Mathematics with a Year in Industry - BSc (Hons)

Introduction to R and investigating data sets. Basic use of R (Input and manipulation of data). Graphical representations of data. Numerical summaries of data.

Sampling and sampling distributions. ?² distribution. t-distribution. F-distribution. Definition of sampling distribution. Standard error. Sampling distribution of sample mean (for arbitrary distributions) and sample variance (for normal distribution) .

Point estimation. Principles. Unbiased estimators. Bias, Likelihood estimation for samples of discrete r.v.s

Interval estimation. Concept. One-sided/two-sided confidence intervals. Examples for population mean, population variance (with normal data) and proportion.

Hypothesis testing. Concept. Type I and II errors, size, p-values and power function. One-sample test, two sample test and paired sample test. Examples for population mean and population variance for normal data. Testing hypotheses for a proportion with large n. Link between hypothesis test and confidence interval. Goodness-of-fit testing.

Association between variables. Product moment and rank correlation coefficients. Two-way contingency tables. ?² test of independence.

This module serves as an introduction to algebraic methods. These methods are central in modern mathematics and have found applications in many other sciences, but also in our everyday life. In this module, students will also gain an appreciation of the concept of proof in mathematics.

This module introduces mathematical modelling and Newtonian mechanics. Tutorials and Maple worksheets will be used to support taught material.

The modelling cycle: General description with examples; Newton's law of cooling; population growth (Malthusian and logistic models); simple reaction kinetics (unimolecular and bimolecular reactions); dimensional consistency

Motion of a body: frames of reference; a particle's position vector and its time derivatives (velocity and acceleration) in Cartesian coordinates; mass, momentum and centre of mass; Newton's laws of motion; linear springs; gravitational acceleration and the pendulum; projectile motion

Orbital motion: Newton's law of gravitation; position, velocity and acceleration in plane polar coordinates; planetary motion and Kepler's laws.

This module is a sequel to Algebraic Methods. It considers the abstract theory of linear spaces together with applications to matrix algebra and other areas of Mathematics (and its applications). Since linear spaces are of fundamental importance in almost every area of mathematics, the ideas and techniques discussed in this module lie at the heart of mathematics. Topics covered will include vector spaces, linear transformations, eigenvalues and eigenvectors, diagonalisation, orthogonality and applications including conics.

This module introduces widely-used mathematical methods for functions of a single variable. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage. Tutorials and Maple worksheets will be used to support taught material.

Complex numbers: Complex arithmetic, the complex conjugate, the Argand diagram, de Moivre's Theorem, modulus-argument form; elementary functions

Polynomials: Fundamental Theorem of Algebra (statement only), roots, factorization, rational functions, partial fractions

Single variable calculus: Differentiation, including product and chain rules; Fundamental Theorem of Calculus (statement only), elementary integrals, change of variables, integration by parts, differentiation of integrals with variable limits

Scalar ordinary differential equations (ODEs): definition; methods for first-order ODEs; principle of superposition for linear ODEs; particular integrals; second-order linear ODEs with constant coefficients; initial-value problems

Curve sketching: graphs of elementary functions, maxima, minima and points of inflection, asymptotes

This module introduces widely-used mathematical methods for vectors and functions of two or more variables. The emphasis is on the practical use of these methods; key theorems are stated but not proved at this stage. Tutorials and Maple worksheets will be used to support taught material.

Vectors: Cartesian coordinates; vector algebra; scalar, vector and triple products (and geometric interpretation); straight lines and planes expressed as vector equations; parametrized curves; differentiation of vector-valued functions of a scalar variable; tangent vectors; vector fields (with everyday examples)

Partial differentiation: Functions of two variables; partial differentiation (including the chain rule and change of variables); maxima, minima and saddle points; Lagrange multipliers

Integration in two dimensions: Double integrals in Cartesian coordinates; plane polar coordinates; change of variables for double integrals; line integrals; Green's theorem (statement – justification on rectangular domains only)

Introduction to Probability. Concepts of events and sample space. Set theoretic description of probability, axioms of probability, interpretations of probability (objective and subjective probability).

Theory for unstructured sample spaces. Addition law for mutually exclusive events. Conditional probability. Independence. Law of total probability. Bayes' theorem. Permutations and combinations. Inclusion-Exclusion formula.

Discrete random variables. Concept of random variable (r.v.) and their distribution. Discrete r.v.: Probability function (p.f.). (Cumulative) distribution function (c.d.f.). Mean and variance of a discrete r.v. Examples: Binomial, Poisson, Geometric.

Continuous random variables. Probability density function; mean and variance; exponential, uniform and normal distributions; normal approximations: standardisation of the normal and use of tables. Transformation of a single r.v.

