# Mathematics and Statistics

The following modules are available to incoming Study Abroad students interested in Mathematics and Statistics.

## MATH101: Calculus

• Terms Taught: Michaelmas Term Only
• US Credits: 2 semester credits
• ECTS Credits: 4 ECTS
• Pre-requisites: A year of general mathematics, including basic calculus.

### Course Description

The course covers: complex numbers, functions and graphs; limits of sequences and sums of infinite series; differentiation, product and chain rules; Taylor series; integration: fundamental theorem of calculus; integration by parts and substitution.

### Educational Aims

This course aims to provide the student with an understanding of functions, limits, and series, and a knowledge of the basic techniques of differentiation and integration. The purpose of this course is to study functions of a single real variable. Some of the topics will be familiar, others will be studied more thoroughly in subsequent courses.

The module begins by introducing examples of functions and their graphs, and techniques for building new functions from old. We consider rational functions and the exponential function. We then consider the notion of a limit, sequences and series and then introduce the main tools of calculus. The derivative measures the rate of change of a function and the integral measures the area under the graph of a function. The rules for calculating derivatives are obtained from the definition of the derivative as a rate of change. Taylor series are calculated for functions such as sin, cos and the hyperbolic functions.

We then introduce the integral and review techniques for calculating integrals. We learn how to add, multiply and divide polynomials and introduce rational functions and their partial fractions. Rational functions are important in calculations, and we learn how to integrate rational functions systematically. The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series, so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parameterize geometrical curves.

### Outline Syllabus

• Arithmetic of complex numbers;
• Polynomials;
• Rational functions and partial fractions;
• Exponential and hyperbolic functions;
• Compositions and inverses;
• Induction;
• Sequences and limits;
• Differentiation;
• Product and Chain rules;
• Maxima and minima;
• Taylor series;
• Complex exponentials and trigonometric functions;
• Definite integral as areas;
• Fundamental theorem of calculus;
• Integration by parts and by substitution;

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH102: Further Calculus

• Terms Taught: Michaelmas Term Only
• US Credits: 2 semester credits
• ECTS Credits: 4 ECTS
• Pre-requisites: A year of general mathematics, including calculus.

### Course Description

The course covers: improper integrals; integration over infinite ranges; Simpson’s rule; functions of two or more real variables; partial derivatives; curves in the plane; implicit functions; the chain rule for differentiating along a curve; stationary points for functions of two real variables; double and repeated integrals; Cavalieri’s slicing principle; volumes.

### Educational Aims

The first part of this course extends ideas of MATH101 from functions of a single real variable to functions of two real variables. The notions of differentiation and integration are extended from functions defined on a line to functions defined on the plane. Partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes.

### Outline Syllabus

• Complex polynomials and complex roots;
• Integration of rational functions;
• Improper integrals
• Integration over infinite ranges;
• Simpson's rule;
• Functions of two or more real variables;
• Partial derivatives;
• Curves in the plane;
• Implicit functions;
• The chain rule for differentiating along a curve;
• Stationary points for functions of two real variables;
• Double and repeated integrals;
• Cavalieri's slicing principle;
• Volumes

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH103: Probability

• Terms Taught:  Lent and Summer Terms Only
• US Credits: 2 Semester credits
• ECTS Credits: 4 ECTS credits
• Pre-requisites: A year of general mathematics

### Course Description

Probability theory is the study of chance phenomena; the concepts of probability are fundamental to the study of statistics. The course will emphasise the role of probability models which characterise the outcomes of different types of experiment that involve a chance or random component. The course will cover the ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models. No previous exposure to the subject will be assumed.

### Educational Aims

• To provide an introduction to probability theory for discrete distributions.
• To introduce students to some simple combinatorics, set theory and the axioms of probability.
• To make students aware of the different probability models used to model varied practical situations.

### Outline Syllabus

• The axioms of probability
• Conditional probability
• Independence
• Discrete Random variables
• Expectation, mean and variance.
• The binomial, Poisson and geometric distributions

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH104: Statistics

• Terms Taught:  Lent and Summer Terms Only
• US Credits: 2 Semester credits
• ECTS Credits: 4 ECTS credits
• Pre-requisites: A year of general mathematics

### Course Description

The course covers: data collection and summary, modelling discrete data, continuous distributions, modelling continuous data, statistical conclusions.

### Educational Aims

The module aims to enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems in which the recorded information is subject to systematic and random variations. Specifically, by the end of the module, students should be able to select and formulate appropriate probability models, to implement the associated statistical techniques, and to draw clear and informative statistical conclusions for a range of simple scientific problems.

The module starts with the description of examples of scientific investigations in which specific questions are of interest, but they are not straightforward to answer, as the available data are subject to systematic and random variations. A range of exploratory data analysis methods for gaining insight into the sources of variations will be introduced. Then a general strategy for the statistical treatment of such problems will be developed, involving aspects of modelling, investigation, and conclusions.

### Outline Syllabus

• Data collection and summary
• Modelling discrete data
• Continuous distributions
• Modelling continuous data

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH105: Linear Algebra

• Terms Taught: Lent / Summer Terms Only
• US Credits: 2 semester credits
• ECTS Credits: 4 ECTS credits
• Pre-requisites:   A year of general mathematics, including matrices

### Course Description

The course covers: matrix algebra; systems of linear equations; determinants; linear transformations, eigenvalues and eigenvectors.

### Educational Aims

The specific aim of this module is to introduce the notion of matrices and their basic uses, mainly in algebra. The main goals are to learn how the algorithm of elementary row and column operations is used to solve systems of linear equations, the concept and use of determinant, and the notion of a linear transformation of the Euclidean space. The course also aims at defining the main concepts underlying linear transformations, namely singularity, the characteristic equation and the Eigen spaces.

### Outline Syllabus

• Matrices: addition and multiplication, transpose and inverse.
• Simultaneous linear equations
• Reduction to echelon form by elementary row operations
• Elementary matrices
• Determinants: expansions about a row or column
• Elementary row and column operations on determinants
• Properties of determinants
• Linear transformations of Euclidean space
• The matrix of a linear transformation
• Non-singular linear transformations
• Eigenvectors and eigenvalues
• The characteristic equation

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH111: Numbers and Relations

• Terms Taught: Michaelmas Term Only
• US Credits: 2 semester credits
• ECTS Credits: 4 ECTS
• Pre-requisites: A year of general mathematic.

### Course Description

The course covers: truth tables; methods of proof; lowest common multiples; prime numbers; the Fundamental Theorem of Arithmetic; the existence of infinitely many prime numbers; applications of prime factorisation; solving congruences; the Chinese Remainder Theorem; equivalence relations; constructions of number systems; the division algorithm; highest common factors; the Euclidean algorithm.

### Educational Aims

This course aims to:

• Introduce students to mathematical proofs;
• To state and prove fundamental results in number theory;
• To generalize the notion of congruence to that of an equivalence relation and explain its usefulness;
• To generalize the notion of a highest common factor from pairs of integers to pairs of real polynomials.

### Outline Syllabus

• Logic: truth tables, methods of proof (direct, contraposition, contradiction), simple examples of mathematical proofs.
• Number theory: division with remainder; highest common factors and the Euclidean algorithm; lowest common multiples; prime numbers; the Fundamental Theorem of Arithmetic and the existence of infinitely many prime numbers; applications of prime factorization.
• Congruences: definition; solving congruences; the Chinese Remainder Theorem.
• Relations: equivalence relations; the sum and product of two congruence classes; constructions of number systems.
• Polynomials: the division algorithm; highest common factors and the Euclidean algorithm.

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH112: Discrete Mathematics

• Terms Taught:  Michaelmas Term Only
• US Credits: 2 semester credits
• ECTS Credits: 4 ECTS
• Pre-requisites: A year of general mathematics, including matrices.

### Course Description

The course covers: vectors; set notation and manipulation; countability; functions; invertibility and composition; pigeonhole principle; generating functions; basic graph theory.

### Educational Aims

The module formulates the language in which students can describe certain mathematical models or enumerative problems in precise mathematical terms, especially using set terminology. Students can then formulate solve such problems using combinatorial proofs. It also equips students with certain tools such as generating functions and graphs which are helpful in solving combinatorial problems.

### Outline Syllabus

• Introduction to set notation;
• Manipulation of sets; inclusion, intersection, union, complements;
• Inclusion-exclusion;
• Countability;
• Functions and composition;
• Injectivity, surjectivity and bijectivity;
• Invertibility of functions;
• Selecting and counting elements from finite sets;
• The pigeonhole principle;
• Graphs and trees;
• Isomorphism, planarity, traversing and colouring of graphs

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH113: Convergence and Continuity

• Terms Taught: Lent / Summer Terms Only
• US Credits: 2 semester credits
• ECTS Credits: 4 ECTS
• Pre-requisites: A year of general mathematics, including basic calculus.

### Course Description

This course is an introduction to the foundations of Real Analysis, including suprema and infima of real numbers, limits of sequences, convergence and continuity of functions.

### Educational Aims

You will learn will learn:

• The structure of the real number system and the notions of supremum and infimum for sets of real numbers
• The mathematical notion of sequences, subsequences, boundedness, limit points, and convergence
• The mathematical notion of continuity and related properties of functions
• How to understand mathematical notation and how to read and write proofs related to the above topics
• How to provide examples and counter-examples to mathematical definitions and statements regarding the above topics

### Outline Syllabus

• Real numbers: Bounds, Maximum and minimum, supremum and infimum, least upper bound principle for Real Numbers.
• Sequences: Limit points and closed sets in Real Numbers. Convergence, monotonicity, boundedness. Cauchy sequences and the completeness of Real Numbers. Subsequences and the Bolzano-Weierstrass theorem.
• Continuity: Real functions: Monotonicity, boundedness and invertibility. Sequential continuity. Epsilon-delta definition of continuity. Intermediate value theorem and applications. Contraction mapping theorem.

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH114: Series, Integration, and Differentiation

• Terms Taught: Lent / Summer Terms Only
• US Credits: 2 semester credits
• ECTS Credits: 4 ECTS
• Pre-requisites: A year of general mathematics, including basic calculus

### Course Description

The course covers: Integration, series and convergence tests, and differentiation.

