamples.
2. Interrogate proofs in geometry and topology, determining whether they are valid, and identifying or fixing gaps.
3. Perform computations in the differential geometry and topology of curves in 2 and 3 dimensions, and use this to analyse global qualitiative geometrical features of these curves.
4. Calculate fundamental groups for a range of topological spaces and use them to distinguish knot complements.

Outline Syllabus

Knots play a fundamental role in many areas of mathematics, from pure topology and algebra through to polymer physics and DNA modelling. In this module, we will develop tools to measure knottedness. This will include:

• Notions of curvature for curves in 2 and 3 dimensional space and the Serret-Frenet formulas, working towards global geometrical results like the 4-vertex theorem (caustics always have at least four cusps) and the Fary-Milnor theorem (knotted curves have total curvature at least 4 pi).
• Knot invariants like the Jones polynomial or colourings: how to prove invariance using Reidemeister moves.
• The fundamental group of a space, in particular the knot group. This will let us give topological proofs of results like the fundamental theorem of algebra (any complex polynomial has a root). We will see Van Kampen’s theorem for computing fundamental groups and use it to deduce presentations for the knot group.

Assessment Proportions

Technical lectures together with supporting online and physical materials (e.g. notes, videos) will introduce students to the basic ideas of the module. Lectures will be broken up with engagement activities to encourage active learning and to help scaffold students. Frequent example-based workshops will give the students more space to explore the ideas in depth: they will work together to build the key skills listed in the learning outcomes: carrying out computations, interrogating proofs and exploring hypotheses through proof and counterexample. Peer-learning activities in workshops will help them to build confidence and communication skills as well as their mathematical ability.

Students will be assessed in two ways: (a) through a final written examination and (b) through five fortnightly courseworks over the semester. Feedback on the courseworks will also play a formative role, helping them to fine-tune their skills over the semester. One of the five courseworks will be a more substantial mini-project: this will allow the students to explore one aspect of the module in greater depth.

MATH6330: Statistical Inference

• Terms Taught: Michaelmas 
• US Credits: 5 US Semester credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability

Course Description

The module aims to develop a deeper understanding of the theoretical underpinnings and application of statistical inference. In particular, this module focusses on frequentist inference and the central role of the likelihood function and maximum likelihood estimators. The module generalises concepts introduced at Level 5, allows a more detailed study of the theoretical properties of maximum likelihood estimators and introduces likelihood-based hypothesis testing. Each of these concepts underpin advanced domain-specific techniques explored in further Level 6 statistics modules. Students will develop both a deeper appreciation of key theoretical concepts and the understanding to apply this knowledge to conduct modelling and inference tasks with sound statistical judgement.

Educational Aims

Upon successful completion of this module students will be able to

1. Construct appropriate likelihood functions to summarise information about unknown parameters from data collected in practical applications;
2. Compute maximum-likelihood estimates for multiple parameter problems, and assess inter-relationships and asymptotic convergence properties;
3. Describe the properties of sufficiency and optimality in the context of unbiased estimators, and maximum-likelihood estimators;
4. Utilise likelihood-based model selection and hypothesis testing to aid statistical reasoning;
5. Use scripts from the R language to plot and interpret likelihood functions for practical problems.

Outline Syllabus

This module extends an understanding of the likelihood function and maximum likelihood estimation to the multiparameter setting – which is fundamental to practical statistical inference.

We will first study the asymptotic properties of the likelihood, and its implications for the distribution of estimators as large samples of data are collected. These theoretical properties provide a basis for quantifying uncertainty in estimates of key parameters and functions of parameters. These ideas will be extended to the profile likelihood which provides a mechanism to remove the effect of nuisance parameters.

The module also explores the role of sufficient and ancillary statistics, and key theoretical results such as the Rao-Blackwell theorem and the Cramér-Rao lower bound. This further develops the understanding of frequentist statistical inference and the principles allowing efficient extraction of information from data.

Our final theme is testing and model selection, where we develop likelihood-based approaches to evaluate the evidence for hypotheses and to select the most appropriate model structures to explain the relationship between variables. We study theoretical results demonstrating the optimality of these methods.

The skills and understanding built throughout the module (in a general multiparameter setting) support students to deploy advanced inference methods for specific classes of data in further Level 6 statistics modules, or beyond their degree.

Assessment Proportions

The learning strategy for this module supports students to build on the mathematical skills and technical understanding of the frequentist method developed in Level 4 and Level 5 statistical concepts and techniques. Students will principally acquire and develop knowledge through synchronous lectures and guided independent study.

Technical and theoretical material will be introduced using lectures, which complement the course’s comprehensive written materials, while providing alternative, more detailed exposition. The lectures will integrate interactive activities and discussion to promote active engagement with the material. Where appropriate lectures will be supplemented by short, pre-recorded, videos.

Students will put technical material into practice in workshops, approaching mathematical and statistical exercises with group-working encouraged and with instructor support. These exercises build towards coursework- and exam-level challenges and provide a formative means for students to assess their understanding, via peer discussion and specimen solutions.

Students will be assessed through both a final written examination (which assesses their consolidated understanding of technical concepts, interpretation of statistical inference, and mathematical reasoning), fortnightly, low stakes, multiple-choice coursework and a class test.

MATH6331: Statistical Learning and Prediction

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability

Course Description

This module aims to build your understanding of the statistical models used to address supervised learning challenges and how these differ from comparable machine learning algorithms. The concepts covered in this module form a crucial component of modern statistics, data science and even AI, providing a way to extract patterns or trends from large and complex data sets. Such findings can then be used to forecast or predict future behaviour or interpolate missing information. On completion of the module you should have a thorough understanding of the mathematical aspects of generalised linear models and regularised regression. You will also have developed your essential skillset through selection and implementation of these methods using suitable statistical software, interpretation of the model output, critiquing modelling choices and communication of your findings.

Educational Aims

Upon successful completion of this module students will be able to…

1. Define the class of generalised linear models; select an appropriate model from this class and justify their choice; implement the model using appropriate statistical software; and interpret the output in the context of the original research question.
2. Explain the limitations of generalised linear models in the context of “large p, small n” data sets, describe methods that address these limitations and justify these methods.
3. Analyse a “large p, small n” data set using either a regularisation method or dimension reduction, assess the findings of the analysis and critique their modelling choices.
4. Explain and use simple machine-learning alternatives to statistical models for prediction and classification.

Outline Syllabus

Pattern identification is a crucial part of almost all data-driven research from clinical trials, psychology and economics to ecology, climate science and physics. Pattern identification also underpins many modern AI technologies. Seen by many as a form of magic, in fact many pattern identification methods are classical statistical models. This module will give you a deeper understanding of how to construct and apply some popular classes of statistical model that can be used for pattern identification and forecasting. You will first explore generalised linear models and apply these to real-world data sets, learning how to interpret and critique the model output and communicate this to non-specialists. In the second half of the module you will discover how generalised linear models have been adapted in response to evolutions in data collection methods. Where once data collection was expensive, it is now much cheaper and so data sets are much larger. As well as understanding what challenges this change provoked, you will learn how new methods, including regularised regression and dimension reduction, evolved to address these challenges. Throughout you will be given opportunities to put theory into practice through data analysis.

Assessment Proportions

The learning strategy for this module is designed to enable the student to understand the mathematical underpinning of the statistical models, to correctly select and implement these models using statistical software and to communicate their choices and findings using both technical and non-technical language.

Technical components will be taught using lectures, supported where appropriate by short, pre-recorded, videos, and comprehensive written materials.

Practical elements will be delivered in computer labs, using worksheets designed to consolidate theory and guide students through putting this theory into practice. The course will use the statistical software package R, though students may opt to use Python instead.

To encourage students to analyse and evaluate their modelling choices and findings, they will be encouraged to keep a running lab-book throughout the course using a suitable scientific tool (e.g. markdown, Rmarkdown, or LaTeX).

The module will have a series of online quizzes which students can use to assess their formative understanding, alongside model solutions which will be provided after each lab session.

Students will be assessed through both a final written examination (to cover the theoretical elements) and a group project (to cover practical elements and communication).

MATH6332: Stochastic Processes

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability

Course Description

This module aims to show how Probability can be used to model the real world through Stochastic Processes; for example the wealth of a gambler, the size of a queue or the number of phone calls at a call centre. The module focuses on Markov chains, an important type of stochastic processes. We will study both discrete and continuous time Markov chains; you will learn how to find stationary and asymptotic distributions as well as other properties of these processes such as expected hitting times.

Educational Aims

Upon successful completion of this module students will be able to:

1. Understand the relevance of stochastic processes as natural models for stochastic phenomena.
2. Understand mathematically the definition of a stochastic process.
3. Use and manipulate probability generating functions for a range of probability calculations.
4. Carry out simple calculations for probabilities of simple random walks.
5. Understand definitions of Markov chains in both discrete and continuous time (including probability transition matrices and in continuous time, transition rate matrices).
6. Have a basic knowledge of the principles of irreducibility and recurrence.
7. Understand stationary and asymptotic distributions for Markov chains in simple examples.
8. Understand simple examples such as those from queues, and transfer techniques and skills to other similar examples.
9. Understand specific types of Markov chains in discrete and continuous time, namely Branching and Poisson processes.

Outline Syllabus

Key topics covered in this module include:

(1) The Bernoulli process and simple random walks.(2) Conditional expectations (in particular, the tower rule) and applications to random walks and the gambler’s ruin problem (in particular, the reflection principle). (3) Probability generating functions and applications.(4) Discrete time Markov chains:

• Time dependent, stationary and asymptotic distributions.
• State classification, periodicity recurrence and transience.
• Reversible Markov chains and detailed balance.
• Expected hitting times.
• Explicit n-step transition formulae.
• Branching processes (discrete time).

(5) Continuous time Markov chains:

• Rate matrix and distribution of holding times.
• Time dependent, stationary and asymptotic distributions.
• Reversible continuous time Markov chains and detailed balance.
• Kolmogorov’s forward/backward equations.
• Poisson processes.
• Branching processes (continuous time).

Assessment Proportions

The module will be delivered through in-person lectures given to the full class where subject specific knowledge will be taught and many worked examples will be given. There will also be a workshop (students split into smaller groups) where students will have the chance to practice the relevant skills and learning outcomes by working through problems themselves, getting useful feedback as they do so.

The students will be assessed through a combination of an end of module exam and five pieces of assessed coursework throughout the semester. Individual as well as overall feedback will be given on each coursework allowing students to develop their skills throughout the module.

MATH6333: Bayesian Statistics

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: Basic probability

Course Description

This module aims to introduce the Bayesian view of Statistics, stressing its philosophical contrasts with classical statistics, its facility for including information other than 'the data' into an analysis, its coherent approach to inference and prediction, and its decision theoretic foundations.

On completion of the module, students will have a mature appreciation of the relative advantages and disadvantages of a Bayesian and classical approaches and be able to complete all the stages in Bayesian data analysis for statistical models in the exponential family.