Joint distributions. Discrete r.v.'s; independent random variables; expectation and its application.

Generating functions. Idea of generating functions. Probability generating functions (pgfs) and moment generating functions (mgfs). Finding moments from pgfs and mgfs. Sums of independent random variables.

Laws of Large Numbers. Weak law of large numbers. Central Limit Theorem.

Real Numbers: Rational and real numbers, absolute value and metric structure on the real numbers, induction, infimum and supremum.

Limits of Sequences: Sequences, definition of convergence, epsilon terminology, uniqueness, algebra of limits, comparison principles, standard limits, subsequences and non-existence of limits, convergence to infinity.

Completeness Properties: Cantor's Intersection Theorem, limit points, Bolzano-Weierstrass theorem, Cauchy sequences.

Continuity of Functions: Functions and basic definitions, limits of functions, continuity and epsilon terminology, sequential continuity, Intermediate Value Theorem.

Differentiation: Definition of the derivative, product rule, quotient rule and chain rule, derivatives and local properties, Mean Value Theorem, L'Hospital's Rule.

Taylor Approximation: Taylor's Theorem, remainder term, Taylor series, standard examples, limits using Taylor series.

This module builds on the Stage 1 Real Analysis 1 module. We will extend our knowledge of functions of one real variable, look at series, and study functions of several real variables and their derivatives.

The outline syllabus includes: Continuity and uniform continuity of functions of one variable, series and power series, the Riemann integral, limits and continuity for functions of several variables, differentiation of functions of several variables, extrema, the Inverse and Implicit Function Theorems.

The concept of symmetry is one of the most fruitful ideas through which mankind has tried to understand order and beauty in nature and art. This module first develops the concept of symmetry in geometry. It subsequently discusses links with the fundamental notion of a group in algebra. Outline syllabus includes: Groups from geometry; Permutations; Basic group theory; Action of groups and applications to (i) isometries of regular polyhedra; (ii) counting colouring problems; Matrix groups.

In this module we will study linear partial differential equations, we will explore their properties and discuss the physical interpretation of certain equations and their solutions. We will learn how to solve first order equations using the method of characteristics and second order equations using the method of separation of variables.

Introduction to linear PDEs: Review of partial differentiation; first-order linear PDEs, the heat equation, Laplace's equation and the wave equation, with simple models that lead to these equations; the superposition principle; initial and boundary conditions

Separation of variables and series solutions: The method of separation of variables; simple separable solutions of the heat equation and Laplace’s equation; Fourier series; orthogonality of the Fourier basis; examples and interpretation of solutions

Solution by characteristics: the method of characteristics for first-order linear PDEs; examples and interpretation of solutions; characteristics of the wave equation; d’Alembert’s solution, with examples; domains of influence and dependence; causality.

Probability: Joint distributions of two or more discrete or continuous random variables. Marginal and conditional distributions. Independence. Properties of expectation, variance, covariance and correlation. Poisson process and its application. Sums of random variables with a random number of terms.

Transformations of random variables: Various methods for obtaining the distribution of a function of a random variable —method of distribution functions, method of transformations, method of generating functions. Method of transformations for several variables. Convolutions. Approximate method for transformations.

Sampling distributions: Sampling distributions related to the Normal distribution — distribution of sample mean and sample variance; independence of sample mean and variance; the t distribution in one- and two-sample problems.

Statistical inference: Basic ideas of inference — point and interval estimation, hypothesis testing.

Point estimation: Methods of comparing estimators — bias, variance, mean square error, consistency, efficiency. Method of moments estimation. The likelihood and log-likelihood functions. Maximum likelihood estimation.

Hypothesis testing: Basic ideas of hypothesis testing — null and alternative hypotheses; simple and composite hypotheses; one and two-sided alternatives; critical regions; types of error; size and power. Neyman-Pearson lemma. Simple null hypothesis versus composite alternative. Power functions. Locally and uniformly most powerful tests. Composite null hypotheses. The maximum likelihood ratio test.

Interval estimation: Confidence limits and intervals. Intervals related to sampling from the Normal distribution. The method of pivotal functions. Confidence intervals based on the large sample distribution of the maximum likelihood estimator – Fisher information, Cramer-Rao lower bound. Relationship with hypothesis tests. Likelihood-based intervals.

This module is an introduction to the methods, tools and ideas of numerical computation. In mathematics, one often encounters standard problems for which there are no easily obtainable explicit solutions, given by a closed formula. Examples might be the task of determining the value of a particular integral, finding the roots of a certain non-linear equation or approximating the solution of a given differential equation. Different methods are presented for solving such problems on a modern computer, together with their applicability and error analysis. A significant part of the module is devoted to programming these methods and running them in MATLAB.