### Educational Aims

In this module the student will learn:

• The concept of integration of continuous functions
• The notion of series and convergence of series and convergence tests
• The relation of series to sequences and to integrals
• The concept of differentiability of functions, and its relation to continuity and integration.
• How to understand mathematical notation and how to read and write proofs related to the above topics.
• How to provide examples and counter-examples to mathematical definitions and statements regarding the above topics

### Outline Syllabus

• Integration: Uniform continuity of continuous functions, Integration of continuous functions,
• Basic estimates, Improper integrals.
• Series: Series versus sequences, convergence tests, examples.
• Differentiation: Difference quotients and limits, continuity of differentiable functions.
• Algebra of differentiation: product rule, chain rule, differentiating inverse functions.
• Mean-value theorem and applications. Fundamental theorem of calculus and applications.

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH115: Geometry and Calculus

• Terms Taught: Lent / Summer Terms Only
• US Credits: 2 semester credits
• ECTS Credits: 4 ECTS
• Pre-requisites: A year of general mathematics, including calculus.

### Course Description

The course covers: linear, separable and homogeneous first-order equations; linear second-order equations with constant coefficients; Laplace transforms.

### Educational Aims

A vast number of naturally occurring phenomena are modelled by differential equations, for which solutions are required to explain these phenomena. This course, which should be particularly useful for science students, sets about obtaining solutions to a number of standard types of differential equations.

### Outline Syllabus

• First-order differential equations: integrable, separable, linear, homogeneous, Bernoulli, with linear coefficients
• Second-order differential equations: reduction of order, linear homogeneous and inhomogeneous with constant coefficients, Cauchy-Euler
• Initial-value problems
• Higher-order differential equations
• Laplace transforms

### Assessment Proportions

• Coursework: 50%
• Exam: 50%

## MATH210: Real Analysis

• Terms Taught: Michaelmas Term Only
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Mathematics majors or appropriate college mathematics.

### Course Description

This course gives a rigorous introduction to the subject with proofs based on formal definitions. Topics include: limits of sequences and convergence of series, limits and continuity of functions; differentiation; the intermediate value theorem; the mean value theorem and applications to the special functions and inequalities; Riemann integration, with applications to inequalities and series; infinite products; uniform convergence; Fourier series and applications.

### Educational Aims

The notion of a limit underlies a whole range of concepts that are really basic in mathematics, including sums of infinite series, continuity, differentiation and integration. After the more informal treatment in the first year, our aim now is to develop a really precise understanding of these notions and to provide fully watertight proofs of the theorems involving them. We also show how the theorems apply to give useful facts about specific functions such as exp, log, sin, cos, including some integrals and other unexpected identities.

### Outline Syllabus

• Limits of sequences: basic results; monotonic sequences; subsequences.
• Infinite series: standard examples; comparison and ratio tests; absolute convergence; power series; Abel summation; double series.
• Limits and continuity of functions.
• Differentiation: the definition and basic results; compositions and inverse functions; differentiation of power series.
• Intermediate value theorem. Boundedness and uniform continuity of functions continuous on a closed interval.
• The mean value theorem; applications to identities and inequalities.
• Definition of the Riemann integral. The fundamental theorem of calculus.
• Inequalities for integrals; application to the estimation of discrete sums; the series for tan ^-1 x and log (1+ x); Euler's constant.
• Infinite products.
• Sequences and series of functions: uniform convergence.

### Assessment Proportions

• Coursework: 20%
• Exam: 80%

## MATH215: Complex Analysis

• Terms Taught: Lent / Summer Terms Only
• US Credits: 4 semester credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Mathematics majors or appropriate college mathematics.

### Course Description

The course covers: polar form for complex numbers; convergence; Cauchy's criterion; continuity of complex functions; differentiability of complex functions; rational functions; differentiability of power series; the exponential function as a power series; line integrals; Cauchy's theorem for a starlike region; Cauchy's estimates; Liouville's theorem; fundamental theorem of algebra; uniform convergence; M-test and Weierstrass theorem; residue theorem and applications; Cauchy-Riemann equations and criterion for differentiability.

### Educational Aims

The purpose of this course is to give an introduction to the theory of functions of a single complex variable together with some basic applications. The treatment will be analytical, and develops ideas from calculus and real analysis. The first part of the course reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The course then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by a bisection argument. The results of function theory are used to evaluate some definite integrals. The course aims to strengthen students' understanding of the geometry of the plane by analysis on starlike regions and by discussion of the Möbius group of transformations. The Cauchy-Riemann equations emphasize the link between real and complex analysis; the maximum modulus theorem for harmonic functions is obtained in this way.

### Outline Syllabus

• The Argand diagram: polar form for complex numbers.
• Convergence: Cauchy's criterion; uniform convergence and the M-test.
• Continuity and differentiability of complex functions; rational functions; differentiability of power series; the exponential function as a power series.
• Line integrals and contours; fundamental theorem of calculus; Cauchy's theorem for a triangle; Cauchy's formula for a disc.
• Formulae for derivatives; Taylor's theorem; examples.
• Cauchy's theorem for a starlike region; Cauchy's estimates.
• Liouville's theorem; fundamental theorem of algebra; zeros and poles; residue theorem and applications.
• Elementary transformations of the plane; Möbius transformations.
• Cauchy-Riemann equations and criterion for differentiability; example of log, Maximum modulus principle

### Assessment Proportions

• Coursework: 20%
• Exam: 80%

## MATH220: Linear Algebra II

• Terms Taught: Michaelmas Term Only
• US Credits: 4 US credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Mathematics majors or appropriate college mathematics.

### Course Description

The course covers: theory of finite dimensional vector spaces; linear dependence, bases, coordinates; matrices and linear equations; linear transformations; matrix of a linear transformation, change of basis, determinants; eigenvalues and eigenvectors, inner products, orthogonal sets.

### Educational Aims

The course begins by introducing the idea of a vector space, showing how it grows naturally out of the ideas developed for studying vectors in two and three dimensions. It shows how the abstraction has a number of advantages, including making precise and underlying assumptions, suggesting concepts of utility in areas where their significance might not otherwise have been recognised, and simplifying matters by trimming away superfluous information. It then goes on to consider the structure-preserving maps between vector spaces and shows how the study of these leads to a deeper understanding of matrices and important matrix questions. The next section is concerned with the effect of changing bases on the matrix representing one of these maps, and examines how we can choose bases so that this matrix is as simple as possible. The final section takes up the theme of scalar products to examine vector spaces in which the concepts of length and angle can be studied, leading to a means of diagonalising real symmetric matrices using geometrically interesting changes of basis.

### Outline Syllabus

• Vector spaces over a field (with emphasis on Rn): subspaces, spanning, linear independence, bases, dimension.
• Linear transformations (with emphasis on geometrical examples): invertibility, matrices of linear transformations, kernel and image, rank of a matrix, applications to linear equations.
• Change of basis: eigenvectors and eigenvalues, characteristic equation, diagonalisation of square matrices.
• Euclidean spaces: orthonormal bases, orthogonal matrices, orthogonal diagonalisation of a real symmetric matrix.
• Jordan Normal Form

### Assessment Proportions

• Coursework: 30%
• Exam: 70%

## MATH225: Abstract Algebra

• Terms Taught: Lent / Summer Terms Only
• US Credits: 4 semester credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Mathematics majors or appropriate college mathematics.

### Course Description

The course covers: an introduction to the basic notions of abstract algebra, covering: binary operations; the definition of a group and examples; the cancellation law; subgroups; cyclic groups and the order of an element; co-sets and Lagrange’s theorem; homomorphisms; normal subgroups; quotient groups; the fundamental isomorphism theorem for groups; groups of permutations; the definition of a ring and examples; subrings and ideals; quotient rings; the fundamental isomorphism theorem for rings; integral domains; fields; polynomial rings; principal ideal domains.

### Educational Aims

The aim of this module is to introduce students to the basics of the theory of groups and rings. The first part of the module emphasizes finite groups which can be considered in terms of their Cayley tables. The module stresses the importance of equivalence relations, which occur at several points. In the second part of the module fundamental concepts of ring theory are introduced. General results and examples are presented, before some important special classes of commutative rings such as integral domains, fields and principal ideal domains are considered in greater detail.

The most important results are:

• Lagrange's theorem;
• The fundamental isomorphism theorem for groups;
• The fundamental isomorphism theorem for rings;
• Ideal structure in the ring of integers and the polynomial ring over a field

### Outline Syllabus

• Binary operations; definition of a group and examples; elementary properties; equivalence relations and modular arithmetic; further examples.
• Subgroups; cyclic groups and order of elements; cosets; Lagrange's theorem and applications.
• Group homomorphisms and isomorphisms; kernel and image.
• Normal subgroups; quotient groups; the fundamental isomorphism theorem for groups.
• Groups of permutations; cycle notation; the sign of a permutation; Cayley's theorem.
• Definition of a ring; examples; rings with identity, commutative rings; subrings; ring homomorphisms and isomorphisms.
• Ideals, quotient rings; the fundamental isomorphism theorem for rings.
• Integral domains and prime ideals
• Fields, polynomial rings, principal ideal domains; ideals in the integers and in the polynomial ring over a field.

### Assessment Proportions

• Coursework: 20%
• Exam: 80%

## MATH230: Probability II

• Terms Taught: Michaelmas Term Only
• US Credits: 4 US credits
• ECTS Credits: 7.5 ECTS credits
• Pre-requisites: Mathematics majors or appropriate college mathematics.

### Course Description

The course covers: events and the axioms of probability; conditional probabilities, independence, Bayes' theorem; discrete random variables; standard distributions; bivariate distributions; expectations: means, variances, correlation; continuous random variables; transformations of random variables; bivariate and multivariate continuous random; joint distribution, conditional distributions; independent random variables; transformations and change of variable; sums of independent random variables; generating functions and application.

### Educational Aims

This course gives a formal introduction to probability and random variables. In the first half we introduce methods for dealing with continuous random variables, building on the work in MATH104 Probability for discrete distributions. We shall use many examples from a variety of statistical applications to illustrate the theoretical ideas.

The second half aims to extend knowledge of probability and distribution theory so that the student should become competent in manipulating functions of one or more random variables, develop probability models for more realistic problems, and discover how distributions that are important in statistical inference are interlinked.