Educational Aims

Upon successful completion of this module students will be able to

1. Solve discrete problems using a Bayesian formulation.
2. Understand the advantages that a prior brings to statistical analysis, appreciate some of the difficulties it creates and show how these difficulties may be resolved.
3. Recognise distributions from the exponential family, create their conjugate priors, derive the posterior distribution, the marginal likelihood and the predictive distribution.
4. To appreciate the role of each of the above distributions in integrating statistical reasoning and to be able to use them to solve problems involving inference, choice and prediction both analytically and using the language R.
5. To be familiar with a range of loss functions, to select an appropriate loss function for a given problem and show how the minimisation of the expected loss leads to a rational action.
6. Understand the large-sample asymptotics of posterior distributions, the consequences for the practical validity of Bayesian inference and the contrast between the Bayesian approach and classical approaches.

Outline Syllabus

Students will appreciate how the entire paradigm of Bayesian statistics follows naturally from the rules of joint, conditional and marginal probability introduced in Year 2.

The bulk of the module focusses on continuous problems. The concepts of prior, posterior and evidence are introduced and are related to each other through the likelihood. Students will analyse various point estimates and through their motivations via loss functions, they will create and interpret credible intervals, comparing the interpretation with that of confidence intervals. They will apply appropriate conjugate priors, and understand the difficulty in expressing prior ignorance, and a possible solution.

Students will evaluate different models and conjectures through the model evidence, and appreciate the intuitiveness of this compared with the classical approach of hypothesis testing, as well as the potential issues. They will derive predictive distributions and understand how these naturally incorporate parameter uncertainty, exploring the difference with the classical counterpart.

Students will learn to formulate simple, real-world decision problems in terms of a loss function and discover a unifying Bayesian framework that points to the best decision.

Finally, students will study the asymptotic properties of posterior distributions and appreciate the reassurances these supply, as well as the similarities and differences compared with the asymptotic distribution of the MLE.

Assessment Proportions

The learning strategy for this module supports students to build on the mathematical skills and technical understanding of probability and statistical models developed in Level 4 and Level 5, extending their understanding to appreciate Bayesian concepts. Students will principally acquire and develop knowledge through synchronous lectures and guided independent study.

Technical and theoretical material will be introduced using lectures which complement the course’s comprehensive written materials, while providing alternative, more detailed exposition. The lectures will integrate interactive activities and discussion to promote active engagement with the material. Where appropriate lectures will be supplemented by short, pre-recorded, videos.

Students will put technical material into practice in workshops (even numbered weeks), approaching mathematical and statistical exercises in groups with instructor support. These exercises build towards coursework- and exam-level challenges and provide a formative means for students to assess their understanding, via peer discussion and specimen solutions.

Students will be assessed through both a final written examination (which assesses their consolidated understanding of technical concepts, interpretation of statistical inference, and mathematical reasoning), fortnightly, low stakes, multiple-choice or handwritten coursework and a class test.

MATH6335: Medical Statistics

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic Probability

Course Description

This module aims to give students an overview of the conceptual and theoretical basis of health investigations including issues of study design, measurement of disease, causality and confounding. All topics in the module will be introduced and motivated in the context of the practical challenges faced in medical research and to emphasize the role that statisticians play in the process. The module first covers randomized controlled trials where confounding can be controlled through randomization and then moves onto observational studies where confounding is typically dealt with at the analysis stage. The final part of the module introduces time-to-event data, which are ubiquitous in medical studies but require special techniques due to the presence of censoring. Throughout, students will be encouraged to develop skills in interpreting statistical results in a way that is understandable to clinicians, patients or other end users.

Health and medicine offer some of the most high-profile applications of statistical methods and the medical sector is one of the largest employers of statisticians within the UK. The module therefore serves to highlight this career option to students on the programme and turn highlights the master’s level training in medical statistics available within the university.

Educational Aims

Upon successful completion of this module students will be able to…

1. Evaluate and critique study designs in terms of relevance/practicality for the investigation at hand and also their hierarchy in terms of evidence of causation.
2. Estimate measures of disease, exposure disease associations and associated measures of uncertainty based upon data from observational studies.
3. Show analytically the impact of design upon inference, for example, stratification, case-control analyses, matching.
4. Evaluate and compare diagnostic methods in terms of NPV, PPV, sensitivity and specificity.
5. Identify survival data and describe the effect of censoring in statistical inference.
6. Construct Kaplan-Meier estimates of the survivor function and compare survival between groups using the log-rank test.

Outline Syllabus

Statistical methods play a central role in health research. This module introduces the key study designs used in health investigations, such as randomised controlled trials and various types of observational study.?

Issues of study design are covered from both a practical and theoretical perspective, aiming to identify the most efficient design which adheres to ethical principles and can be carried out in a feasible amount of time, or using a feasible number of patients. Various approaches to controlling for confounding will be discussed, including both design and analysis-based methods. The module will explore different types of response data, including introducing time-to-event data and the resulting challenges presented by censoring.?

Throughout, real-world studies and published articles will be used to illustrate the concepts, and reference will be made to the ICH guidelines for pharmaceutical research and STROBE guidelines for epidemiological studies.

Assessment Proportions

Lectures give an overview of the core concepts of the module and cover the underlying theory in detail through step-by-step derivations using the visualiser.

Workshops and computer-lab sessions are designed to reinforce the content covered in lectures by giving students an opportunity to complete related problems or exercises either by hand or using statistical software. The material in the lecture notes is supplemented by articles from the medical literature which either go into the topics in more depth, provide a different perspective on the material, or help to provide further practical context. Interactive R Shiny apps are also provided to illustrate key topics such as minimization, power calculations, confounding and receiver operator characteristic curves.

The coursework consists of short exercises designed to consolidate the theory covered in lectures and for students to practise correctly interpreting the results of analyses. The module project requires students to write a concise statistical report in a form that aligns with industry standards. The final written examination aims to assess all learning outcomes of the module, but ensures coverage of the theoretical aspects.

MATH6337: Environmental Statistics

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic Probability

Course Description

This module aims to equip students with the knowledge and skills necessary for analysing a variety of environmental data sets. Spatial dependence is a key feature of many environmental data sets, and the Gaussian process will be introduced as a model for continuous spatial processes, such as air temperature. Students will learn about the properties of the Gaussian process and implement this model for spatial data analysis, before investigating methods for point-reference data, such as earthquake or wildfire locations. They will also learn about ideas from natural hazard risk management, which seeks to mitigate the effects of events such as flooding or storms, in a manner that is proportionate to the risk. They will learn basic concepts from extreme value theory, including the appropriate distributions for extremes, and how to use these as statistical models for estimating the probability of events more extreme than those in the data set.

Educational Aims

Upon successful completion of this module students will be able to…

1. Explain the different types of spatial data and formulate a stochastic model for spatially continuous data.
2. Estimate and interpret spatial dependence summaries for spatially continuous data.
3. Understand the key features of the Gaussian process, fit the Gaussian process to spatially continuous environmental data and interpret the model fit.
4. Understand and interpret simple point process models for analysing point-reference data.
5. Understand the importance of modelling the tail of a distribution for risk assessment, which distributions are appropriate for modelling extremes and why.
6. Fit extreme value models to extreme environmental data and interpret the model fit. Estimate and interpret return levels as an environmental risk measure.

Outline Syllabus

The module will begin with an introduction to the concept of spatial dependence between observations of a process at different locations. You will see different examples of spatially-referenced data, with a particular focus on introducing environmental data that can be modelled via continuous spatial processes such as temperature, accumulated rainfall, or wind speeds. We will discuss formulation of a model for such data via spatially-indexed stochastic processes. You will learn about the spatial Gaussian process model and its properties, as well as how to fit this to spatially continuous data and interpret the output. You will learn how to use this model to estimate features of environmental processes at unobserved locations.

You will also learn about techniques for analysing point-reference data, such as earthquakes or wildfire locations. We will introduce point process models, with a focus on the Poisson process, and how to interpret its intensity function.

For natural hazard risk assessment, it is important to understand the behaviour of extremes of environmental processes. You will learn about the distributions that are suited for modelling extremes, specifically the generalized extreme value and generalized Pareto distributions. The course will cover the probabilistic arguments for why they are the appropriate distributions, and how to fit these to data. You will learn to interpret the fitted models and use them to estimate risk measures such as return levels.

Assessment Proportions

Key concepts and technical material and will be covered in lectures, which will also include some practical demonstrations of data analysis to back up the material. When appropriate, there will be opportunities for students to think about and discuss short questions in the lectures.

Practical aspects will be taught in computer labs, using the R programming software. The students will have worksheets to cover data analysis techniques learned in recent lectures and will be equipped to get a good start on the tasks via the practical demonstrations seen in lectures. Students will be encouraged to use RMarkdown (or similar) to record their analyses from labs. Stretch questions will be provided to assist with guided independent study outside of formal contact hours. Two hour labs will be scheduled in even weeks, with a one hour lab in the final teaching week to offer supervised time to discuss stretch questions from earlier labs as needed.

There will be a worksheet available of theoretical questions for formative self-assessment, which will also assist students in exam preparation.

Formal assessment will be via exam (focussing on theoretical components, but also including interpretation of model output), and coursework. There will be two coursework exercises (10% each) based on the two main components of the module, focussing on practical and data analysis skills.

MATH6341: Optimisation for Machine Learning

• Terms Taught: Michaelmas
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: Basic linear algebra

Course Description

Optimisation is the hidden engine behind the remarkable success of modern AI. This module aims to give students a detailed introduction to the mathematical tools that underpin modern approaches to optimisation. It will develop students’ appreciation of how concepts such as convexity, smoothness, duality and curvature inform the design of practical algorithms, and how these ideas are used to train models efficiently at scale. Alongside subject knowledge, you will also gain the knowledge necessary to analyse algorithms rigorously and adapt them to the practical challenges encountered in AI, data science, and beyond.

Educational Aims

Upon successful completion of this module students will be able to

• Formulate machine learning problems as optimisation problems, identifying objective functions, constraints, and regularisation terms, and explaining how modelling choices and data properties influence the resulting optimisation objectives.
• Classify and analyse optimisation objectives, including convex and non-convex objectives, smooth and non-smooth functions, and finite-sum and stochastic formulations, and explain how these properties affect algorithm design and theoretical guarantees.
• Derive, analyse, and critically compare core optimisation algorithms for machine learning, such as gradient-based, second-order, stochastic, proximal, and adaptive methods, including convergence rates, complexity bounds, and practical trade-offs between theory and real-world performance.
• Design, implement, and evaluate optimisation algorithms using industry-standard software libraries, translating theoretical concepts into working code, visualising optimisation dynamics, and assessing algorithmic behaviour on representative machine learning tasks under realistic computational constraints.