Introduction: Importance of numerical methods; short description of flops, round-off error, conditioning

Solution of linear and non-linear equations: bisection, Newton-Raphson, fixed point iteration

Interpolation and polynomial approximation: Taylor polynomials, Lagrange interpolation, divided differences, splines

Numerical integration: Newton-Cotes rules, Gaussian rules

Numerical differentiation: finite differences

Introduction to initial value problems for ODEs: Euler methods, trapezoidal method, Runge-Kutta methods.

This module introduces the basic ideas to solve certain ordinary differential equations, like first order scalar equations, second order linear equations and systems of linear equations. It mainly considers their qualitative and analytical aspects. Outline syllabus includes: First-order scalar ODEs; Second-order scalar linear ODEs; Existence and Uniqueness of Solutions; Autonomous systems of two linear first-order ODEs.

This module will present a new perspective on Newton's familiar laws of motion. First we introduce variational calculus with applications such as finding the paths of shortest distance. This will lead us to the principle of least action from which we can derive Newton's law for conservative forces. We will also learn how symmetries lead to constants of motion. We then derive Hamilton's equations and discuss their underlying structures. The formalisms we introduce in this module form the basis for all of fundamental modern physics, from electromagnetism and general relativity, to the standard model of particle physics and string theory.

Indicative syllabus:

Review of Newton mechanics: Newton's law; harmonic and anharmonic oscillators (closed and unbound orbits, turning points); Kepler problem: energy and angular momentum conservation

Lagrangian Mechanics: Introdution to variational calculus with simple applications (shortest path - geodesic, soap film, brachistochrone problem); principle of least action: Euler-Lagrange equations (Newtonian mechanics with conservative forces); constraints and generalised coordinates (particle on a hoop, double pendulum, normal modes); Noether's theorem (energy and angular momentum conservation)

Hamiltonian Dynamics: Hamilton's equations; Legendre transform; Hamiltonian phase space (harmonic oscillator, anharmonic oscillators and the mathematical pendulum); Liouville's theorem; Poisson brackets.

Constructing suitable models for data is a key part of statistics. For example, we might want to model the yield of a chemical process in terms of the temperature and pressure of the process. Even if the temperature and pressure are fixed, there will be variation in the yield which motivates the use of a statistical model which includes a random component. In this module, we study how suitable models can be constructed, how to fit them to data and how suitable conclusions can be drawn. Both theoretical and practical aspects are covered, including the use of R.

The main aim of this module is to give an introduction to the basics of differential geometry, keeping in mind the recent applications in mathematical physics and the analysis of pattern recognition. Outline syllabus includes: Curves and parameterization; Curvature of curves; Surfaces in Euclidean space; The first fundamental form; Curvature of surfaces; Geodesics.

Can we square a circle? Can we trisect an angle? These two questions were studied by the Ancient Greeks and were only solved in the 19th century using algebraic structures such as rings, fields and polynomials. In this module, we introduce these ideas and concepts and show how they generalise well-known objects such as integers, rational numbers, prime numbers, etc. The theory is then applied to solve problems in Geometry and Number Theory. This part of algebra has many applications in electronic communication, in particular in coding theory and cryptography.

The security of our phone calls, bank transfers, etc. all rely on one area of Mathematics: Number Theory. This module is an elementary introduction to this wide area and focuses on solving Diophantine equations. In particular, we discuss (without proof) Fermat's Last Theorem, arguably one of the most spectacular mathematical achievements of the twentieth century. Outline syllabus includes: Modular Arithmetic; Prime Numbers; Introduction to Cryptography; Quadratic Residues; Diophantine Equations.

Students spend a year (minimum 900 hours) doing paid work in an organisation outside the University, usually in an industrial or commercial environment, applying and enhancing the skills and techniques they have developed and studied in the earlier stages of their degree programme.

The work they do is entirely under the direction of their industrial supervisor, but support is provided by the SMSAS Placement Officer or a member of the academic team. This support includes ensuring that the work they are being expected to do is such that they can meet the learning outcomes of the module.

Participation in this module is dependent on students obtaining an appropriate placement, for which support and guidance is provided through the School in the year leading up to the placement. It is also dependent on students progressing from Stage 2 of their studies.

Students who do not obtain a placement will be required to transfer to the appropriate programme without a Year in Industry.

