糖心Vlog

• 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.

• 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

• 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.

• 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.

• 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.

• 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.

• 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).

• 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.

• 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.

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

• Terms Taught: Michaelmas
• US Credits: 5 US Semester Credits
• ECTS Credits: 10 ECTS
• Pre-requisites: None

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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