Outline Syllabus

The module begins by framing machine learning problems as optimization problems. We introduce empirical risk minimisation, regularisation, constraints, and common loss functions arising in supervised and unsupervised learning. Practical considerations such as data noise, over-parameterisation, stochasticity, and computational budgets are discussed to motivate the structure of real-world objectives and the gap between idealised formulations and deployable models. We then study the mathematical properties of optimization objectives that underpin algorithmic design and analysis. Core functional classes are introduced, including convex and strongly convex objectives, smooth and non-smooth functions, composite objectives, and finite-sum and expectation-based formulations. Students learn how convexity, smoothness, curvature, and regularity assumptions influence convergence guarantees, and what variations of these assumptions arise naturally in modern machine learning.

Building on this foundation, the module develops common algorithmic classes of optimizers used in machine learning. We derive gradient descent, stochastic gradient methods, momentum and acceleration schemes, proximal algorithms, and adaptive methods, and analyse their convergence rates and iteration complexities under different objective assumptions. Emphasis is placed on understanding the modelling choices behind stochastic gradients, mini-batching, step-size schedules, and variance reduction. The module also focuses on implementation and experimentation. Students design and implement optimisation algorithms using standard software libraries, visualising optimization performance and comparing methods on representative machine learning tasks. Through guided coding exercises and a summative project, students investigate trade-offs between convergence speed, stability, generalisation, and computational efficiency, and learn to translate theoretical insights into practical algorithmic decisions.

Assessment Proportions

The module is delivered through lectures and example classes. Lectures introduce theoretical concepts and solution methods, supported by worked examples. Example classes provide guided practice with problems, reinforcing understanding and preparing students for assessment tasks. Assessment will consist of

• Written examination (80%), assessing conceptual understanding and ability to employ taught analytical techniques under timed conditions.
• Coursework (20%), comprising four practical programming assignments to implement optimisation algorithms in Python and conduct numerical experiments on representative machine learning problems.

Coursework provides opportunities for feedback and skill development, supporting students in mastering complex problem-solving techniques and assessing progress throughout the module.

MATH6345: Industry-inspired Project

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic probability / some computational background

Course Description

This module aims to give students the opportunity to apply the knowledge and skills gained through their degree to tackle a substantial problem from industry, the government or the third sector. At the same time, it provides a platform for students to improve their research, group working, writing and presenting skills.

Educational Aims

Upon successful completion of this module students will be able to…

1. Work independently and as part of a group, under supervision, researching and implementing possible routes to the solution of a substantial problem.
2. Work efficiently as part of a team, understanding their own strengths and those of the other team members.
3. Appreciate the importance of mathematics, mathematical modelling and/or artificial intelligence in solving substantial problems in industry, the government or the third sector.
4. Prepare a report with an abstract, good structure, appropriate citations and professional-standard graphics.
5. Design and deliver an oral presentation on a mathematics-related topic and create an informative, readable poster on a related topic.
6. Understand and be able to follow good academic practice when preparing reports, presentations and posters.

Outline Syllabus

Problem solving, gathering relevant information, writing, presenting and working as part of a group are essential in almost every professional career. Students will develop and practice these skills as they undertake an industry-linked group project and an individual extension. They will also integrate prose, mathematics, computer code and its output into a clear, well-structured and appropriately linked Markdown document - a key skill in reproducibly documenting their research.

• Writing a scientific report; possible formats and styles.
• Abstracts; their purpose and how to write them.
• Citing and searching literature.
• Creating and delivering a good presentation.
• Useful Markdown features for reports and slide presentations.
• Producing high-quality graphics.
• Designing and creating posters: important features and the Markdown mechanics.
• Tackling a substantial real-world problem as part of a group and individually, working independently, under the supervision of an academic.

Assessment Proportions

In the first week of the module, students will attend lectures and participate in computer labs (6 hours of contact time in total) where will they learn and practice writing an abstract, searching literature and citing appropriately, producing high-quality graphs and using various advanced Markdown features that enable them to transfer these concepts to their own document. This sets them up to complete the first two pieces of coursework, which will be due shortly after. The assessments will be formative in their design but carry summative credit (totalling 15%). In Week 2 the students will start on their group project and, over the next few weeks they will also attend a session on writing and structuring a larger report and another recapping (from Years 1 and 2) the key points in creating a good presentation. The group presentations will occur around Week 5 or 6; these are worth 20%. Assessment may purely be at the group level, with feedback from a pair of academics. However, if automated IT systems permit, we are hoping that some of the marks and feedback can be from individual-level marking and feedback by all the other students in the session, moderated by an academic, as this encourages more student engagement all round. The group project report (worth 40%) will be due several weeks before the end of the module, probably around the start of Week 9. Around the same time as handing in the group report, students will attend a two-hour session on creating a poster. This, together with their own guided research over the next few weeks on a student-specific project extension, will enable them to complete the final poster-based assessment (25%).

MATH6346: Dynamic Modelling

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic computational skills

Course Description

This module aims to introduce students to Markov jump processes – as stochastic continuous-time dynamic models – and demonstrate how they can be related to deterministic differential equations, and to show how to apply these models to real-world dynamical systems. It is being offered as part of the undergraduate maths programme to build on knowledge of modelling dynamical systems from previous years (e.g. on ODEs and their applications) and to develop students’ skills in programming and modelling of real-world systems.

Educational Aims

Upon successful completion of this module students will be able to…

1. Justify the need for Markov jump processes (MJPs) as a fundamental class of dynamic mathematical model within and beyond the natural sciences;
2. Investigate behaviour of MJPs using analytical and numerical simulation methods;
3. Relate MJPs to large-population limit approximations, including differential equations;
4. Employ MJPs and differential equations to model and understand a range of real-world dynamical systems;
5. Critique both their own and others' choice of mathematical model for specific real-world examples.

Outline Syllabus

Models of dynamical systems are fundamental to much of our understanding of the physical and natural world. This module will introduce a new class of continuous-time dynamic model, Markov jump processes, and investigate how they can be used to model real-world systems such as those found in biology, ecology and epidemiology. Students will learn how to simulate Markov jump processes and will study methods for understanding their properties and behaviours. They will also see how Markov jump processes, which are stochastic, can be related to deterministic differential equations and how this can be used to aid analysis and understanding of dynamical systems.

Assessment Proportions

The learning and teaching strategy for this module is focused around active learning on the part of the students. Lectures will develop students understanding of the theory underlying Markov jump processes and ordinary differential equations and their ability to apply it to real-world problems by introducing and explaining fundamental concepts and methods and interactively going through examples in which they are applied. The timetable will alternate between weeks with 3 one-hour lectures and weeks with 1 one-hour lecture and 1 two-hour workshop, to allow sufficient time to do small group work and give feedback. Students learning in lectures will be supported through reading the notes, seeing additional explanations of the theory on a visualiser, watching and participating in live code demonstrations, questioning and answering, and doing anonymous online quizzes (with discussion of answers). The last activity will provide a means of formative assessment and feedback to enable students to gauge their learning and where they have gaps in understanding. Comprehensive module notes will be provided, and lectures will consist of interactively going through these with discussion and additional explanation. Students will be expected to read the notes in advance of lectures and come prepared to discuss them. Practical sessions will consist of working in small groups on analytical and numerical simulation problems designed to build understanding of theory and putting it into practice, and presenting solutions to the class. Students will be formatively assessed in this way and given feedback to support them to complete the coursework. The course will use the R and Python programming languages, and students will have a choice of which programming language they want to use (or can use both). Students will be encouraged to keep an electronic lab book (e.g. as a Rmarkdown document or Jupyter notebook) to log their work and learning. Students will be assessed via fortnightly coursework problem sheets (20%) that will involve coding tasks as well as written answers, an extended piece of group coursework on applying what they have learnt to model a real-world system whilst developing collaborative working skills (20%), and a final written exam (60%), to assess individual achievement of all the ILOs.

MATH6347: Mathematics of Generative Modelling

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Basic Linear algebra

Course Description

This module offers a rigorous introduction to the mathematics underlying modern generative AI, with a particular focus on diffusion and flow-based models. Building on students’ prior knowledge of probability, analysis, and differential equations, the module develops a unified view of how probability distributions evolve in time through ordinary and stochastic differential equations, partial differential equations for density evolution, and transport on spaces of probability measures. Motivated by state-of-the-art image, audio, and scientific generative models, the module systematically derives denoising diffusion models and flow-matching methods from first principles, explaining how continuity equations, Fokker–Planck equations, reverse-time SDEs, and probability-flow ODEs lead directly to practical generative AI algorithms. Alongside the theoretical development, students will implement small-scale generative models in Python, gaining practical experience with score matching, numerical SDE/ODE solvers, and the design choices (noise schedules, guidance, solver accuracy) that determine sample quality, stability, and computational cost. By the end of the module, students will be equipped to read contemporary research on generative modelling, reproduce key derivations, and build and assess prototype models on low-dimensional and image-based datasets.

Educational Aims

Upon successful completion of this module students will be able to:

1. Explain and derive the core mathematical objects that govern probability evolution in generative models, including continuity and Fokker–Planck equations, reverse-time SDEs, and probability-flow ODEs.
2. Derive and analyse the training objectives and sampling algorithms for denoising diffusion models and flow-based models (e.g. score matching, denoising score matching, flow matching), and relate these to maximum likelihood estimation.
3. Design, implement, and critically evaluate small-scale generative models in Python, including discrete- and continuous-time diffusion models and simple flow-matching architectures, selecting appropriate solvers, noise/step-size schedules, and evaluation metrics (e.g. NLL, bits-per-dimension, coverage) under computational constraints.

Outline Syllabus

The module begins with the mathematical description of how probability distributions evolve over time. We introduce continuity equations and the Fokker–Planck equation, and show how they correspond to underlying stochastic dynamics. From this, we derive reverse-time stochastic differential equations and the associated probability-flow ODE, establishing the link between generative modelling and dynamical systems. We then develop denoising diffusion models in discrete time: forward noising processes, reverse-time parameterisation, and the standard training objective. Students will see how this objective approximates maximum likelihood, and how score matching and denoising score matching arise as practical surrogates. Continuous-time formulations and their numerical realisation via Euler–Maruyama, predictor–corrector and ODE-based samplers are covered, together with the role of noise and step-size schedules. In parallel, we study flow-based models: continuous normalising flows and flow matching. We interpret these as fitting a velocity field along a path between base and target distributions and connect them to optimal transport and Schrödinger bridges. Comparisons between diffusion-style and flow-style samplers are emphasised. The final part of the module focuses on fast sampling and practical deployment. Topics include sampler distillation and consistency models, guidance mechanisms, and the trade-offs between sample quality, mode coverage, stability, and computational cost. Throughout, short coding exercises and a summative project require students to implement prototype samplers on low-dimensional data and simple image datasets and to justify their design decisions quantitatively.?