### Outline Syllabus

• Review of basic results in discrete probability.
• Continuous random variables, probability distribution functions, cumulative distribution functions.
• Expectation and variance of continuous distributions. Higher order moments, skewness and kurtosis.
• Standard distributions: uniform, exponential, gamma, normal, chi-squared, and their inter-relationships and justification as probability models.
• Joint distribution of vector random variables; that is, systems of two or more random variables, marginal and conditional distributions. Expectations and variances of vector variables.
• Properties of linear combinations of random variables.
• Transformations of random variables: motivation, univariate and bivariate methods.
• Limit theory: convergence of variables, laws of large numbers, Central Limit Theorem.
• Multivariate normal distribution.

### Assessment Proportions

• Coursework: 20%
• Exam: 80%

## MATH235: Statistics II

• Terms Taught: Lent / Summer Terms Only
• US Credits: 4 US Credits
• ECTS Credits: 7.5 ECTS credits
• Pre-requisites:
• Mathematics majors or appropriate college mathematics.
• Previous studies in statistics and probability, equivalent to MATH230, is recommended

### Course Description

The course covers: parametric statistical inference via likelihood; probability models, the likelihood function, maximum likelihood estimates and likelihood intervals; likelihood ratio tests; properties of direct likelihood methods; large sample properties; data from experimental and observational studies; response and explanatory variables; the linear model; parameter estimation by least squares; their distributional properties; use and interpretation of T and F tests; connection with ANOVA; practical exercises.

### Educational Aims

This course aims for students:

• To appreciate the importance of statistical methodology in making conclusions and decisions.
• To recognize the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.
• To appreciate the central role of the likelihood function in statistical inference.
• To appreciate the role of statistics in making sense of uncertainty.

### Outline Syllabus

Hypothesis testing and Estimation

• Estimates and Estimators
• Paired and unpaired t-tests
• ANOVA
• Confidence Intervals

Regression

• Least squares estimation
• Parameter testing and confidence intervals
• Model comparison
• Model checking
• Model interpretation

Likelihood Theory

• Maximum Likelihood estimation
• Distributions of maximum likelihood estimators; Fisher information
• Confidence intervals of parameters
• Information suppression and sufficiency

### Assessment Proportions

• Coursework: 30%
• Exam: 70%

## MATH240: Project Skills

• Terms Taught: Full Year students only
• US Credits: 4 US credits
• ECTS Credits: 7.5 ECTS credits
• Pre-requisites: Mathematics majors or appropriate college mathematics.

### Course Description

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

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

Students will gain an excellent grasp of LaTeX, learning to prepare mathematical documents; display mathematical symbols and formulae; create environments; and present tables and figures. Scientific writing, communication and presentations skills will also be developed. Students will work on short and group projects to investigate mathematical or statistical topics, and present these in written reports and verbal presentations.

### Educational Aims

This module aims to teach and enhance skills, including both subject-related and transferable skills, appropriate to Part II students in Mathematics and Statistics. These skills include the preparation of mathematical documents and presentation materials, scientific writing, oral presentations and group work.

### Outline Syllabus

The module consists of 5 components:

• LaTeX. Use of LaTeX to prepare mathematical documents; text and mathematical symbols, displayed formulae, numbering, environments, lists, page and document layout, sections and table of contents, tables and figures, slides for oral presentations.
• Scientific Writing. Style, conventions, good practice, clarity, logical presentation.
• Communication and Presentation Skills. Communication skills, oral communication skills, presenting scientific material verbally, group working.
• Individual Project. Investigation of a mathematical or statistical topic, production of a written report.
• Group Project. Group investigation of a mathematical or statistical topic under the direction of a supervisor, production of a written report, presentation of the conclusion.

### Assessment Proportions

• Coursework: 100%

## MATH245: Computational Mathematics

• Terms Taught: Lent  / Summer Terms Only
• US Credits: 4 US credits
• ECTS Credits: 7.5 ECTS credits
• Pre-requisites:
• Mathematics majors or appropriate college mathematics.
• Knowledge of R or an equivalent programming language.

### Course Description

The use of computers is one of the main drivers of progress in modern society. The aim of this module is to develop the use of computers as a tool for problem solving in mathematics. The module uses the programming language R, which was introduced in the first year, with an emphasis on adopting good programming practices that are transferable to other coding. languages and settings. A focus of the module is algorithmic design for mathematical problem solving using the principle that good algorithms should be accurate, efficient, robust and stable. For example, algorithms to carry out numerical differentiation are used to analyse how to minimise errors when designing an algorithm, and the concept of stability is illustrated through ODE-solving algorithms. The module ends with the application of the techniques learned to modelling real-world problems, such as epidemics.

### Educational Aims

The aims of this module are to:

• develop familiarity with the use of a computer as a tool for solving problems;
• convey the importance of being precise and unambiguous (a computer will execute the program you wrote, not the program you "obviously" meant to write!);
• build an understanding of the trade-offs to be made when using computers, for example between accuracy and computational time;
• highlight the use of computational mathematics to model real-world problems which are either analytically intractable or very time consuming

### Outline Syllabus

• Programs and Algorithms: good programming practices, debugging, programming in R, R Markdown, designing algorithms.
• Errors: sources of error, rounding errors, truncation errors.
• Complexity: efficiency, space complexity, time complexity, P vs NP.
• Iteration: fixed points, the contraction mapping theorem, rates of convergence, applications.
• Stability: propagation errors, sensitivity to initial conditions, examples from mathematical programming.

### Assessment Proportions

• Coursework: 40%
• Exam: 60%

## MATH313: Probability Theory

• Terms Taught: Lent 11-15 / Summer Terms only.
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH210 and MATH230.

### 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. The course links second year analysis courses with statistics courses. The characteristic function is used to study the distributions of sums of independent variables. The results are illustrated in applications to random walks and to statistical physics.

### Educational Aims

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. The characteristic function is used to study the distributions of sums of independent variables. The results are illustrated in applications to random walks and to statistical physics.

The aim is to teach probability theory within a general axiomatic approach which emphasizes rigour and mathematical analysis. This module links second year analysis courses with statistics courses. The course also shows how one can describe various probabilistic models within this framework and particularly emphasizes the standard distributions such as the Poisson and Gaussian. A significant application is to ideal gases.

### Outline Syllabus

• Sigma algebras of sets; probability measures; countable additivity.
• Random variables and the cumulative distribution function. Transformations of random variables and probability density functions.
• Expectation and variance. Chebyshev's inequality. Cauchy-Shwarz inequality.
• Convergence of random variables; weak law of large numbers. Borel Cantelli lemmas.
• Characteristic function of a probability measure. Fourier integrals; applications to sums of independent random variables. Inversion theorem.
• Applications of characteristic functions including the Central Limit Theorem.

### Assessment Proportions

• Coursework: 20%
• Exam: 80%

## MATH314: Lebesgue Integration

• Terms Taught: Lent 16-20 / Summer Term only.
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH210.

### Course Description

The aim of this course is to introduce the Lebesgue integral for functions on the real line. The course features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems. The results are illustrated in classical convergence problems including Fourier integrals.

### Educational Aims

The aim of this course is to introduce the Lebesgue integral for functions on the real line. The course features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems. The results are illustrated in classical convergence problems including Fourier integrals.

• To develop a rigorous approach to the notions of length and area by expressing these analytically
• To provide a deeper understanding of the concept of a real number
• To introduce the Lebesgue integral as a tool, enabling students to study advanced topics in mathematical analysis and its applications.

### Outline Syllabus

• Lebesgue's definition of the integral. Integral of a step function. Subsets of the real line; open sets and countable sets. Measure of an open set. Measurable sets and null sets. Lebesgue's area measure in the plane.
• Integrable functions. Lebesgue's integral of a bounded measurable function. Lebesgue's bounded convergence theorem. Lebesgue's integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
• Applications of the convergence theorems. Wallis's product for P. Gaussian integral. Some classical limit inversion results. The Fourier cosine integral.
• Inversion formula for the Fourier cosine transform: Properties of the Fourier transform; Plancherel's formula; Applications of the Fourier transform.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH316: Metric Spaces

• Terms Taught: Michaelmas Term Only Weeks 1 - 5
• US Credits: 4 semester credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Mathematics Majors or appropriate College Mathematics. MATH210 and MATH220.

### Course Description

The course gives an introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications. It offers a deeper understanding of continuity, leading to an introduction to abstract topology. The course provides firm foundations for further study of many topics including geometry, Lie groups and Hilbert space, and has applications in many others, including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

### Educational Aims

This course introduces the student to the mathematical notions of distance and topology, from a modern perspective. This leads to an understanding of the basic ideas and concepts of abstract distance and topology. It will equip students for a broad range of further topics which depend on metric and topological ideas.

### Outline Syllabus

• Metrics and embedding metric spaces in normed spaces
• Lipschitz maps
• Uniform continuity
• Complete metric spaces
• Cantor intersection theorem
• Banach Fixed point theorem
• Compact sets and the Heine-Borel theorem
• Distinguishing topological spaces

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH317: Hilbert Spaces

• Terms Taught: Michaelmas Term only, Weeks 6-10
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Equivalent to MATH210 and MATH220

### Course Description

A Hilbert space is a linear space with the additional feature of an ‘inner product’, which allows us to extend the notions of distance and angle to much more general settings than ordinary geometrical space, for example, to infinite-dimensional spaces of functions. The concepts of linear algebra and analysis combine to produce a powerful theory. The notion of orthogonality is applied to best approximations, bases and linear operators.

### Educational Aims

This course introduces the student to an area of Mathematics in which the concepts of linear algebra, analysis and geometry are harnessed together. It is shown how this leads to powerful and elegant generalizations of earlier results, many of which are fundamental to modern applications of analysis.

### Outline Syllabus

• Normed linear spaces: definition and examples. Sequences and series. Closest points. Convex sets. Continuity and norms of linear mappings. The closure of a set.
• Inner products. The Cauchy-Schwarz inequality and the derived norm. Examples. Linear mapping on inner product spaces.
• Orthogonality. Finding the closest point in a linear subspace. Orthonormal sets. The Gram-Schmidt process. Bessel's inequality. Fourier series.
• Completeness. Theorem on closest points in a closed, convex subset. Orthogonal complements. Representation of linear functionals. Isometry of all separable Hilbert spaces.
• The adjoint of a linear operator. Kernel and range. Quadratic forms. The spectral theorem in finite dimensions.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH318: Differential Equations

• Terms Taught: Michaelmas Term only, Weeks 1-5
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH210

### Course Description

Differential equations arise throughout the applications of mathematics and consequently the study of them has always been recognised as a fundamental branch of the subject. This course aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations and theoretical results about the existence, uniqueness and nature of solutions.