Students spend a year (minimum 900 hours) doing paid work in an organisation outside the University, usually in an industrial or commercial environment, applying and enhancing the skills and techniques they have developed and studied in the earlier stages of their degree programme. Employer evaluation, personal and professional reviews and on-line blogs are assessed under MAST5801 (Industrial Placement Experience) which is a co-requisite of this module. The assessment of this module draws on the experience gained in MAST5801 and is assessed through a Placement Report and Presentation.

The placement work they do is entirely under the direction of their industrial supervisor, but support is provided by the SMSAS Placement Officer or a member of the academic team. This support includes ensuring that the work they are being expected to do is such that they can meet the learning outcomes of this module.

Participation in the placement year, and hence in this module, is dependent on students obtaining an appropriate placement, for which support and guidance is provided through the School in the year leading up to the placement. It is also dependent on students progressing satisfactorily from Stage 2 of their studies.

Students who do not obtain a placement or who fail module MAST5801 (Industrial Placement Experience) will be required to transfer to the appropriate programme without a Year in Industry and any marks obtained on this module will not contribute to their final degree classification.

Introduction: Principles and examples of stochastic modelling, types of stochastic process, Markov property and Markov processes, short-term and long-run properties. Applications in various research areas.

Random walks: The simple random walk. Walk with two absorbing barriers. First–step decomposition technique. Probabilities of absorption. Duration of walk. Application of results to other simple random walks. General random walks. Applications.

Discrete time Markov chains: n–step transition probabilities. Chapman-Kolmogorov equations. Classification of states. Equilibrium and stationary distribution. Mean recurrence times. Simple estimation of transition probabilities. Time inhomogeneous chains. Elementary renewal theory. Simulations. Applications.

Continuous time Markov chains: Transition probability functions. Generator matrix. Kolmogorov forward and backward equations. Poisson process. Birth and death processes. Time inhomogeneous chains. Renewal processes. Applications.

Queues and branching processes: Properties of queues - arrivals, service time, length of the queue, waiting times, busy periods. The single-server queue and its stationary behaviour. Queues with several servers. Branching processes. Applications.

Marks on this module can count towards exemption from the professional examination CT4 of the Institute and Faculty of Actuaries. Please see http://www.kent.ac.uk/casri/Accreditation/index.html for further details.

A time series is a collection of observations made sequentially in time. Examples occur in a variety of fields, ranging from economics to engineering, and methods of analysing time series constitute an important area of statistics. This module focuses initially on various time series models, including some recent developments, and provides modern statistical tools for their analysis. The second part of the module covers extensively simulation methods. These methods are becoming increasingly important tools as simulation models can be easily designed and run on modern PCs. Various practical examples are considered to help students tackle the analysis of real data.The syllabus includes: Difference equations, Stationary Time Series: ARMA process. Nonstationary Processes: ARIMA Model Building and Testing: Estimation, Box Jenkins, Criteria for choosing between models, Diagnostic tests.Forecasting: Box-Jenkins, Prediction bounds. Testing for Trends and Unit Roots: Dickey-Fuller, ADF, Structural change, Trend-stationarity vs difference stationarity. Seasonality and Volatility: ARCH, GARCH, ML estimation. Multiequation Time Series Models: Spectral Analysis. Generation of pseudo – random numbers, simulation methods: inverse transform and acceptance-rejection, design issues and sensitivity analysis.

Marks on this module can count towards exemption from the professional examination CT6 of the Institute and Faculty of Actuaries. Please see http://www.kent.ac.uk/casri/Accreditation/index.html for further details.

This is a practical module to develop the skills required by a professional statistician (report writing, consultancy, presentation, wider appreciation of assumptions underlying methods, selection and application of analysis method, researching methods).

Software: R, SPSS and Excel (where appropriate/possible). Report writing in Word. PowerPoint for presentations.

• Presentation of data

• Report writing and presentation skills

• Hypothesis testing: formulating questions, converting to hypotheses, parametric and non-parametric methods and their assumptions, selection of appropriate method, application and reporting. Use of resources to explore and apply additional tests. Parametric and non-parametric tests include, but are not limited to, t-tests, likelihood ratio tests, score tests, Wald test, chi-squared tests, Mann Whitney U-test, Wilcoxon signed rank test, McNemar's test.

• Linear and Generalised Linear Models: simple linear and multiple regression, ANOVA and ANCOVA, understanding the limitations of linear regression, generalised linear models, selecting the appropriate distribution for the data set, understanding the difference between fixed and random effects, fitting models with random effects, model selection.

• Consultancy skills: group work exercise(s)

Sampling: Simple random sampling. Sampling for proportions and percentages. Estimation of sample size. Stratified sampling. Systematic sampling. Ratio and regression estimates. Cluster sampling. Multi-stage sampling and design effect. Questionnaire design. Response bias and non-response.