Assessment Proportions

The module is designed to balance rigorous mathematical development with hands-on implementation. Lectures provide the theoretical backbone: we derive continuity and Fokker–Planck equations, reverse-time SDEs, and probability-flow ODEs on the board, and then show how these results translate into the core algorithms of denoising diffusion and flow matching. Short in-lecture derivation exercises and worked examples consolidate these ideas. Weekly labs and workshops focus on implementation. Students will code simple SDE/ODE solvers, implement score matching and denoising objectives in low-dimensional settings, and progressively build small diffusion and flow-matching models using Python and standard libraries (e.g. PyTorch). These sessions emphasise the connection between the mathematics developed in lectures and practical design choices: parameterisations, noise schedules, solver settings, and guidance strategies. The assessment strategy aligns with the programme’s emphasis on strong mathematical foundations, reproducible computation, and critical interpretation of results. It deviates the standard assessment structure within the school, through a computational project which aims to foster computational?fluency?and prepare students for data-driven roles beyond Lancaster. An end-of-module exam tests students’ understanding of key derivations, definitions, and analytical results. A substantial group (or small-team) project requires students to implement and compare generative models on a modest dataset, documenting their methodology, mathematical reasoning, and empirical findings in a Jupyter notebook and short technical report. Regular online quizzes provide low-stakes formative feedback on core concepts and definitions throughout the term. Formative feedback is provided through lab interaction, quiz solutions, and informal discussion of project plans. Summative feedback on the project and exam highlights both mathematical strengths and areas for improvement in modelling and implementation.

MATH6365: Mathematical Education Placement

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level real analysis

Course Description

Build on the knowledge gained in the MATH6360: Mathematical Education module and put that theory into practice. Designed with employability in mind, this module will support students as they participate in a semester-long part-time placement in a local primary or secondary school. During that placement students will have the opportunity to take part in classroom observation and assistance, the development of classroom resources, the provision of one-on-one or small group support and possibly even teaching sections of lessons to the class as a whole.? This module is intended for students who are interested in, or curious about, a career in teaching or educational research. During regular meetings, the module convenor will help you to develop your skills in critical reflection as you relate your placement experiences with theoretical frameworks. Alongside your personal development, it is also our intention that you provide genuine assistance to local teachers, by bringing your mathematical knowledge and enthusiasm into their classroom to help enthuse future mathematicians.?

Educational Aims

Upon successful completion of this module students will be able to…

1. Demonstrate experience of teaching methods, classroom management and lesson preparation for Mathematics teaching in schools
2. Demonstrate awareness of the different needs of individuals in a learning situation
3. Develop and present mathematical learning materials suitable for the school in which they are placed and for the age group which they are teaching
4. Reflect critically on their classroom experiences and the effectiveness of Mathematical teaching and learning methods
5. Relate their experiences to relevant educational literature
6. Work collaboratively with teachers, classroom assistants, and other students

Outline Syllabus

The student will be expected to spend half a day a week for most weeks?of Semester 2. It is intended that this will be, in the first?instance, a practical and experiential module,?and wherever possible or appropriate, students’ own?ideas and learning will feed back into the content of their activity as they?become more experienced.?This will involve classroom observation and assistance, the development of classroom resources, the provision of one-on-one or small group support and?possibly the opportunity to teach sections of lessons to the class as a whole. Prior to the start of the placement, the student will be required complete safeguarding and child protection training. While in school, the teachers will act as the main source of guidance?but, in addition, students will have regular tutorials with a module convenor. This module is primarily intended for students who are considering a career in teaching. Suitability may be assessed by interview prior to enrolment on this module.

Assessment Proportions

The regular taught sessions will primarily delivered through small group tutorials. The main summative assessment in this module will be an essay, in which students will be expected to critically reflect on their classroom experience, and relate it to educational literature. There will also be an assessed logbook, which students will be expected to complete throughout the duration of their placement, and will include gathering some relevant literature references. And also students will be expected to design an annotated lesson plan, potentially suitable for the classroom in which they have been placed. This lesson may or may not be delivered to pupils, but its design will nevertheless be assessed, as will their justification for their design choices.

MATH7410: Operators and Spectral Theory

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level real analysis

Course Description

This module provides an introduction to an important topic in the field of modern analysis. A bounded operator is a generalisation of a matrix or linear transformation, where the underlying space is no longer required to be finite-dimensional. Fundamental ideas from analysis are used to extend well-known algebraic concepts, such as eigenvalues and the adjoint, to the infinite-dimensional setting. Students will encounter important families of operators, including unitary, self-adjoint and non-negative operators. The notion of a function of an operator will be extended from polynomial to continuous functions, provided that the operator is self-adjoint. The course culminates in a generalisation of a familiar result from linear algebra, that any Hermitian matrix can be diagonalised. An appropriate version of this statement is shown to hold for any compact self-adjoint operator.

Educational Aims

Upon successful completion of this module students will be able to

1. prove that an operator is bounded and find its norm, adjoint and, if invertible, inverse.
2. identify orthogonal projections, isometries, partial isometries and unitary operators, through their algebraic and geometric characterisations.
3. explain the rudiments of the spectral theory of Banach algebras.
4. calculate the spectrum of various operators.
5. describe and apply the continuous functional calculus for a bounded self-adjoint operator.
6. explain the concept of operator order, and show that certain operators are non-negative.
7. state the spectral theorem for compact self-adjoint operators.
8. develop written and oral presentation skills.

Outline Syllabus

This module begins with a review of some basic results about matrices, corresponding to finite-dimensional operator theory. We will also review some basic Hilbert space theory. We will then transition to bounded operators on Hilbert spaces, which are a generalisation of matrices. Notions that the students are already familiar with in the finite dimensional case will be introduced for bounded operators on infinite dimensional Hilbert spaces. This includes the adjoint of an operator, the kernel-adjoint-range relation, isometries, unitary and normal operators and partial isometries. We will continue with discussing spectral theory for bounded operators. In particular, we will examine the spectrum and the resolvent set of an operator. We will also show that the spectrum of a bounded operator is non-empty and compact. We will then discuss the question, how to define a function of a bounded self-adjoint operator. This will lead us to developing the continuous functional calculus for a bounded self-adjoint operator. This raises the question, how the spectrum of an operator is changed when we apply a function to it. This is the content of the spectral mapping theorem, which will be proved. A matrix is called non-negative, if all its eigenvalues are non-negative. The concept of non-negativity will be extended to bounded self-adjoint operators. This allows us to define an order on these operators. Moreover, by the functional calculus, we will be able to find a square root of a non-negative operator. This allows us to also find a polar decomposition for bounded linear operators. A familiar result from linear algebra states that any Hermitian matrix can be diagonalised. We will show that an appropriate version of this statement is shown to hold for any compact self-adjoint operator.

Assessment Proportions

This module will primarily be taught using lectures, supported by comprehensive lecture notes written by the lecturer. In the lectures, the lecturer will motivate ideas, definitions and results and put them in context, mathematically and historically. The lecturer will also show how proofs are constructed in real time and explain the reasoning behind calculations. Some parts may be covered in the lecture notes only. Students will be encouraged to participate actively in the lectures by including time for questions. The module will have weekly short pieces of formative assessment in the form of written coursework. The frequent assessments will ensure that students engage actively with the course content on a regular basis and practise and apply the taught material, and they will receive prompt written feedback on their work, ensuring that any issues can be addressed quickly. Solutions to coursework exercises are handed in in Weeks 2-11, marked and returned in the next workshop. The first half of the workshop is used for students to present their coursework to the whole class and the lecturer. Students will be chosen in turns, and it is estimated that in every class 2-4 students will present their solution to one specific question and over the course of the module every student will present 3-5 times. The purpose of this component is to improve student engagement, develop presentation skills and to test authenticity of the handed in coursework. The second half of the workshop will be used to solve questions related to the material taught in the most recent lectures. Students will work on their own or in small groups, supported by the lecturer. Students can also clarify questions about feedback on their latest coursework. The lecturer will also hold weekly office hours, where students can ask questions and receive individual help.

MATH7415: Measure and Integration

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level (Linear) algebra

Course Description

The aim of this course is to introduce general measure theory and the Lebesgue integral. The course features a classical approach to the construction of Lebesgue measure on the line and in higher dimensions and measures on more general domains, and to the definition of the integral. Standard convergence theorems and the Radon-Nikodym theorem are proved. Measure theory is an important pillar of mathematics which forms the basis of advanced probability, hence it deserves to be offered at least to Year 4 students. The connections to probability theory will be highlighted and elaborated.

Educational Aims

Upon successful completion of this module students will be able to:

1. Use the definition and fundamental properties of the outer Lebesgue measure, Lebesgue measurable sets, and Lebesgue measure to prove further mathematical facts about these objects.
2. Engage with the general measure theoretic framework, namely, the abstract notions of sigma-algebra and measure and apply them to prove further theorems and construct examples and non-examples
3. Apply and justify the use of the fundamental integration theorems, such as the Monotone Convergence Theorem and the Dominated Convergence Theorem, to compute limits of specific Lebesgue integrals.
4. Apply advanced topics, which includes the space of square integrable functions, the L^2 space, Fubini’s theorem, for higher-dimensional Lebesgue integrals and Radon-Nikodym's theorem.
5. Develop concrete examples (and non-examples) for the abstract mathematical definitions introduced in this module.
6. Develop ease working with abstract mathematical notions.
7. Develop written and oral presentation skills.

Outline Syllabus

The idea of a measure is motivated using volumes in R^d and formalised in the Lebesgue outer measure and the actual Lebesgue measure. We review metric and topological spaces, which then allows us to define the concept of Borel sigma algebras and general sigma algebras on metric spaces. Outer measures on Borel sigma algebras are introduced and the corresponding measures defined using Caratheodory’s characterisation, leading us to measure spaces. We show that Lebesgue measures are special cases of these but look at several other examples, too. We next look at real-valued functions on a measure space and define what it means for a function to be measurable. Once we have that, we can introduce the integral of a function on a measure space, starting with simple functions, then looking at nonnegative functions by approximating them with simple functions from below and eventually arbitrary R-valued measurable functions. We will prove Fatou’s lemma and standard convergence theorems, namely the monotonic and the dominant convergence theorem about how to compute the integral of a limit of a sequence of functions and apply this to examples. We discuss several specific examples. We introduce the L^2-space as an example of a Hilbert space, which connects us with MATH6315. We then look at various theorems about the integral, e.g. we prove Fubini’s theorem about product measures. We introduce transformations of measures and show how to compute the corresponding integral using the determinant rule, which is a generalisation of the method of substitution. We will also prove Radon-Nikodym's theorem, leading to density functions. We apply this to several interesting measures coming from probability theory such as Gaussian measures, which connects well to MATH5230 and MATH7434.