### Educational Aims

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

• Solve first order and elementary types of second order linear ordinary differential equations;
• Use several general methods for second order linear equations and find series solutions;
• Understand Wronskians, the uniqueness theorem and Sturm's separation and comparison theorems for zeros of solutions;
• Understand Sturm-Liouville systems and solve certain examples;
• Solve constant coefficient first order systems of equations and sketch solution curves for 2x2 systems;
• Understand Picard's existence and uniqueness theorem and use Picard iteration to produce approximate solutions.

### Outline Syllabus

• First order linear equations: integrating factors. Linear equations with constant coefficients, with proof that the method gives all solutions.
• Methods for general second-order linear equations: the substitution y=uv, where u is a known solution of L(y)=0; series solutions.
• Theory of second-order linear equations. Wronskians. Uniqueness theorem. Separation of zeros of linearly independent solutions. Theorems on zeros of solutions.
• Boundary value problems. Green's functions. Sturm-Liouville theory.
• Systems of first-order equations. Pairs of first-order equations with constant coefficients; solution curves.
• Picard's existence theorem for systems of first-order equations and for equations of higher order. Picard iteration.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH319: Linear Systems

• Terms Taught: Lent / Summer Terms Only,  Weeks 11 - 15
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS Credits
• Pre-requisites: MATH215 and MATH220

### Course Description

Linear systems is engineering mathematics. In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input. Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The course shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables us to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute. The course then describes feedback control for linear systems. The main result describes all the linear controllers that stabilize an (A,B,C,D) system. The course also describes digital systems.

### Educational Aims

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

• Derive linear differential equations from block diagrams;
• Express linear differential equations in matrix form;
• Manipulate linear differential equations involving matrices;
• Use the exponential of a matrix to solve linear matrix differential equations;
• Understand the (A,B,C,D) system with constant complex matrices;
• Calculate the transfer function of a (A,B,C,D) system, by hand and by appropriate computing software;
• Understand the concept of a bounded-input bounded-output system;
• Understand the notion of stability, and how it relates to the poles of the transfer function;
• Use linear algebra to solve an (A,B,C,D) system;
• Compute the stabilizing controllers of a single input single output linear system.

### Outline Syllabus

Linear operators. Diagrams.Reduction of order for linear differential equations.

The exponential of a square matrix. Exponential of diagonalisable matrices and Jordan canonical forms.

Solving the basic linear differential equation by means of the exponential. Inputs and outputs. The notion of MIMO and SISO. Notion of a linear system (A,B,C,D) for constant complex matrices. The transfer function of (A,B,C,D). The damped harmonic oscillator. BIBO systems and spectral conditions ensuring boundedness of exponential matrices. Lyapunov's condition. Verifying Lyapunov's condition in examples. The space of functions of exponential type and the Laplace transform. Properties of the Laplace transform and Laplace convolution. Solving (A,B,C,D) by the transfer function. Realization of rational functions as transfer functions. The ring of stable rational functions. Coprime factorization in the space of stable rational functions. BIBO systems characterized by stable rational transfer functions. The need for controllers. Notion of feedback control for SISO. Every rational SISO system has a rational controller. Youla's parametrization of the rational controllers of SISO.

### Assessment Proportions

• 10% Coursework
• 70% Exam
• 20% Project

## MATH321: Groups and Symmetry

• Terms Taught: Michaelmas Term only, Weeks 6-10
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH225

### Course Description

The course includes: conjugacy, conjugacy classes; centralizers, conjugacy and normal subgroups; permutations; external and internal direct product classification of finite abelian groups; groups actions; orbits and stabilizers; the orbit-stabilizer theorem classification and symmetry groups of platonic solids; Sylow’s theorems and applications; elementary results on p-groups.

### Educational Aims

The aim of this module is to build on the theory of groups as introduced in the 2nd year module MATH225:

Groups and Rings. Emphasis will be given to finite groups. The most important results covered will be as follows:

• The classification of finite abelian groups.
• The orbit-stabilizer theorem.
• The Jordan-Holder theorem.
• The classification and symmetry groups of the Platonic solids.
• Sylow's theorems.

We shall first consider a way of comparing the elements of a group and show how a group may be built up from smaller components using 'direct products'. Next we shall treat situations in which a group 'acts' on a set by permuting its elements; after identifying the five Platonic solids, we shall use group actions to determine their symmetry groups. Finally we shall prove some interesting and important results, known as the 'Sylow theorems', relating to subgroups of certain orders.

To continue developing students algebraic understanding and their ability to reason from stated axioms and definitions, and introduce them to the use of algebraic ideas in the study of symmetry and geometry.

### Outline Syllabus

• Conjugacy; conjugacy classes; centralizers; conjugacy and normal subgroups; conjugacy for permutations.
• External and internal direct products;classification of finite abelian groups.
• Group actions; orbits and stabilizers; the orbit-stabilizer theorem; classification and symmetry groups of Platonic solids.
• Series of groups; Jordan-Holder theorem; simplicity of An.
• Sylow's theorems and applications; elementary results on p-groups.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH322: Commutative Algebra

• Terms Taught: Michaelmas Term Only
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH111 & MATH225

### Course Description

The course introduces two important classes of commutative rings: Euclidean domains and unique factorisation domains, and then studies the properties of such rings, especially factorisation of polynomials over them. Topics include: Euclidean domains (ED), every ED is a principal ideal domain (PID), highest common factors, irreducible elements and prime elements, unique factorization domains (UFD), every PID is a UFD, primitive polynomials, Gauss’ Lemma, Eisenstein’s Irreducibility Criterion.

### Educational Aims

This is a theory-based course whose aim is to study factorization in integral domains, in particular in polynomial rings, and to present some applications of this theory. It is intended to build upon the material encountered in the second-year courses in Rings and Linear Algebra, and to provide an introduction to the fourth year Galois Theory course.

The course presents students with a hierarchy of properties that certain integral domains possess. The weakest of these properties leads to a factorization theory which is analogous to the prime factorization of integers. The aim of the course is to study these properties and their relationship, to determine which integral domains possess them, and in the positive case to understand how the "prime factorizations" mentioned above can be found in practice.

### Outline Syllabus

• Euclidean domains: motivation, definition, examples; Euclidean domains are principal ideal domains.
• Invertible and associated elements; highest common factors: definition; Bézout's Theorem and Euclidean algorithm.
• Irreducible elements: definition, examples (especially in polynomial rings); prime elements; maximal ideals in principal ideal domains; primitive polynomials and Eisenstein's Irreducibility Criterion.
• Unique factorization domains: motivation; invertible and irreducible elements; definition, examples; principal ideal domains are unique factorization domains.
• Factorization of polynomials over an integral domain: Gauss's Lemma and Gauss's Factorization Theorem; examples and applications; the polynomial ring over a unique factorization domain is a unique factorization domain.
• The field of fractions of an integral domain; the characteristic of a unital ring, finite fields.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH323: Algebraic Curves

• Terms Taught: Lent / Summer Terms Only
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH225 (Coursework equivalent to MATH240 is recommended)

### Course Description

This module gives you a solid foundation in the basics of algebraic geometry. You will explore how curves can be described by algebraic equations, and learn how to understand and use abstract groups in dealing with geometrical objects (curves). You will also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally you'll learn to perform algebraic computations with elliptic curves.

### Educational Aims

This course is an introduction to elliptic curves, and hence to algebraic geometry. It also presents applications and results of the theory of elliptic curves. The course also provides a useful link between concepts from algebra and geometry.

The student should learn the basics of algebraic geometry; understand and use abstract groups in dealing with geometrical objects (curves), know the notions and the main results pertaining to elliptic curves.

### Outline Syllabus

• Algebraic geometry: Polynomial rings, Affine spaces, Projective spaces, Affine and projective plane,
• Projective transformations.
• Algebraic curves: Parametrisation of the projective line, Rational points on curves of degree 2 and 3.
• Intersection multiplicity and singularity, Bézout's theorem.
• Elliptic curves and the group law: Normal and Weierstrass's forms of cubic curves, Invariants of cubic curves, Singular cubic curves, Graphs, Chord-tangent composition law, The group law of an elliptic curve.
• Results on elliptic curves: Nagell-Lutz theorem, Changing the field, Elliptic functions, Mordell's theorem.

### Assessment Proportions

• Coursework: 10%
• Exam: 70%
• Project: 20%

## MATH325: Representation Theory of Finite Groups

• Terms Taught: Lent / Summer Terms Only (Weeks 11 - 15).
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH220, MATH225 (Coursework equivalent to MATH321 is recommended)

### Course Description

The primary aim of this course is to provide an introduction to representation theory. The main part of the course treats the ordinary representations of finite groups, for which two traditional approaches are taken: representations via courses and via group homomorphisms into matrix groups; the correspondence between them being stressed. Various classical results will be covered, in particular, Maschke's theorem, Schur's lemma, and the classification of irreducible representations of finite abelian groups.

### Educational Aims

At the end of the course the students should be able to demonstrate subject specific knowledge, understanding and skills and have the ability to:

• Understand the basics of representation theory -understand the concept of CG-module, the use of matrix groups in the study of the representations of an abstract finite group, and the correspondence between group representations and CG-modules.
• Know and be able to apply the main results such as Schur's lemma and Maschke's theorem.
• Know how to find the irreducible representations of finite abelian groups.
• Know the notions of group algebra and composition factors.

### Outline Syllabus

• R-modules and R-homomorphisms for a unital ring R: submodules, direct sums and quotient modules;
• the kernel and the image of an R-homomorphism; isomorphism theorems for R-modules.
• Representations of Finite groups; correspondence between \CG-modules and the representations of a Finite group G.
• Maschke's theorem and complete reducibility.
• \CG-homomorphisms; Schur's lemma; spaces of \CG-homomorphisms and their dimensions.
• The representations of Finite abelian groups.
• The group algebra \CG of a Finite group G and \CG-modules; composition factors; the regular representation of a Finite group, and the decomposition of the group algebra.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH326: Graph Theory

• Terms Taught: Michaelmas Term only, Weeks 6-10
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH220

### Course Description

The course aims to introduce students to graph theory, a substantial modern area of mathematics with many applications in pure and applied sciences. Graph theory originated in Euler's 1736 paper on the seven bridges of Konigsberg. Concepts are introduced alongside examples and, where possible, discussion of how the concepts can be applied in real world situations.

### Educational Aims

Graph theory is the cornerstone of discrete mathematics. This course will introduce a range of fundamental topics in graph theory. Graphs are mathematical structures used to model pairwise relations between objects, for example networks of communications. Students will develop an appreciation for a range of discrete mathematical techniques. Emphasis will be placed on topics linking graph theory to linear algebra and to topology.

### Outline Syllabus

• Basics of graph theory
• Structural notions such as connectivity
• Properties of trees and methods for counting them
• Graph minors, minor closed families and characterisations
• Matrices associated to graphs, their eigenvalues, determinants and properties of graphs
• The Tutte-polynomial, the chromatic polynomial and graph colourings
• Planar graphs and graphs on surfaces of higher genus, non-orientable surfaces

### Assessment Proportions

• Coursework: 20%
• Exam: 80%

## MATH327: Combinatorics

• Terms Taught: Lent Term only, Weeks 11 - 15
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH112 & MATH220

### Course Description

Historically, combinatorics has its roots in mathematical recreations, puzzles and games. However, many problems in combinatorics that were previously studied simply for amusement or for their aesthetic appeal, are today of great importance in many areas of pure and applied sciences. For example, applications of combinatorics include topics as diverse as codes, circuit design, molecular assembly and drug design, and algorithm complexity. Moreover, in pure mathematics, combinatorial problems and methods frequently arise in areas such as algebra, probability, topology and geometry. It has thus become essential for workers in many scientific fields to have some familiarity with this subject.

### Educational Aims

• The course provides an introduction to some of the key concepts and methods in combinatorics and graph theory, the cornerstones of discrete mathematics.
• In particular, the course will introduce students to various important counting methods and counting coefficients, to some key concepts, methods and algorithms in graph theory and to the theory of combinatorial designs.
• Some applications of the results and methods will also briefly be discussed.
• The course aims to introduce students to discrete mathematics, a fundamental part of mathematics with many applications in computer science and other pure and applied sciences.

### Outline Syllabus

• Counting: basic techniques; the fundamental counting coefficients; Stirling numbers; recursions; a review of generating functions.
• Graphs and Algorithms: graphs and directed graphs; trees; Eulerian and Hamiltonian graphs; matchings; networks; colourings.
• Combinatorial Designs: Latin squares; block designs; Steiner systems.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH328: Number Theory

• Terms Taught: Lent / Summer Terms only (Weeks 16 - 20)
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH111 (Coursework equivalent to MATH225 may be helpful)

### Course Description

The course gives an all round introduction to the concepts, results and methods of number theory, including topics such as the following: Greatest common divisors, congruence, prime numbers, arithmetic functions, the divisor and phi functions, the Euler-Fermat theorem and applications to coding, quadratic residues, Dirichlet series, convolutions, the Möbius function, sums of two squares, partial sums of arithmetic functions. Connections between these topics will be emphasised, and results will be illustrated by numerical examples.

### Educational Aims

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

• Use modular arithmetic to solve simple problems or reformulate advanced ones;
• Prove that various types of prime numbers occur in infinite families;
• State, prove and apply the Euler-Fermat theorem;
• Identify when a number can be written as a sum of two squares;
• State and prove properties of Mersenne, Fermat and perfect numbers
• Use convolution and the Möbius function to examine relationships between arithmetic functions

### Outline Syllabus

• Division, congruence and greatest common divisors. Chinese remainder theorem. Unique prime factorization, with applications. Infinitude of the primes. Factorizing large numbers. Mersenne and Fermat numbers.
• Arithmetic functions. The divisor function, sum of divisors and Euler's phi function
• Groups of residue classes coprime to an integer. The Euler-Fermat theorem, application to coding. Carmichael numbers. Quadratic residues.
• Dirichlet series and convolutions. The Euler product. The Möbius function. Partial sums of the divisor function and Euler's phi function.
• Numbers that can be expressed as sums of two squares. Pythagorean triples and Fermat's theorem for n=4.
• Further topics chosen from: Sums of four squares. Chebyshev's bounds for the frequency of primes. Zeros of polynomials mod p and primitive roots.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH329: Geometry of Curves and Surfaces

• Terms Taught: Lent / Summer Term only, (Weeks 16 - 20)
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH115 or MATH220

### Course Description

This course is an introduction to the study of smooth curves and surfaces in three-dimensional space. Various geometrical properties of these objects will be encountered; some familiar ones, such as length and area, and some less familiar, such as torsion and curvature. The meaning of these quantities will be explored and their values will be calculated for a variety of examples, applying techniques from calculus and linear algebra.

### Educational Aims

• To provide an introduction to the differential geometry of curves and surfaces in three-dimensional space.
• To allow the student to appreciate geometric concepts, by showing how ideas from calculus can be used to compute familiar and novel quantities which allow the description of three-dimensional curves and surfaces

### Outline Syllabus

• Parameterised curves in three-dimensional space. Arc-length. Tangent, normal and binormal. Serret-Frenet formulae.
• Smooth surfaces in three-dimensional space; regularity and tangent planes. The unit normal.
• The length of a curve on a surface. The first fundamental form. Surface area. Isometric surfaces.
• Geodesic curvature of a curve in a surface. Geodesics. The geodesic equations (derivation non-examinable).
• Geodesics as extremal paths (proof omitted). Geodesics on surfaces of revolution.
• Normal curvature of a curve in a surface. The second fundamental form. Principal curvatures and directions.
• Gaussian curvature. Gauss's Theorema Egregium.
• Further topics chosen from: Mean curvature. Minimal surfaces. The Gauss-Bonnet theorem.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH330: Likelihood Inference

• Terms Taught: Michaelmas Term only, Weeks 1-5
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH235

### Course Description

This course aims to present the key tools for statistical inference, stressing the fundamental role of the likelihood function. The approach taken will be quite geometrically based, showing how the classical methods are simple geometrical summaries of the likelihood function at its maximum position. The role of likelihood methods in substantive applications will be emphasised and illustrated.

### Educational Aims

The course aims to present the key tools for statistical inference, stressing the fundamental role of the likelihood function. The approach taken will be quite geometrically based, showing how the classical methods are simple geometrical summaries of the likelihood function at its maximum position. The role of likelihood methods in substantive applications will be emphasised and illustrated.

This course aims for students:

• To gain skills in problem solving and critical thinking.
• To appreciate the importance of communicating technical ideas at an appropriate level.
• To appreciate the importance of making evidence-based decisions.

### Outline Syllabus

• The likelihood function and maximum likelihood estimation.
• Comparing estimators and the Cramér-Rao lower bound.
• Multiparameter problems and examples.
• Properties of the likelihood.
• Asymptotics: the distribution of maximum likelihood estimators
• The multivariate normal distribution.
• Model selection and the generalised likelihood test.
• Graphical and geometrical interpretations.
• Use of the profile likelihood for dealing with nuisance parameters.
• Parameter functions.
• Substantial applications of the above topics e.g. to ANOVA testing and extensions outside the Gaussian family.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH331: Bayesian Inference

• Terms Taught: Michaelmas Term only, Weeks 1-5
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Equivalent of  MATH 330 Likelihood Inference;
• Equivalent to MATH 235 Statistics
• Mathematics Majors or appropriate College Mathematics.

### Course Description

This course 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 its decision theoretic foundations.

### Educational Aims

At the end of the course the students should be able to demonstrate subject specific knowledge,understanding and skills and have the ability to:

• To be able to resolve some well-known discrete paradoxes using a Bayesian formulation
• To understand the advantages that a prior brings to statistical analysis, appreciate some of the difficulties it creates and show how these difficulties are resolved.
• To recognise distributions from the exponential family, derive their conjugate priors, the posterior distributions, the marginal likelihoods and the predictive distributions.
• To appreciate the role of each of the above distributions in integrating statistical reasoning and to simulate or calculate them using the language R from simple datasets.
• To be familiar with the 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
• To be able to carry out a variety of strategies for dealing with multi-parameter problems using directed graphs and the language R
• To be able to contrast the Bayesian and classical approaches toward making inferences, choosing models and predicting

### Outline Syllabus

• Bayesian updating of belief
• The formulation of a prior belief for a Bayesian analysis
• Bayesian decision theory and the role of the utility function in Bayesian estimation
• The predictive and marginal distributions for model checking and selection and model selection
• Multi-parameter models
• Comparison with likelihood methods
• The asymptotic approximation for multi-parameter and non-conjugate models
• The use of all the above in substantive applications

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH332: Stochastic Processes

• Terms Taught: Michaelmas Term only, Weeks 6-10
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Equivalent to MATH 230 Probability

### Course Description

The course aims to show how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced. By the end of the course you should be able to use conditioning arguments to calculate probabilities and expectations of random variables for stochastic processes; to calculate the distribution of a Markov process at different time points; to determine whether a Markov process has an asymptotic distribution and to calculate it; and to understand how stochastic processes are used as models.

### Educational Aims

On successful completion of this module students will be able to

• Understand the relevance of stochastic processes as natural models for stochastic phenomena;
• Understand mathematically the definition of a stochastic process;
• Use and manipulate generating functions for probability calculations;
• Carry out simple calculations for probabilities of simple random walks;
• Understand finite-dimensional and infinitesimal definitions of continuous time Markov processes;
• Have a basic knowledge of the principles of irreducibility and recurrence;
• Understand equilibrium distributions for Markov chains in simple examples;
• Understand simple examples such as those from queues, and transfer techniques and skills to other similar examples.

### Outline Syllabus

• The Bernoulli process and the simple random walk.
• Conditional expectations and applications to random walks and the Gambler's ruin. The reflection principle.
• Generating functions and their applications.
• Markov chains in discrete time: time-dependent state distribution, stationary distribution, limit theorems, reversible Markov chains, expected hitting times and explicit n-step formulae.
• Markov chains in continuous time: time-dependent state distribution, limit theorems, queuing networks, Poisson processes, birth-death and immigration models.