Assessment Proportions

This module will primarily be taught using lectures, supported by comprehensive lecture notes written by the lecturer. In the lectures, the lecturer will motivate ideas, definitions and results and put them in context, mathematically and historically. The lecturer will also show how proofs are constructed in real time and explain the reasoning behind calculations. Some parts may be covered in the lecture notes only. Students will be encouraged to participate actively in the lectures by including time for questions. The module will have weekly short pieces of formative assessment in the form of written coursework. The frequent assessments will ensure that students engage actively with the course content on a regular basis and practise and apply the taught material, and they will receive prompt written feedback on their work, ensuring that any issues can be addressed quickly. Solutions to coursework exercises are handed in in Weeks 2-11, marked and returned in the next workshop. The first half of the workshop is used for students to present their coursework to the whole class and the lecturer. Students will be chosen in turns, and it is estimated that in every class 2-4 students will present their solution to one specific question and over the course of the module every student will present 3-5 times. The purpose of this component is to improve student engagement, develop presentation skills and to test authenticity of the handed in coursework.

The second half of the workshop will be used to solve questions related to the material taught in the most recent lectures. Students will work on their own or in small groups, supported by the lecturer. Students can also clarify questions about feedback on their latest coursework. The lecturer will also hold weekly office hours, where students can ask questions and receive individual help.

MATH7420: Galois Theory

• Terms Taught: Michaelmas 
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level algebra

Course Description

This module aims to introduce the core ideas for Galois Theory, which is the study of roots of polynomials and symmetries of these roots. In the modern understanding, this is understood to be broadly equivalent to the study of field extensions. Galois Theory occupies a central position in abstract mathematics, connecting various other fields such as group theory, number theory and algebraic geometry. The main aim of the course will be to understand The Fundamental Theorem of Galois Theory and various consequences of it, such as Galois’s criterion for solvability by radicals and the insolubility of the general quintic polynomial.

Educational Aims

Upon successful completion of this module students will be able to:

1. carry out basic calculations involving polynomials and field extensions, such as checking irreducibility, computing the degree and finding the basis of a field extension, finding the minimal polynomial of a field element, constructing the splitting field of a polynomial, and checking whether a field extension is normal or separable;
2. prove basic statements about finite, algebraic, transcendental, normal, separable and Galois field extensions;
3. correctly state the Fundamental Theorem of Galois Theory and apply the ideas from its proof to solve mathematical problems of appropriate difficulty;
4. compute the Galois group of a field extension of small degree, and explicitly describe the Galois correspondence for a Galois extension;
5. demonstrate understanding of elementary facts about solvable groups, composition series and simple groups (in particular, that the alternating group of degree n is simple for n>4);
6. determine the Galois group of a polynomial: by hand for small degree, and using algebra software for degrees up to about 10;
7. prove basic statements about radical extensions, normal closures, cyclic and abelian extensions;
8. correctly state Galois's criterion for solubility by radicals and apply both the result and the ideas from its proof to solve related problems of appropriate difficulty, including the insolvability of the general equation of degree >4;
9. prove that an irreducible rational polynomial of prime degree with two non-real roots is not solvable by radicals, and apply the ideas of the proof to solve similar problems of appropriate difficulty.

Outline Syllabus

Field extensions: simple, finite, algebraic and transcendental extensions, classifying extensions, degree of an extension, splitting fields, separable and normal extensions. The Galois correspondence: the Galois group of an extension, Galois extensions, the fundamental Theorem of Galois Theory. Finite group theory: solvable groups, simple groups, composition series. Galois groups of polynomials and computational methods for determining Galois groups. Solvability by radicals: radical extensions, cyclic and abelian extensions, Galois's criterion for solvability by radicals, the field of symmetric rational expressions, solution of general quadratics, cubics and quartics, rational equations which are not solvable by radicals.

Assessment Proportions

A complete set of lecture notes will be provided to students in advance of the module. These notes, which will largely focus on the abstract theory (with a few examples), will be explained and expanded on during the lectures (2 per week). Students will learn to apply the knowledge gained from the lecture notes in the examples classes/workshops (1 per week), which will involve a mixture of supervised study time and worked examples at the whiteboard. Continuous assessment will be provided via five sets of coursework exercises, set approximately once a fortnight. To help the students prepare for the coursework, unassessed “workshop exercises” will frame the examples classes/workshops. Towards the end of the course, a special session will focus on computational methods for determining Galois groups.

MATH7421: Number Theory

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level algebra

Course Description

This module aims to introduce the use of analytic and algebraic techniques for studying problems in number theory. For example, we will see how methods from analysis can be useful in studying the distribution of prime numbers, the asymptotics of arithmetic functions and the solubility of Diophantine equations. We will also study the solubility of Diophantine equations using methods from algebra, particularly the concept of unique factorisation in certain rings (or lack of). By the end of this module, students will have developed a working knowledge of analytic and algebraic number theory which will prepare them for further study in the area.

Educational Aims

Upon successful completion of this module students will be able to…

1. Precisely formulate questions relating to the natural numbers which are central to modern research.
2. Use analytic techniques to make partial progress towards the above questions and engage with modern research in analytic number theory.
3. Use algebraic techniques to make partial progress towards the above questions and engage with modern research in algebraic number theory.

Outline Syllabus

The first half of the module introduces the use of real and complex analysis in the study of problems concerning integers. A main source of such problems will concern prime numbers: their distribution amongst the integers and their behaviour under addition. By studying arithmetic functions and associated Dirichlet series, we will demonstrate the infinitude of the primes and obtain estimates for how frequently they occur. Other topics may include: the Hardy-Littlewood circle method (for demonstrating the solubility of Diophantine equations); probabilistic methods (such as the Hardy-Ramanujam estimate for the number of divisors); sieve methods (for instance bounding the number of twin primes from above); combinatorial methods (e.g. the use of Schnirelmann density to show that every positive integer can be written as a sum of primes). The second half of the module introduces the use of algebraic techniques in the study of Diophantine equations. The aim is to describe the arithmetic of algebraic number fields – generalisations of the integers where unique factorisation may fail. Students will learn to prove theorems about integral bases, and about unique factorisation into ideals. They will learn to calculate class numbers, and to use the theory to solve simple Diophantine equations.

Assessment Proportions

The learning strategy for this module is designed to enable the student to develop their techniques for solving problems in number theory, and to communicate these solutions in a rigorous and readable manner. The problem-solving strategies will be modelled in lectures and written materials. The student will practice such problems for themselves using worksheets, assisted by experienced workshop tutors. The student will have the opportunity to submit a selection of their answers for feedback, helping them improve their techniques in number theory.

MATH7425: Lie Groups and Lie Algebras

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level algebra

Course Description

The first time you meet groups, they tend to be finite groups: the symmetry group of a triangle, or a cube, or a permutation group. Lie theory is the study of continuous groups of transformations, like rotations of 2-, 3-, 4- or higher-dimensional spaces. It underpins most of modern geometry and particle physics, with applications from solving differential equations to understanding matter made of quarks to classifying polynomials in pure algebra. This module will explore the foundations of this powerful subject, and equip students with the skills to perform the complex calculations needed to understand the applications.

Educational Aims

Upon successful completion of this module students will be able to…

1. Determing the validity of simple, previously unseen claims in the theory of Lie groups and Lie algebras, and verify this by finding proofs or constructing counterexamples.
2. Interrogate proofs in Lie theory, determining whether they are valid, and identifying or fixing gaps.
3. Perform computations with Lie groups, Lie algebras and their representations.
4. Apply the results of computations to prove theorems in other areas of mathematics.

Outline Syllabus

The focus of the module will be on matrix Lie groups and Lie algebras, since this makes the subject more immediately accessible and allows students to quickly grasp the fundamental ideas. We will cover:

• Concepts of Lie groups and Lie algebras, concrete examples like the unitary, special unitary, orthogonal and rotation groups.
• The relationship between a Lie group and its Lie algebra and the exponential map. The Baker-Campbell-Hausdorff formula.
• Spin: SU(2) and rotations.
• Representations of Lie groups and Lie algebras. The case of U(1) and weight spaces.
• Applications to other areas including invariant theory, geometry, physics.
• The examples of SU(2) and SU(3). Root systems.

Assessment Proportions

Unlike most other mathematics modules, this module is 100% coursework based. This is because Lie theory is a vast topic, and students come to it with a variety of backgrounds (theoretical physics, algebra, geometry, analysis,…). Project work can be tailored to a student’s individual needs, and they can focus on the aspects of the module which interest them most, rather than all learning exactly the same material for an exam. Technical lectures together with online and physical supporting materials (e.g. notes, videos) will introduce students to the basic ideas of the module. Lectures will be broken up with engagement activities to encourage active learning and to help scaffold students. Frequent example-based workshops will give the students more space to explore the ideas in depth: they will work together to build the key skills listed in the learning outcomes: carrying out computations, interrogating proofs and exploring hypotheses through proof and counterexample. Peer-learning activities in workshops will help them to build confidence and communication skills as well as their mathematical ability. Students will be assessed through four fortnightly courseworks over the module and a more substantial final project. Feedback on the courseworks will also play a formative role, helping them to fine-tune their skills over the semester. For the project students will be expected to produce an individual written project and participate in an oral assessment, giving them an opportunity to demonstrate their ability to communicate advanced mathematics in a range of formats.

MATH7426: Combinatorics

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability /statistics

Course Description

Combinatorics is a core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous (for example graphs, lattices, designs and codes). While combinatorics is a huge subject - with deep and important connections to many areas of modern mathematics - it is a very accessible one. This module aims to introduce students to the fundamental topics of combinatorial enumeration (sophisticated counting methods) and combinatorial design theory (Latin squares and block designs). Students will also be introduced to additional combinatorial topics from areas such as set systems, error-detecting and error-correcting codes, and combinatorial geometry. Throughout the course, students will develop an understanding of how combinatorial results and methods may be applied both within pure mathematics and in wider practical contexts.

Educational Aims

Upon successful completion of this module students will be able to…

1. solve various counting and partition problems using the basic counting principles of combinatorics, the fundamental counting coefficients, generating functions and other advanced counting methods.
2. understand the theory behind combinatorial designs and construct examples such as Latin squares and block designs.
3. analyse and construct combinatorial structures such as set systems, codes, or discrete geometric objects, and appreciate their uses in solving problems in both theoretical and applied contexts.
4. construct rigorous proofs of combinatorial results and demonstrate an understanding of the relationships between different combinatorial concepts.

Outline Syllabus

This module introduces students to central themes in modern combinatorics, combining classical and advanced counting techniques with structural and applied aspects of discrete mathematics. The first part of the module will cover fundamental methods in combinatorial enumeration, including the basic principles of counting, the use of fundamental counting coefficients such as binomial and Stirling numbers, recursive relations, and an introduction to generating functions and their applications in solving counting problems. The second core theme is combinatorial design theory, where students will study the structure and construction of objects such as Latin squares, block designs, and Steiner systems. These topics are central to both pure and applied combinatorics and provide concrete examples of how combinatorial ideas manifest in structured systems. Beyond these core topics, the module will introduce further areas of combinatorics. Students will encounter important results on set systems, such as those involving intersecting families and matching conditions. There will also be coverage of coding theory, including the basic ideas behind constructing and analysing error-detecting and error-correcting codes. Finally, selected topics from combinatorial geometry will be explored, focusing on classical theorems about convex sets and the combinatorial structure of polytopes.