### Assessment Proportions

• Coursework: 20%
• Exam: 80%

## MATH333: Statistical Models

• Terms Taught: Lent / Summer Terms Only. Weeks 16-20.
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Equivalent of MATH 330 Likelihood Inference.
• Coursework equivalent to MATH240 is recommended.

### Course Description

The course introduces a class of well known statistical models for regression problems. The class includes linear regression for normal data, generalised linear models for non-normal data. By the end of the course you should be able to formulate sensible models for different sets of data, taking account of the constraints on the data, and to explore and analyse the data using R.

### Educational Aims

• To understand the theoretical basis of generalized linear models and to apply to a diverse range of practical problems.
• To understand the effect of censoring in the statistical analyses and to use appropriate statistical techniques for lifetime data.
• To relate modern statistical models and methods to real life situations and use relevant computer software for statistical analysis.

### Outline Syllabus

Normal linear models

• Normal distribution
• Effects of covariates
• Likelihood inference

Logistic regression

• Binary data
• Binomial distribution

Log-linear models and Poisson regression

• Count data
• Poisson distribution

Generalized linear models (GLMs)

• Exponential family distributions, the linear predictor, link function, likelihood inference
• Model selection based on deviance.
• The iteratively reweighted least squares (IRWLS) method.
• Application of data sets.

* The R language is required for the laboratory sessions of MATH333. Combined major or minor students who do not take the whole of MATH390 should, if at all possible, attend the practical course in the R language which is given as part of MATH390, usually in week 6 or 7 of the summer term of their second year.

Single major mathematics students are required to write the project report using LaTeX, and so are combined major students who have taken the whole of MATH390. Other students are encouraged to audit the LaTeX course in MATH390 to enable them to use this to write their project, but they are permitted to use any other suitable word processor.

### Assessment Proportions

• Coursework: 20%
• Exam: 50%
• Project: 30%

## MATH334: Time series analysis

• Terms Taught: Lent / Summer Terms Only, Weeks 11-15
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Equivalent of MATH330 Likelihood Inference.
• Coursework equivalent to MATH240 is recommended

### Course Description

This course aims to provide an introduction to recent developments in statistics. This may include statistical methods for analysing time series, multivariate data with emphasis on the financial applications, change-point analysis and stochastic volatility models.

### Educational Aims

• To provide an introduction to recent developments in statistics. This may include statistical methods for analysing time series, multivariate data with emphasis on the financial applications, change point analysis and stochastic volatility models.
• To allow the student to appreciate statistical methods and data analysis concepts as well as the use of the statistical software R.

### Outline Syllabus

• Topics taken from the following: Time Series, Volatility Modelling, Multivariate Analysis, Change Point Methods. The choice of topics and level of depth will reflect the lecturer's personal research interests.
• Time Series: descriptive statistics for stationary time series; linear models for stationary time series; AutoRegressive (AR), Moving Average (MA), and AutoRegressive Moving Average (ARMA) models; harmonic regression; periodogram; linear models for nonstationary and seasonal time series; ARIMA, EWMA predictors; Box Jenkins models.
• Volatility Modelling: conditional heteroscedasticity; Value at Risk; ARCH and GARCH models; QMLE and other robust estimation procedures; asymptotics; diagnostics; various stochastic volatility models.
• Multivariate Analysis: general introduction to multivariate data; exploratory data analysis; multivariate normal distribution and associated inference; principal components analysis; classification using cluster analysis.
• Change Point: Methods: general introduction to change point problems; CUSUM; likelihood based approaches; penalised likelihood; current trends in change point methods.

* The R language is required for the laboratory sessions of MATH334. Combined major or minor students who do not take the whole of MATH390 should, if at all possible, attend the practical course in the R language which is given as part of MATH390, usually in week 6 or 7 of the summer term of their second year. Single major mathematics students are required to write the project report using LaTeX, and so are combined major students who have taken the whole of MATH390. Other students are encouraged to audit the LaTeX course in MATH390 to enable them to use this to write their project, but they are permitted to use any other suitable word processor.

• Exam: 70%
• Project: 30%

## MATH335: Medical Statistics: study design and data analysis

• Terms Taught: Lent / Summer Terms Only
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Equivalent of MATH235 Statistics and MATH 240 Project Skills.

### Course Description

This course aims to understand the conceptual and theoretical basis of health investigations including measures of disease, study design, causality, confounding and measures of disease-exposure association (mortality and morbidity). The course develops a firm understanding of key analytical methods and procedures used in studies of disease aetiology, disease screening/diagnosis and clinical trials. It includes the understanding of the effect of censoring in the statistical analyses and the use of appropriate statistical techniques for time to event data.

### Educational Aims

• To understand the conceptual and theoretical basis of health investigations including measures of disease, study design, causality, confounding and measures of disease-exposure association (mortality and morbidity). Firm understanding of key analytical methods and procedures used in studies of disease aetiology. disease screening/diagnosis and clinical trials including the impact of key study designs upon inference.
• To understand the effect of censoring in the statistical analyses and to use appropriate statistical techniques for time to event data.
• To relate statistical models and methods to address 'real' health research questions and to use relevant computer software for statistical analysis. To critique published articles related to health research and discuss in context.

### Outline Syllabus

• Introduction to health research: Overview of aims of health research disease: - detection, - control, - prevention and - aetiology, causality, types of outcome, confounding, study designs and design hierarchy.
• Distribution and determinants of disease: measures of disease (risk, odds, incidence, prevalence etc), observational designs (cohort, case-control, - nested, case-control and matching), exposures, association measures: odds ratios, relative risks and associated measures of uncertainty. Impact of design upon inference/estimation.
• Disease surveillance: diagnostic testing, disease screening, evaluation: NPV, PPV sensitivity, specificity, ROC etc.
• Clinical Trials: intervention studies and causality, measurements and end points, bias and replication, treatment allocation: randomisation and control, group comparisons: t-tests, use of baseline measurements.
• Survival analysis: time to event data, censoring, the survivor and hazard function. Kaplan-Meier estimate of the survivor function, parametric models, sub-group comparisons.

### Assessment Proportions

• Coursework: 10%
• Exam: 70%
• Project: 20%

## MATH336: Mathematics for Artificial Intelligence

• Terms Taught: Michaelmas Term only, Weeks 6-10
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Students should be familiar with multivariate random variables and matrix algebra.
• Equivalent to MATH220, MATH235 and MATH245

### Course Description

This module aims to introduce students to some of the theoretical and practical elements of machine learning and multivariate statistics. The specific focus will be on, multivariate data representation/visualisation,feature extraction and dimensionality reduction, e.g. through Principal Component Analysis (PCA),multivariate data classification using e.g. discriminant analysis and Support Vector Machines (SVMs).Apart from learning the theoretical aspects of the above methods, the students will also learn how to apply them in practice using R.

### Educational Aims

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

• Appreciate the need for multivariate statistical analysis
• Represent and visualise high-dimensional data
• Understand the concept of dimensionality reduction and its use in feature extraction from high-dimensional data
• Comprehend, analyse, apply and interpret methods for multivariate data classification

As well as learning the theoretical aspects of the above methods, the students will also learn how to apply them in practice using R.

### Outline Syllabus

The course will cover the following topics in multivariate statistics, using the classical problem of data classification as a running example.

1. Mathematical representation and visualisation of multivariate data, motivation for multivariate analysis, the classification problem
2. Dimensionality reduction and Principal Component Analysis
3. Linear Discriminant Analysis
4. Support Vector Machines

* The R language is required for the laboratory sessions.

### Assessment Proportions

• Coursework: 30%
• Exam: 70%

## MATH345: Mathematics for Stochastic Finance

• Terms Taught: Lent and Summer Term only, Weeks 16-20
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH230 & MATH332

### Course Description

This module will give a simple introduction to mathematical finance. This includes some financial terminology and the study of European and American option pricing with respect to different models. More precisely, we consider two discrete models, the binomial Model and finite market model, and one continuous model, the Black Scholes model. We also introduce some probabilistic terminology, which is required to study the properties of these models. This includes martingales, stopping times and an overview over the mathematical theory of Brownian motion, including stochastic integration with respect to Brownian motion.

### Educational Aims

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

• Construct binomial tree model
• Determine the associated risk-neutral probability
• Describe the basic concepts of investment strategy analysis, including self-financing, arbitrage free
• Price forward contracts and perform price calculations for stocks without and with dividend payments
• Calculate the price of various European and American options in a binomial tree model
• Construct a replicating portfolio in a binomial tree model
• Perform calculations with the Black-Scholes call option price formula
• Be able to prove various steps in the derivation of the Black-Scholes call option price formula

### Outline Syllabus

• Discrete financial models: Binomial Model and Finite market model
• Continuous financial model: Black Scholes model
• European and American option pricing
• Conditional expectation
• Filtrations
• Martingales
• Stopping times
• Brownian motion
• Black-Scholes formula
• Stochastic integration with respect to Brownian motion

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH361: Mathematical Education

• Terms Taught: Full Year course, Weeks 6-16
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Mathematics Majors or appropriate College Mathematics.

### Course Description

The aims of this course are to provide you with opportunities to gain insights into a range of issues relating to mathematics education in general, and consider theories about teaching and learning in particular. This will be achieved in practical, activity based workshops and in seminars.

### Educational Aims

The aims of this course are as follows:

• To enable students to reflect on mathematics, in particular on its history, application and significance in human culture;
• To foster in students a deeper understanding of philosophies on the learning of mathematics and their application to mathematics education;
• To consider the position of mathematics within the curriculum;
• To consider the purposes of assessment, and the advantages and disadvantages of various forms there of.

### Outline Syllabus

• Mathematics as a problem solving discipline;
• Using ICT to do, learn and teach mathematics;
• Different ways that children learn mathematics;
• Different styles and strategies a mathematics teacher might use to enhance learning;
• Significant developments and government initiated publications since 1980, e.g. the national Curriculum (DCSF 2008);
• Assessing children's mathematical achievements.