Assessment Proportions

This module aims to develop students’ deep understanding of combinatorial theory and practice through a blend of lectures, small-group workshops, coursework, and a final exam. Combinatorics, as a core part of discrete mathematics, provides an ideal setting for students to practise logical reasoning and structured argumentation. Lectures will be the main vehicle for teaching and will be supported by detailed written materials. They offer a space where mathematical reasoning can be made explicit: even short proofs or counting arguments often involve subtle chains of logic, and the lectures aim to unpack these step by step. Emphasis will be placed not only on results and techniques, but also on how arguments are constructed and why particular approaches are taken. Carefully chosen examples and problems will illustrate key ideas.

Students will consolidate their understanding through weekly coursework, alternating between written assignments and online quizzes. There will be a total of five written assessments and five online quizzes. (Each written assignment is expected to take approximately 3 hours to complete, while each online quiz should take around 1 hour.) These allow students to engage regularly with problem-solving and proof-writing, and to receive timely feedback on their work. This learning will be further reinforced through six workshops focused on working through examples and solving problems collaboratively. Occasionally, the coursework will pose more challenging or open-ended problems that would not be appropriate for timed exams; accordingly, the coursework contributes 20% of the final mark, supporting learning without undue pressure. A two-hour final examination will assess the full range of module content, focusing on students’ ability to apply techniques and reason independently under exam conditions. Both coursework and the final exam are designed to align with the learning outcomes and to give students multiple opportunities to develop and demonstrate proficiency in combinatorial thinking.

MATH7430: Estimation and Inference

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability/ statistics

Course Description

This module will unpack the most important frameworks for modern statistical inference. Broadly, statistical inference refers to the process of training a statistical model by using data to both estimate the unknown model parameters and quantify the uncertainty in the estimates. Alongside model design and construction, inference is core to both the development of innovative statistical techniques and their application in data science and AI. This module will equip students with an understanding of both frequentist and Bayesian inference, connecting the two through the likelihood function. Similarities and differences between the two methods will be thoroughly examined. The module aims to build proficiency both in the implementation of each approach and the interpretation of the output. Upon successful completion of the module, the student should have the ability to undertake statistical inference on a very broad class of statistical models.

Educational Aims

Upon successful completion of this module students will be able to…

1. Describe the mathematical and practical differences between frequentist and Bayesian inference, critique and justify the use of each.
2. Identify and implement all steps required to obtain parameter estimates and confidence or credibility intervals for a variety of statistical models.
3. Interpret the output of a fitted statistical model and use this output for hypothesis testing and model selection.
4. Undertake statistical prediction using either a frequentist or Bayesian framework.
5. Identify when an analytical solution is unavailable and, in simple instances, construct a suitable solution via numerical optimisation or Monte Carlo, as appropriate.

Outline Syllabus

This module will cover both likelihood and Bayesian inference. Although philosophically quite different, both frameworks provide a mathematically principled way to learn about statistical models from a sample of data. Statistical inference is not just an essential part of any statistical analysis but is also a component of many machine learning and AI algorithms. You will first be introduced to likelihood inference and the asymptotic sampling theory which makes this such a versatile framework for parameter estimation, uncertainty quantification and prediction. You will practice constructing likelihood functions for a variety of models and using these to obtain maximum likelihood estimates and confidence intervals, either graphically, analytically or numerically. For multi-parameter models, you will also cover profile likelihoods and model selection using the deviance test. The module then switches to Bayesian inference, where you will become familiar with prior, posterior and posterior predictive distributions, obtaining these analytically wherever possible. You will explore different types of prior, including Jeffrey’s prior, conjugate priors and uninformative priors. You will then use Bayesian loss functions to derive suitable Bayesian point estimates and credibility intervals, allowing a direct comparison with the maximum likelihood approach covered earlier in the module. Lastly you will discover how Monte Carlo simulation can be used for parameter estimation when classical methods fail. The understanding that you acquire over the course of this module will underpin the rest of your statistical training.

Assessment Proportions

The learning strategy for this module is designed to enable the student (a) to understand the mathematical theory statistical inference framework, (b) to correctly implement both likelihood and Bayesian inference for a variety of statistical models, analytically and using statistical software and (c) to communicate their choices and findings using both technical and non-technical language. Technical components will be taught using lectures, supported where appropriate by short, pre-recorded, videos, and comprehensive written materials. Practical elements will be delivered in computer labs, using worksheets designed to consolidate theory and guide students through putting this theory into practice. The course will use the statistical software package R, though students may opt to use Python instead. To encourage students to analyse and evaluate their modelling choices and findings, they will be encouraged to keep a running lab-book throughout the course using a suitable scientific tool (e.g. markdown, Rmarkdown, or LaTeX). Students can assess their formative understanding using model solutions provided after each lab. Students will be assessed through both a final written examination (to cover the theoretical elements) and two short pieces of coursework to cover different practical elements and written communication (one on maximum likelihood and one on Bayes).

MATH7431: Advanced Statistical Modelling

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: Advanced ststistical Modeling 

Course Description

This module aims to equip students with a comprehensive understanding of modern statistical modelling approaches, spanning classical parametric and contemporary nonparametric techniques. Beginning with linear and generalised linear regression, the module advances to flexible modelling frameworks such as Generalised Additive Models (GAMs), splines, and smoothing methods commonly used in time series and spatial statistics. Students will gain practical experience with nonparametric machine learning methods, such as classification and regression trees, random forests, and boosting, alongside model selection strategies such as cross-validation. The ultimate goal is to develop students' abilities to construct and assess statistical models for complex, nonlinear, and high-dimensional data, thereby preparing them for advanced research or applied analytical roles. Additionally, the module offers students opportunities to enhance their transferable skills, including teamwork, oral presentation, critical thinking, and academic research writing.

Educational Aims

Upon successful completion of this module students will be able to:

1. Demonstrate a mathematical understanding of fundamental statistical models, for instant linear and generalised linear models. Formulate, interpret, implement, and critically evaluate them for various data types.
2. Understand and apply appropriate nonparametric regression models and be able to explain the difference between the parametric and nonparametric approaches, and machine learning techniques. Possible models include splines, GAMs, classification and regression trees, random forest, and boosting techniques, as well as model selection using, for instance, cross-validation.
3. Use appropriate methods to fit models using a range of software packages.
4. Design and carry out an independent statistical analysis using advanced modelling approaches, clearly communicating methodological decisions and results in written form.
5. Be able to present work as part of a group to conduct statistical research, successfully present this work both orally and via a written report in the style of a scientific publication.

Outline Syllabus

This module provides a rigorous foundation in modern statistical modelling techniques, combining theoretical development with hands-on practical implementation. It focuses on methods for analysing data where classical modelling assumptions may not hold, particularly in the presence of nonlinearity, or complex structure. A central theme is the use of nonparametric and flexible modelling frameworks that avoid strict distributional assumptions and are particularly effective at uncovering intricate relationships within the data. Topics covered fall naturally into two categories: Statistical models: This includes linear and generalised linear models (GLM). GLMs provide the starting point for nearly all complex statistical methods. GAMs will be studied as an alternative modelling approach when traditional assumptions (e.g. normality, homoscedasticity, linearity) are not valid. Splines will be central for the complex representations of the data. Applications of the nonparametric models will be on time series data and spatial data. Machine learning models: This includes classification and regression trees (CART). CART and Boosting are essential in modern data analysis because they form the foundation of some of the most powerful, flexible, and widely used machine learning algorithms today. They are central to solving complex non-linear prediction problems. Alongside these two categories, the course focuses on the practical implementation of methods and the development of research skills. This includes time for students to develop their scientific writing skills, including good practice on how to search and cite literature for academic papers. Students will get a chance to peer-review each other’s work before embarking on a substantial data-driven project involving the methods and techniques introduced in the first half of the course and a group project and presentation relating to the content of the second half.

Assessment Proportions

Formative Assessment:

• In-lab formative exercises promote collaborative learning and will enable students to reflect on their work via peer review.

Summative Assessment:

• A project (20%) will assess students’ abilities to effectively solve problems with a real-world context using statistical methods alongside their ability to present a written report.
• A group presentation (10%) will assess students’ ability to work effectively in a group to solve problems, as well as their ability to present results orally. Peer review will be used as a method to provide formative feedback on aspects of this project.
• An end-of-module examination (70%) combines problem-solving questions with interpretation of output to assess against the non-coding-based learning outcomes.

MATH7432: Computing and Algorithms for Statistics

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability

Course Description

Contemporary statisticians work with large data sets and intractable models. Often, effective inference is feasible only with modern computing hardware, strong programming skills, and advanced computational algorithms. This module provides students the opportunity to learn a range of techniques and associated algorithms relevant to statistics and artificial intelligence and to enhance their R and Python programming abilities through hands-on implementation. Students will gain competence in constructing basic programmes using the fundamental building blocks of statistical computing and making appropriate use of large language models to support classical programming. Students will implement numerical optimisation techniques, such as gradient descent, and learn how to apply them to intractable statistical inference problems. The module will explore the toolkit of algorithms used for advanced statistical inference, such as bootstrapping and Markov chain Monte Carlo, enabling students to gain proficiency in their implementation and application.

Educational Aims

Upon successful completion of this module students will be able to…

1. Implement basic programs and demonstrate understanding of programming structures and logic. Work with large data sets and/or intractable models to draw appropriate statistical inferences.
2. Appropriately utilise large language models to support statistical programming.
3. Understand and explain the mathematical basis of, and directly implement a range of optimisation and advanced statistical inference techniques, such as gradient descent, bootstrapping and MCMC.
4. Select the most appropriate technique for a given statistical inference problem.
5. Create reproducible reports using Markdown.
6. Confidently take a problem statement; select and code appropriate statistical techniques to solve the problem; critically evaluate the results; and communicate findings effectively.

Outline Syllabus

This module interweaves techniques required for advanced statistical inference with the development of mathematical programming in R and Python. The module will enhance the range of techniques at your disposal and enable you to choose an appropriate approach for a given statistical problem, understanding its strengths and weaknesses relative to other possibilities. You will develop the ability to code and validate each technique, refining as necessary, and to interpret the results. Over the course of the module, you will build a coherent understanding of the computational methods used in modern quantitative work. The course investigates gradient descent, stochastic and momentum-based optimisation algorithms, bootstrap approaches and Markov chain Monte Carlo methods. You will compare the properties, assumptions, and computational efficiencies of these approaches while developing the foundational programming concepts, such as control flow, structures and functional design, that support them. There is a key focus on reproducibility throughout the course; students will create a portfolio of short reports that mix mathematics, prose, code, and its processed output using Markdown. The module will also introduce the ethical and responsible use of large language models as supportive tools for coding, documentation, and reflection.

Assessment Proportions

Formative Assessment:

• Weekly online quizzes check understanding of recently covered material and reveal common misconceptions.
• In-lab formative exercises promote collaborative learning and expose students to a diversity of coding styles.

Summative Assessment:

• Mid-term computing test (20%) – an in-person examination that assesses the students’ ability to code and to formulate and implement algorithmic solutions to real-world problems.
• Four short coursework assignments (20%) – that assess the ability of students to code the algorithms taught and apply them to real problems and write the exercise up in a Markdown report.
• End-of-module exam (60%) – an in-person, pen-on-paper exam to test knowledge and understanding and mathematical intuition of the procedures introduced in the module.

MATH7434: Probability Theory

• Terms Taught: Lent / Summer 
• US Credits: 5 Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability / statistics

Course Description

The aim of this course is to develop an analytical and axiomatic approach to the theory of probabilities. The notion of a probability space is introduced and illustrated by simple examples, featuring both discrete and continuous sample spaces. Random variables and the expectation are then used to develop a probability calculus, which is applied to achieve laws of large numbers for sums of independent random variables. Characteristic function method is used to study the distributions of sums of independent variables. The results are illustrated in applications to random walks, in particular, the Poisson approximation and central limit theorems are proven, with upper bounds for the accuracy of both approximations.

Educational Aims

On successful completion of this module students will be able to:

1. understand the role of probability axioms in probabilistic models;
2. carry out standard calculations in the calculus of probabilities;
3. use the standard tools of random variables and cumulative distribution functions to describe probabilistic models;
4. understand the concept of weak convergence of probability distributions;
5. understand scientific literature in statistics and statistical physics that uses these concepts.

Outline Syllabus

The focus of the module will be a rigorous approach to stochastic modelling. We will cover:

• Sigma algebras of sets; probability measures; countable additivity.
• Random variables and the cumulative distribution function. Transformations of random variables and probability density functions.
• Construction of expectation for general random variables. Mean value and variance. Chebyshev's inequality. Cauchy-Schwartz inequality.
• Convergence of random variables; statement of convergence theorems for expectation. Weak law of large numbers.
• Almost sure convergence. Borel-Cantelli Lemmas. Strong Law of Large Numbers.
• Weak convergence. Equivalence of weak convergence and convergence of distribution functions. Characteristic functions.
• Poisson approximation, rate of convergence.
• The Central Limit Theorem, accuracy of the normal approximation.
• Multivariate normal distribution and multivariate central limit theorem.

Assessment Proportions

The primary method of delivering the course material are live, interactive lectures, which serve as the central component of the teaching process. These lectures are carefully designed to provide students with a thorough understanding of the course material and to cover key topics in a structured manner. Additionally, comprehensive and detailed written lecture notes are made available to ensure that students have a reliable resource for review and reference outside of class.? Frequent example-based workshops will give the students more space to explore the ideas in depth: they will work together to build the key skills listed in the learning outcomes: carrying out computations, interrogating proofs and exploring hypotheses through proof and counterexample. Peer-learning activities in workshops will help them to build confidence and communication skills as well as their mathematical ability. Students will be assessed through five fortnightly courseworks and five fortnightly Moodle quizzes over the semester. Feedback on the courseworks will also play a formative role, helping them to fine-tune their skills over the semester. Finally students will be assessed by a written examination during the examination period.?

MATH7435: Clinical Trials

• Terms Taught: Lent / Summer
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: 3rd level probability / statistics

Course Description

This module is designed to introduce students to key statistical concepts relevant to the design and analysis of clinical trials. By the end of the course, students will understand the fundamental components of clinical trials, apply principles of robust study design, and analyse and interpret trial data to draw valid scientific conclusions.

Educational Aims

Upon successful completion of this module students will be able to…

1. understand the basic elements of clinical trials
2. recognise and use principles of good study design
3. analyse and interpret study results to make correct scientific inferences.

Outline Syllabus

Clinical trials are planned experiments on human beings designed to assess the relative benefits of one or more forms of treatment. For instance, we might be interested in studying whether aspirin reduces the incidence of pregnancy-induced hypertension; or we may wish to assess whether a new immunosuppressive drug improves the survival rate of transplant recipients. The module will provide a definition and estimation of treatment effects. Furthermore, cross-over trials, issues of sample size determination, and equivalence trials are covered. There is an introduction to flexible trial designs that allow modifications to key aspects of the study with based on interim data during an ongoing trial. Finally, other relevant topics such as meta-analysis and accommodating confounding at the design stage are briefly discussed. Topics covered will include:

• Clinical trials fundamentals: trial terminology, principles of sound study design and ethics;
• Defining and estimating treatment effects: continuous and binary data;
• Crossover trials: motivation, design issues and analyses;
• Sample size determination; continuous and binary data;
• Equivalence and Non-inferiority trials;
• Systematic reviews and Meta Analysis;
• Group sequential designs: Motivation, general framework, comparison of design;
• Adaptive designs: Sample size review, response-adaptive designs, and other adaptive designs.

Assessment Proportions

The learning strategy for this module is designed to enable the student to understand the statistics underpinning of the design and analysis of clinical trials, to determine the different approaches that can be taken in addressing clinical questions related to the effectiveness of treatments and other types of interventions. The module will be delivered by weekly lectures and lab sessions. Lecture slides and detailed notes covering the course material will be provided. To support and reinforce students' understanding, examples and guided learning questions are also provided where appropriate in the lectures. Lab sessions will provide students with hands-on experience in working on different designs and analysing practical datasets following the choice of designs with R. These sessions are designed to deepen students’ understanding of the subject and to build their competence and confidence in applying these techniques to real-world data. Students will be assessed through both a final written examination (to cover the theoretical elements) and a coursework project (to cover practical elements and communication).

MATH7436: Epidemiology and Disease Modelling

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability/statistics

Course Description

This module aims to:

1. Provide an introduction to the field of epidemiology and epidemiological study design;
2. Show how to analyse different studies to calculate measures of risk and rate of disease appropriately;
3. Introduce infectious disease models and apply them to data to estimate key epidemiological quantities;
4. Develop awareness and conceptual understanding of more advanced methods used both in epidemiology and more widely such as causal inference.
5. Develop a cadre of modern statisticians capable of responding to disease outbreaks such as Covid19.

Educational Aims

Upon successful completion of this module students will be able to…

1. Define and calculate appropriate measures of disease prevalence, incidence, cumulative incidence and mortality;
2. Describe the key statistical issues in the design of cross-sectional surveys, case-control studies, and cohort studies and their advantages and disadvantages.
3. Discuss and address strategies for dealing with bias and confounding;
4. Evaluate diagnostic and screening tests in terms of design and analysis issues;
5. Understand the inevitability and difficulties of analysing data from observational or opportunistic studies, such as during disease outbreaks.
6. Use disease transmission models to describe the dynamics of various infectious diseases, fit such models to data, predict outbreak trajectories, and estimate disease reproduction numbers.

Outline Syllabus

Epidemiology and disease modelling play a crucial role in maintaining public health. This module will introduce both non-communicable disease epidemiology and infectious disease epidemiology, starting with the fundamental concepts of measures of disease occurrence and risk and likelihood inference for epidemiological parameters, and going all the way to mathematical modelling of infectious diseases. Along the way, epidemiological study design and analysis, causal inference and disease screening will be covered. The last part of the module will look at infectious disease models and how they can be fitted to data, including estimation of reproduction numbers of infectious diseases.

Assessment Proportions

The learning and teaching strategy for this module is focused around active learning on the part of the students, as reflected by an alternating pattern of 2-hour lecture and 1-hour practical and 1-hour lecture and 2-hour practical per week. Lectures will develop students’ understanding of the theory underlying epidemiology and disease modelling, with practical sessions extending their ability to apply it to example datasets. Thus lectures will be used to introduce and explain definitions and methods, with practicals embedding this knowledge in real-world examples. Students’ learning in lectures will be supported through doing anonymous online quizzes (with discussion of answers) throughout the module, working in groups on problems, questioning and answering, reading slides, seeing additional explanations of the theory on a visualiser, and watching live code demonstrations. These activities will provide a means of formative assessment and feedback that enable students to gauge their learning and where they have gaps in understanding. Lectures will be taught by slides and comprehensive module notes will be provided. The students will be expected to read the notes in advance of lectures and come prepared to discuss them, to enhance their learning in the lectures. Practical sessions will be held in computer labs, using problem sheets designed to build understanding of theory and guide students through putting it into practice. The course will use the R and Python programming languages, and students will have a choice of which programming language they want to use (or can use both). Students will be encouraged to keep an electronic lab book (e.g. as a Rmarkdown document or Jupyter notebook) to log their work and learning. Students will be assessed via an individual project (20%) and a final written exam (80%), to assess individual achievement of all the ILOs.

MATH7437: Survival and Longitudinal Statistics

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability.

Course Description

This module aims to provide students with advanced knowledge and practical skills for the statistical analysis of complex data structures commonly encountered in medical and social research. It focuses on two key areas: the analysis of the survival data related to the time to event (e.g. death or time to relapse of symptoms that are particularly common in medical applications) and the modelling of hierarchical data structures, including longitudinal (data collected over time from same patient) and multilevel data (data collected from the same area or hospital), where dependence among observations must be properly addressed. Students will explore non-parametric, semi-parametric, and flexible parametric methods for survival data, alongside linear and generalised linear mixed effects models for dependent data. Emphasis is placed on understanding marginal vs. conditional modelling approaches, the role of random effects, complementing the classical linear regression models. The practical implementation of these methods will be using R for robust and interpretable data analysis.

Educational Aims

Upon successful completion of this module students will be able to…

1. Understand the unique features and statistical challenges surrounding data collected from multi-level, longitudinal and time-to-event studies;
2. Using an appropriate level of mathematical details, define and critique linear (LMMs) and generalised linear mixed effects models (GLMMs) for a variety of example scenarios and for different types of response variable.
3. Select, define mathematically, and justify appropriate forms of dependence for the random effects component of a LMM or a GLMM.
4. Formulate and compare non-parametric, semi-parametric and parametric statistical methods for survival analysis including the Kaplan-Meier estimator and Cox’s proportional hazards model.
5. Design a suitable strategy to analyse both (a) multi-level and longitudinal and (b) time to event data, including exploratory analysis, model selection and model checking.
6. Undertake a full data analysis of both (a) multi-level and longitudinal and (b) time to event data by implementing appropriate techniques using suitable statistical software.
7. Appraise the statistical analysis of both (a) multi-level and longitudinal and (b) time-to-event data and interpret the output in the context of the original research question.