### Assessment Proportions

• Coursework: 100% (2 assignments 20% Michaelmas / 80% Lent)

## MATH411: Operator Theory

• Terms Taught: Lent / Summer Terms Only
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Equivalent to MATH317 or MATH417 Hilbert Spaces

### Course Description

The aims of this course are to build on the theory of Hilbert spaces, to analyse a variety of Hilbert space operators, together with spectral theory for self adjoint operators, and basic integral operators.

### Educational Aims

At the end of the module students should be able to:

• Define and give examples of various types of bounded linear operator on Hilbert space;
• Determine invertibility of various specific operators (e.g. diagonal operators, shifts);
• Define and determine the adjoint of specific operators;
• Understand the invertibility criteria for self-adjoint operators;
• Understand the rudiments of the spectral theory of Banach algebras;
• Understand the spectral theorem for compact self-adjoint operators.

### Outline Syllabus

• Review of finite-dimensional operator theory. Review of Hilbert Space
• Bounded operators on Hilbert space. Inverse operators, adjoint operators, duality of range and kernel.
• Special operators, unitary operators, multiplication operators, projections, compact operators and shift operators.
• Elementary Banach algebra. Continuous functional calculus for self adjoint operators. Spectral theory for bounded operators.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH412: Topology and Fractals

• Terms Taught: Lent / Summer Terms Only
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Equivalent to MATH 210 Real Analysis.

### Course Description

The aims of this course are to understand the definitions and the basic properties (such as closedness, connectedness, fractal dimension) of a variety of fractals. These include, the Cantor set and Cantor dust, the Sierpinski sieve, the von Koch (snowflake) curve, number system fractals, fractals defined by iterated function systems

### Educational Aims

This course aims to:

• Develop students' knowledge and skills in analytical arguments in set theory and in analysis.
• To analyse diverse fractal sets rigorously.

The course gives an introduction to topology in the Euclidean spaces Rn, embracing fundamental terms such as connectedness, total disconnectedness and compactness.

These notions are illustrated with fractal sets and the basic theory of these sets is developed, including their generation by iterated function systems and identification of fractional dimensions.

### Outline Syllabus

• The middle thirds Cantor set. Negligible sets. Elementary topology in R2. Disconnected and totally disconnected sets. Ways in which sets and fractals are similar: homeomorphism and lipeomorphism.
• Deletion fractals: the Sierpinski gasket, the Sierpinski carpet, Cantor dust, the Menger sponge, and others.
• Fractals, what they are and their features. The snowflake curve, the von Koch curve, dragon curves, Heighway's dragon.
• Hausdorff distance and limits of sequences of sets.
• Number system fractals. Affine transformations and self-similarity. Iterated functions systems. Similarity dimension (two versions).

General Cantor sets.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH413: Probability Theory

• Terms Taught: Lent / Summer Terms only, Weeks 11-15
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Equivalent to MATH 210 Real Analysis and MATH 230 Probability (excl. MATH 313).

### 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. The course links second year analysis courses with statistics courses. The characteristic function is used to study the distributions of sums of independent variables. The results are illustrated in applications to random walks and to statistical physics.

### Educational Aims

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

• Understand the abstract notion of a probability space;
• Deduce properties of measure from the axioms;
• Verify that certain discrete probability spaces satisfy the axioms;
• Verify that certain collections of sets are sigma-algebras;
• Understand the abstract notion of a random variable;
• Derive fundamental properties of the cumulative distribution function from the axioms of measure;
• Understand the basic properties of probability density functions;
• Find the probability density function of the x2 random variable using the method of distribution functions;
• Prove basic properties of the expectation of simple random variables;
• Prove basic properties of expectation and variance;
• Derive Chebyshev's inequality from the axioms;
• Calculate the expectation of certain functions of continuous random variables;
• Understand the technical definition of independence;
• Understand different probabilistic notions of convergence;
• Prove comparative strengths of probabilistic modes of convergence;
• Prove the weak law of large numbers and apply it to simple examples;
• Prove the Borel-Cantelli Lemmas, and apply them to simple examples;
• Calculate the characteristic functions of some random variables, and understand its properties;
• Prove addition rules for the characteristic functions;
• Prove replicating properties for sums of independent identically distributed random variables;
• Prove the integrated cosine inversion theorem, and apply it in special cases;
• Understand Maxwell's calculation of the speed distribution of an ideal gas;
• Understand basic properties of the Cauchy distribution relating to laws of large numbers;
• Understand the proof of the Central Limit Theorem.

### Outline Syllabus

• Sigma algebras of sets; probability measures; countable additivity.
• Random variables and the cumulative distribution function. Transformations of random variables and probability density functions.
• Expectation and variance. Chebyshev's inequality. Cauchy-Schwartz inequality.
• Convergence of random variables; weak law of large numbers. Borel-Cantelli lemmas.
• Characteristic function of a probability measure. Fourier integrals; applications to sums of independent random variables. Inversion theorem.
• Applications of characteristic functions including the Central Limit Theorem.

### Assessment Proportions

• Coursework: 20%
• Exam: 80%

## MATH414: Lebesgue Integration

• Terms Taught: Lent/Summer Term only, Weeks 16-20
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Equivalent to MATH210 Real Analysis (excl. MATH314).

### Course Description

The aim of this course is to introduce the Lebesgue integral for functions on the real line. The course features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems. The results are illustrated in classical convergence problems including Fourier integrals.

### Educational Aims

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

• Understand the basic concepts of Lebesgue's definition of the integral;
• Appreciate Dirichlet's comb function;
• Prove basic properties of countable sets;
• Understand Archimedes' axiom and Cantor's uncountability theorem;
• Prove the structure theorem for open sets;
• Prove covering lemmas for open sets;
• Understand the statement (not proof) of Heine-Borel theorem;
• Understand the concept and prove basic properties of outer measure;
• Understand inner measure;
• Prove Lebesgue's theorem on countable additivity of measure;
• Understand the notion of a measurable function, and prove continuous functions are measurable;
• Define the integral of a bounded measurable function;
• Extend the definition of the integral to unbounded functions and understand its properties;
• Prove the bounded monotone convergence theorem, and deduce the dominated and monotone convergence theorems;
• Apply the convergence theorems in standard examples such as the series for the zeroth order
• Bessel's function or Riemann's zeta function;
• Derive Wallis's product for p and deduce the Gaussian integral;
• Establish that Fourier transform integrals are continuous;
• To calculate the Fourier integral of the Gaussian probability density;
• Understand the statement of the Plancherel formula.

### Outline Syllabus

• Lebesgue's definition of the integral. Integral of a step function. Subsets of the real line; open sets and countable sets. Measure of an open set. Measurable sets and null sets.
• Integrable functions. Lebesgue's integral of a bounded measurable function. Lebesgue's bounded convergence theorem. Lebesgue's integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
• Applications of the convergence theorems. Wallis's product for Pi. Gaussian integral. Some classical limit inversion results. The Fourier cosine integral.
• Inversion formula for the Fourier cosine transform: Properties of the Fourier transform; Plancherel's formula; Applications of the Fourier transform

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH416: Metric Spaces

• Terms Taught: Michaelmas Term only, Weeks 1-5
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH210 and MATH220

### Course Description

The course gives an introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications. It offers a deeper understanding of continuity, leading to an introduction to abstract topology. The course provides firm foundations for further study of many topics including geometry, Lie groups and Hilbert space, and has applications in many others, including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

### Educational Aims

This course introduces the student to the mathematical notions of distance and topology, from a modern perspective. This leads to an understanding of the basic ideas and concepts of abstract distance and topology. It will equip students for a broad range of further topics which depend on metric and topological ideas.

### Outline Syllabus

• Metrics and embedding metric spaces in normed spaces,
• Lipschitz maps,
• Uniform continuity,
• Complete metric spaces,
• Cantor intersection theorem,
• Banach Fixed point theorem,
• Compact sets and the Heine-Borel theorem,
• Distinguishing topological spaces.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH417: Hilbert Spaces

• Terms Taught: Michaelmas Term only, Weeks 6-10
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Equivalent to MATH 220 Linear Algebra and MATH 210 Real Analysis (excl. MATH 317).

### Course Description

This course introduces you to an area of Mathematics in which the concepts of linear algebra, analysis and geometry are harnessed together. It is shown how this leads to powerful and elegant generalisations of earlier results, many of which are fundamental to modern applications of analysis.

### Educational Aims

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

• Define the notions norm and inner product;
• Define and give examples of equivalent and inequivalent norms;
• Evaluation norms of linear operators and functionals;
• Understand and apply the Cauchy-Schwarz inequality;
• Use orthogonality to find best approximations in spaces of sequences or functions;
• Understand the existence proof for orthonormal basses, and their application to isomorphisms and Parseval's identity;
• Understand the existence proof for closest points and the applications to orthogonal projections and the Riesz representation theorem;
• Define the adjoint of an operator and understand the proof of the spectral theorem for self adjoint operators in finite dimensions.

### Outline Syllabus

• Normed linear spaces: definition and examples. Sequences and series. Closest points. Convex sets.
• Continuity and norms of linear mappings. The closure of a set.
• Inner products. The Cauchy-Schwarz inequality and the derived norm. Examples. Operators on inner product spaces.
• Orthogonality. The closest point to a closed subspace. Orthonormal sets. The Gram-Schmidt process.
• Bessel's inequality. Fourier series.
• Completeness. Theorem on closest points in a closed, convex subset. Orthogonal complements.
• Isomorphism of all infinite-dimensional separable Hilbert spaces.
• Kernel and range of an operator. Riesz-Fréchet Theorem.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH423: Algebraic Curves

• Terms Taught: Lent / Summer Terms Only
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Equivalent to MATH 225 Groups and Rings
• Coursework equivalent to MATH240 is recommended (excl. MATH 323).

### Course Description

This course is an introduction to elliptic curves, and hence to algebraic geometry. It also presents applications and results of the theory of elliptic curves. The course also provides a useful link between concepts from algebra and geometry, and introduces in a specific context some ideas that have motivated the development of modern algebraic geometry.

### Educational Aims

At the end of the course the students should be able to demonstrate subject specific knowledge, understanding and skills and have the ability to use:

• Algebraic and projective varieties,
• Elliptic curves in various forms, and their group law,
• Main notions and results pertaining to elliptic curves (Bzout, Nagel-Lutz, Mordell).

Also, students must know how to perform some elementary computations using the algebra of the group of an elliptic curve and the geometry of the curve, find rational points.

### Outline Syllabus

• Algebraic geometry: Polynomial rings, Affine spaces, Projective spaces, Affine and projective plane, Projective transformations.
• Algebraic curves: Parametrisation of the projective line, Rational points on curves of degree 2 and 3, Intersection multiplicity and singularity, Bézout's theorem.
• Elliptic curves and the group law: Normal and Weierstrass's forms of cubic curves, Invariants of cubic curves, Singular cubic curves, Graphs, Chord-tangent composition law, The group law of an elliptic curve.
• Results on elliptic curves: Nagel-Lutz theorem, Changing the field, Elliptic functions, Mordell's theorem.

### Assessment Proportions

• Coursework: 10%
• Exam: 70%
• Project: 20%

## MATH424: Galois Theory

• Terms Taught: Michaelmas term only  (Weeks 1 - 10)
• US Credits: 4 Semester Credits
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH225, MATH322
• Coursework equivalent to MATH321 recommended

### Course Description

With this module students will apply Galois theory in a number of important situations, including deciding when a polynomial equation is solvable by radicals, and which regular n-gons are constructible with ruler and compasses;

Students will also appreciate the historical development of Galois Theory and its influence on the subsequent development of pure mathematics as a whole.

### Educational Aims

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

• Compute the degree of a finite extension;
• Understand basic results about algebraic extensions and compute minimal polynomials;
• Find splitting fields;
• Understand basic results about normal extensions;
• Compute the Galois group of any of a wide class of extensions and set up the Galois correspondence between its subgroups and the subfields of the extension.
• Understand field extensions of constructible numbers.
• Understand and use basic results concerning simple groups, solvable groups and composition series, in particular, know that the symmetric group of degree n is soluble only if n
• Find radical extensions and normal closures, and know basic results relating to them;
• Know what is meant by an abelian and by a cyclic extension, and prove results relating to them;
• Find the Galois group of a polynomial;
• Understand Galois's criterion for solubility by radicals and its applications to the general equation of degree n and to rational polynomial equations;
• Appreciate the application of Galois theory to ruler and compass constructions, particularly to the problem of determining which regular n-gons are so constructible.

### 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.
• 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

• Coursework: 30%
• Exam: 70%

## MATH425: Representation Theory of Finite Groups

• Terms Taught: Lent / Summer Terms only, (Weeks 11 - 15)
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Equivalent to MATH 225 Groups and Rings;
• Coursework equivalent to MATH 220 Linear Algebra and MATH 321 Groups and Symmetry is recommended (excl MATH 325).

### Course Description

This course sets out to give an appreciation of the significance of the Fundamental Theorem of Galois Theory through a consideration of its historical context, and through a number of important applications, including the solvability of polynomial equations.

### Educational Aims

The primary aim of this course is to provide an introduction to representation theory.

The main part of the course treats the ordinary representations of finite groups, for which two traditional approaches are taken: representations via modules and via group homomorphisms into matrix groups; the correspondence between them being stressed. Various classical results will be covered, in particular, Maschke's theorem, Schur's lemma, and the classification of irreducible representations of finite abelian groups.

The student should learn the basics of representation theory. In particular, the concept of ordinary group representations and the pertaining important results.

### Outline Syllabus

• R-modules and R-homomorphisms for a unital ring R: submodules, direct sums and quotient modules; the kernel and the image of an R-homomorphism; isomorphism theorems for R-modules.
• Representations of finite groups; correspondence between CG-modules and the representations of a finite group G.
• Maschke's theorem and complete reducibility.
• CG-homomorphisms; Schur's lemma; spaces of CG-homomorphisms and their dimensions.
• The representations of finite abelian groups.
• The group algebra of a finite group G and CG-modules; composition factors; the regular representation of a finite group, and the decomposition of the group algebra.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH426: Lie Groups and Lie Algebras

• Terms Taught: Michaelmas Term only, Weeks 1-10
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites:
• Mathematics Majors or appropriate College Mathematics.
• Equivalent to MATH225 (Coursework equivalent to MATH115 or MATH329 and MATH321 or MATH325 may be helpful)

### Course Description

Lie groups and Lie algebras form an indispensable part of the toolkit of any pure and applied mathematician; they are also widely used in theoretical physics and even chemistry. It is, therefore, important that students have at least some initial exposure to this fundamental and exciting area. This course will allow the students to appreciate the subtle and pervasive interplay between algebra and geometry and appreciate the unified nature of mathematics. The abstract nature of the course will give students a taste of modern research in pure mathematics.

### Educational Aims

At the end of the course the students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras. They will be able to demonstrate subject specific knowledge, understanding and skills and have the ability to:

• Construct a Lie algebra associated with a given Lie group.
• Integrate a Lie algebra to a Lie group, in the nilpotent and simply connected cases.
• For a given homomorphism of Lie groups construct homomorphisms between their Lie algebras.
• Recover a homomorphism between two Lie groups from the tangent homomorphism of their Lie algebras.
• Analyse various examples of Lie groups and Lie algebras, particularly those associated with matrices and vector fields on manifolds.
• Compute the adjoint representations of matrix Lie groups and matrix Lie algebras.
• Discern patterns of Lie groups an Lie algebras when working with various algebraic and geometric objects: associative and commutative algebras, modules, derivations, automorphisms etc.

### Outline Syllabus

• One-parameter groups of transformations, vector fields. Matrix groups.
• Lie algebras, and their example coming from matrix groups.
• The exponential map. Campbell-Hausdorff formula.
• The Lie algebra of a Lie group and a reconstruction of a Lie group from a Lie algebra; the Lie correspondence.
• Adjoint representations of a Lie group and of a Lie algebra. Lie subgroups and quotient groups.

### Assessment Proportions

• Coursework: 40%
• Essay: 30%
• Presentation: 30%

## MATH432: Stochastic Processes

• Terms Taught: Michaelmas Term only, Weeks 6-10
• US Credits: 4 Semester Credits.
• ECTS Credits: 7.5 ECTS
• Pre-requisites: Equivalent to MATH 230 Probability (exclusion: MATH 332 Stochastic Processes).

### Course Description

The course aims to show how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced. By the end of the course the you should be able to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables for stochastic processes; to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times; to determine whether a Markov process has an asymptotic distribution and to calculate it; and to understand how stochastic processes are used as models.

### Educational Aims

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

• Understand the relevance of stochastic processes as natural models for stochastic phenomena;
• Understand mathematically the definition of a stochastic process;
• Use and manipulate generating functions for probability calculations;
• Carry out simple calculations for probabilities of simple random walks;
• Understand finite-dimensional and infinitesimal definitions of continuous time Markov processes;
• Have a basic knowledge of the principles of irreducibility and recurrence;
• Understand equilibrium distributions for Markov chains in simple examples;
• Understand simple examples such as those from queues, and transfer techniques and skills to other similar examples.

### Outline Syllabus

• The Bernoulli process and the simple random walk.
• Conditional expectations and applications to random walks and the Gambler's ruin. The reflection principle.
• Generating functions and their applications.
• Markov chains in discrete time: time-dependent state distribution, stationary distribution, limit theorems, reversible Markov chains, expected hitting times and explicit n-step formulae.
• Markov chains in continuous time: time-dependent state distribution, limit theorems, queuing networks, Poisson processes, birth-death and immigration models.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH440: Stochastic Calculus for Finance

• Terms Taught: Lent / Summer Terms only, (Weeks 16-20)
• US Credits: 4 Semester Credits.
• ECTS Credits:   7.5 ECTS
• Pre-requisites: Equivalent to MATH332 or MATH432 (Coursework equivalent to MATH313/MATH413 is recommended)

### Course Description

Stochastic processes are widely used to model uncertainty in areas ranging from finance to the physical sciences. This module aims to show how stochastic calculus can be used to formulate and solve problems arising from these areas of application.

### Educational Aims

Stochastic Calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discrete-time stochastic processes, in particular defining key concepts such as martingales and stopping times. This leads on to the exploration of continuous-time processes, in particular Brownian motion. The final section covers integration with respect to Brownian motion, and the derivation of Ito's formula, a stochastic analogue of the chain rule. This allows the definition and solution of stochastic differential equations (SDEs), the stochastic analogue to ordinary differential equations (ODEs).

### Outline Syllabus

Continuous-time Stochastic Processes: Brownian motion, filtrations, martingales, stopping times, optional stopping theorem. Stochastic Integration: total variation, Riemann-Stieltjes integral, quadratic variation, Ito integral, Ito's formula, stochastic differential equations (SDEs), solutions of simple SDEs. Applications to finance: Black-Scholes Formula and pricing of simple options.

### Assessment Proportions

• Coursework: 10%
• Exam: 90%

## MATH445: Mathematics for Stochastic Finance

• Terms Taught: Lent / Summer terms only (weeks 16-20)
• US Credits: 4 Semester credits
• ECTS Credits: 7.5 ECTS Credits
• Pre-requisites: Mathematics Majors or appropriate College Mathematics. Equivalent to MATH230  MATH332

### Course Description

This module will give a simple introduction to mathematical finance. This includes some financial terminology and the study of European and American option pricing with respect to different models. More precisely, we consider two discrete models, the binomial Model and finite market model, and one continuous model, the Black Scholes model. We also introduce some probabilistic terminology, which is required to study the properties of these models. This includes martingales, stopping times and an overview over the mathematical theory of Brownian motion, including stochastic integration with respect to Brownian motion.

### Educational Aims

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

Construct binomial tree model

Determine the associated risk-neutral probability

Describe the basic concepts of investment strategy analysis, including self-financing, arbitrage free

Price forward contracts and perform price calculations for stocks without and with dividend payments

Calculate the price of various European and American options in a binomial tree model

Construct a replicating portfolio in a binomial tree model

Perform calculations with the Black-Scholes call option price formula

Be able to prove various steps in the derivation of the Black-Scholes call option price formula

### Outline Syllabus

Discrete financial models: Binomial Model and Finite market model

Continuous financial model: Black Scholes model

European and American option pricing

Conditional expectation

Filtrations

Martingales

Stopping times

Brownian motion

Black-Scholes formula

Stochastic integration with respect to Brownian motion

Coursework: 10%

Exam: 90%