Outline Syllabus

The first part of the module develops advanced statistical modelling techniques for longitudinal and hierarchical (multilevel) data. Such data arise in many fields - including medicine, education, psychology, and ecology - where repeated measurements are collected over time or clustered within groups. A consequence of collecting data in this way is that data within groups or on an individual will be correlated, which means that the familiar linear and generalised linear regression models are unsuitable. We introduce linear and generalised linear mixed effects models. These models incorporate additional structure that allows us to both identify variability between individuals or groups and obtain more accurate uncertainty estimates when data are not independent. For data collected over time, we will also explore how to incorporate time series models (Gaussian processes) in the mixed effect model. The second part of this module considers survival data and the associated concept of censoring. In many clinical studies, subjects are monitored until a particular outcome is observed: this can be death (hence survival), or it could be some other clinical marker (e.g. full recovery, or normal blood cell count). In such studies, subjects often exit the study before the event is observed, e.g. they drop-out of the trial, die from other causes, or are lost to follow up. Observations on such subjects are referred to as being “censored”. Statistical methods for survival analysis provide us with a tool kit with which to extract meaning from any censored data set. Foundational concepts on survival and hazard functions will be followed by nonparametric methods such as the Kaplan-Meier estimator. We then derive the semiparametric setup of the Cox proportional hazards model. Finally, we explore how parametric models can be used to predict survival times. Diagnostic methods discussed will include Schoenfeld and other residuals, testing the proportional hazards assumption and detecting changes in covariate effects.

Assessment Proportions

This module is designed for students to achieve a deep understanding on nonparametric analysis based on Kaplan-Meier estimate, forming partial likelihood for Cox proportional hazards model, constructing parametric models for survival data and diagnostics procedure and building complex statistical models (mixed effect models and longitudinal models). The main mode of delivery will be weekly lectures. Lecture slides and comprehensive lecture notes containing details of the material will be provided. A number of worked out examples on theoretical exercises and data analysis will be provided to consolidate student's understanding. Students will be assessed through formative and summative assessments designed to align with the learning outcomes. Formative Assessment: Lab sessions: students will be provided practical data set analysis by applying various survival and longitudinal methods they have learnt through lectures and theoretical exercises. This will help them to build a solid knowledge and understanding of the subject and to achieve competence and confidence in working with relevant data. Model solutions will be provided after the labs to aid their learning. Summative Assessment:

• The two works will evaluate survival analysis and longitudinal analysis separately. Both will assess students’ abilities to solve problems with a real-world context using statistical methods.
• An end-of-module examination that combines theoretical elements with interpretation of output, to assess against the non-coding-based learning outcomes.

MATH7439: Stochastic Calculus for Finance

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability/statistics.

Course Description

This advanced module in probability is on a topic which is essential preparation for any students interested in pursuing a PhD in probability. The application of stochastic calculus to areas such as finance also makes this very attractive topic to potential employers.

Educational Aims

On successful completion of this module students will be able to:

1. Justify and critique the use of stochastic models for real life applications.
2. Use the stochastic calculus framework to formulate and solve problems involving uncertainty.
3. Identify martingales and apply the optional stopping theorem to simple examples.
4. Derive basic properties of Brownian motion.
5. Apply Ito's formula to functions of Brownian motion.
6. Solve simple stochastic differential equations (SDEs).

Outline Syllabus

The focus of the module will be on stochastic modelling based on continuous-time stochastic processes. We will cover:

• Discrete time Stochastic Processes: conditional expectation, filtrations, martingales, stopping times, optional stopping theorem.
• Continuous time Processes: Brownian motion, path properties, martingale properties, hitting distributions.
• Stochastic Integration: total variation, Stieltjes integral, quadratic variation, definition of Ito integral, Ito's formula, definition of stochastic differential equation (SDE), solution of simple SDEs.
• Black-Scholes model.
• Exotic options.

Assessment Proportions

The primary method of delivering the course material are live, interactive lectures, which serve as the central component of the teaching process. These lectures are carefully designed to provide students with a thorough understanding of the course material and to cover key topics in a structured manner. Additionally, comprehensive and detailed written lecture notes are made available to ensure that students have a reliable resource for review and reference outside of class.? Frequent example-based workshops will give the students more space to explore the ideas in depth: they will work together to build the key skills listed in the learning outcomes: carrying out computations, interrogating proofs and exploring hypotheses through proof and counterexample. Peer-learning activities in workshops will help them to build confidence and communication skills as well as their mathematical ability. Students will be assessed through five fortnightly courseworks. Feedback on the courseworks will also play a formative role, helping them to fine-tune their skills over the semester. Finally students will be assessed by a written examination during the examination period.?

MATH7445: Hidden-Process Models

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability/some computational background.

Course Description

This module aims to give students an exposure to a selection of key latent-process models used in contemporary machine learning, AI and statistics. In particular, it will introduce the Gaussian process and several important classes of state space model. The Gaussian process is a core tool in spatial statistics, it is the driving engine behind modern experimental design and it is the go-to model for cheaply emulating complex systems such as climate models. State space models represent processes that evolve over time but which are only partially or noisily observed; they include the dynamic linear model and hidden Markov models.

Educational Aims

Upon successful completion of this module students will be able to…

1. Demonstrate a mathematical understanding of a variety of hidden-process models and distinguish the meanings and impacts of their parameters.
2. Explain the associated methods for inference and prediction.
3. Evaluate a range of applications and construct an appropriate hidden-process model for each.
4. For previously unseen data sets and scenarios, fit and use the model and interpret the output in the context of the application.

Outline Syllabus

Many of the more advanced statistical models and machine-learning procedures are built around a latent, or hidden, random process. This provides much greater flexibility than is seen in traditional tools, such as the linear model, yet naturally enhances the quantification of uncertainty in estimates and predictions. This module introduces you to a selection of the most important latent-variable models including Gaussian processes, state space models and their intersection, the dynamic linear model. These more advanced statistical models often provide a natural motivation for the introduction of more advanced statistical methods and algorithms, such as variational Bayes and the Kalman filter. You will discover how hidden-process models can be applied in areas such as spatial statistics, system emulation, modern experimental design and time series. Using simulation, you will learn about each model's properties and how these are affected by different choices of parameters and other settings. You will explore both classical and Bayesian approaches and see the advantages and disadvantages of each. You will understand why new methods are often required for inference and prediction when using latent-variable models and delve into these new techniques. The models and their uses will be motivated and developed through example data sets and applications in areas such as health, engineering and the environment. Topics covered fall naturally into two categories: Gaussian processes: their properties and parameters, their use in emulation, spatial statistics and Bayesian Optimisation. Techniques including variational Bayes and advanced Monte Carlo algorithms will help solve some tractability issues. State-space models: this large class of time-series models includes the dynamic linear model, dynamic non-linear models and hidden Markov models. Tractability issues will be solved through techniques including the Kalman filter and the forward-backward algorithm.

Assessment Proportions

Each week will cover a particular subtopic, with two lectures on the subtopic, followed by a computer lab to allow students to connect the theory behind the relevant concepts and methods to their practical use. The main summative assessment will be through a final exam, contributing 80% and assessing mathematical, conceptual and algorithmic aspects of the module as well as appropriate choices of methodology and interpretation of output. The 20% summative/formative coursework will consist of Moodle quizzes and handwritten coursework, with a quiz and a coursework for each of the two main topics (Gaussian processes and state space models). The Moodle quizzes will connect to the relevant computer labs and will assess and encourage learning of the more practical aspects of the module. The written coursework will assess and encourage learning of the more conceptual and mathematical side.

MATH7446: Machine Learning

• Terms Taught: Lent/Summer
• US Credits: 5
• ECTS Credits: 10
• Pre-requisites: 3rd level probability/statistics/ some computational background.

Course Description

This module aims to give students an exposure to a selection of key machine learning methods used in contemporary data science, AI and statistics. In particular, it will introduce fundamental techniques for unsupervised learning, focusing on dimension reduction methods, clustering algorithms, and graphical models. Matrix decomposition techniques like Principal Component Analysis and Non-negative Matrix Factorisation serve as useful tools for dimensionality reduction, while clustering methods such as K-means and Gaussian Mixture Models enable the discovery of natural groupings in data. Graphical models provide a framework for representing complex dependence structures.

Educational Aims

Upon successful completion of this module students will be able to

1. Demonstrate a mathematical understanding of a variety of machine learning methods and models and distinguish the meanings and impacts of their parameters.?
2. Evaluate a range of applications and construct an appropriate machine learning method for each.?
3. For previously unseen data sets and scenarios, fit and use the model and interpret the output in the context of the application.?

Outline Syllabus

Machine learning lies at the heart of modern artificial intelligence, providing powerful tools that can identify hidden patterns, reduce complexity, and identify structure in vast amounts of data. This module introduces you to fundamental machine learning methods that form the backbone of contemporary data science and AI systems. You will explore how mathematical foundations in matrix theory enable sophisticated techniques for extracting meaning from high-dimensional data. You will discover how unsupervised learning approaches reveal structure without requiring labelled data, transforming complex datasets into more accessible representations. Through hands-on implementation, you will evaluate the strengths and limitations of different algorithms, from classical techniques to modern neural network-based methods. Throughout the module, you will work on modern machine learning problems such as recommendation systems, image recognition and noise filtering and get practical experience with the full machine learning pipeline. Topics covered fall naturally into three categories: Dimension reduction: You will explore how matrix decompositions serve as powerful tools for visualisation and feature extraction. Considering both distance-based methods like Principal Component Analysis and its sparse version, and model-based methods like Exploratory Factor Analysis, you will develop a toolkit for multivariate analysis and high-dimensional data. Clustering: This section covers techniques for discovering natural groupings within unlabelled data, including K-means and Gaussian Mixture Models with the expectation-maximisation (EM) algorithm. Graphical models: You will examine how conditional independence structures can be modelled through graphs, enabling efficient representation of complex relationships. From Bayesian networks to the graphical Lasso, you will learn to capture meaningful interactions while maintaining computational tractability.

Assessment Proportions

The module is structured around weekly subtopics, each comprising two lectures that establish theoretical frameworks and mathematical foundations, followed by a computer lab where students apply these concepts to real-world datasets. This approach bridges theory and practice, allowing students to gain hands-on experience with implementing machine learning algorithms and interpreting their outputs. Assessment consists of a final examination (80%) that evaluates students’ understanding of mathematical principles, algorithmic procedures, and the appropriate selection and interpretation of machine learning methods. The remaining 20% comprises formative and summative coursework divided between Moodle quizzes and written assignments. Each of the three main topics (dimension reduction, clustering, and graphical models) will have associated practical quizzes that reinforce computer lab content and assess implementation skills, while written coursework will deepen understanding of theoretical concepts. This module aligns with the programme's learning, teaching, and assessment strategy by emphasising both theoretical understanding and practical implementation skills essential for statisticians and machine learning professionals. The assessment structure supports the programme’s focus on developing critical thinking through mathematical reasoning and problem-solving capabilities. The combination of individual written work and computer-based applications mirrors the programme’s commitment to preparing students for both research and industry environments.