General principles of experimental design: blocking, randomization, replication. One-way ANOVA. Two-way ANOVA. Orthogonal and non-orthogonal designs. Factorial designs: confounding, fractional replication. Analysis of covariance.

Design of clinical trials: blinding, placebos, eligibility, ethics, data monitoring and interim analysis. Good clinical practice, the statistical analysis plan, the protocol. Equivalence and noninferiority.

Multivariate normal distribution, Inference from multivariate normal samples, principal component analysis, mixture models, factor analysis, clustering methods, discrimination and classification, graphical models, the use of appropriate software.

Statistics methods contribute significantly to areas such as biology, ecology, sociology and economics. The real data collected does not always follow standard statistical models. This module looks at modern statistical models and methods that can be utilised for such data, making use of R programs to execute these methods.

Indicative module content: Motivating examples; model fitting through maximum likelihood for specific examples; function optimization methods; profile likelihood; score tests; Wald tests; confidence interval construction; latent variable models; EM algorithm; mixture models; simulation methods; importance sampling; kernel density estimation; Monte Carlo inference; bootstrap; permutation tests; R programs.

This module provides an overview of analytical careers in finance and explores the mathematical techniques used by actuaries, accountants and financial analysts. Students will learn about different types of financials assets, such as shares, bonds and derivatives and how to work out how much they are worth. They will also look at different types of debt and learn how mortgages and other loans are calculated. Developing these themes, the module will explain how to use maths to make financial decisions, such as how much an investor should pay for a financial asset or how a company can decide which projects to invest in or how much money to borrow. Risk management is a vital part of most mathematical careers in finance so the module will also cover different mathematical techniques for measuring and mitigating financial risk. Extension topics may include complex derivatives, economic theories of finance and the dangers of misusing mathematics. The module provides an opportunity to apply complex mathematical techniques to important real-world questions and is excellent preparation for those considering a financial career.

Introduction to financial mathematics: Key uses of mathematics in finance; key practitioners of financial mathematics.

Financial valuation and cash flow analysis: Discounting, Interest rates and time requirements, Future and Present value. Project Evaluation.

Characteristics and valuation of different financial securities: Debt capital, bonds and stocks, valuation of bonds and stocks.

Loans and interest rates: term structure of interest rates, spot and forward rates, types of loan, APR, loan schedules.

Capital structure and the cost of capital: Gearing, WACC, understanding betas.

Additional topics that may be covered: arbitrage and forward contracts, efficient markets hypothesis, pricing and valuing forward contracts, option pricing and the Black Scholes model, credit derivatives and systemic risks, limitations of mathematical modelling.

Linear PDEs. Dispersion relations. Review of d'Alembert’s solutions of the wave equation. Review of Fourier transforms for solving linear diffusion equations.

Quasi-linear first-order PDEs. Total differential equations. Integral curves and integrability conditions. The method of characteristics.

Shock waves. Discontinuous solutions. Breaking time. Rankine-Hugoniot jump condition. Shock waves. Rarefaction waves. Applications of shock waves, including traffic flow.

General first-order nonlinear PDEs. Charpit's method, Monge Cone, the complete integral.

Nonlinear PDEs. Burgers' equation; the Cole-Hopf transformation and exact solutions. Travelling wave and scaling solutions of nonlinear PDEs. Applications of travelling wave and scaling solutions to reaction-diffusion equations. Exact solutions of nonlinear PDEs. Applications of nonlinear waves, including to ocean waves (e.g. rogue waves, tsunamis).

Revision of complex numbers, the complex plane, de Moivre's and Euler's theorems, roots of unity, triangle inequality

Sequences and limits: Convergence of a sequence in the complex plane. Absolute convergence of complex series. Criteria for convergence. Power series, radius of convergence

Complex functions: Domains, continuity, complex differentiation. Differentiation of power series. Complex exponential and logarithm, trigonometric, hyperbolic functions. Cauchy-Riemann equations

Complex Integration: Jordan curves, winding numbers. Cauchy's Theorem. Analytic functions. Liouville's Theorem, Maximum Modulus Theorem

Singularities of functions: poles, classification of singularities. Residues. Laurent expansions. Applications of Cauchy's theorem. The residue theorem. Evaluation of real integrals.

Possible additional topics may include Rouche’s Theorem, other proofs of the Fundamental Theorem of Algebra, conformal mappings, Mobius mappings, elementary Riemann surfaces, and harmonic functions.

In this module we study the fundamental concepts and results in game theory. We start by analysing combinatorial games, and discuss game trees, winning strategies, and the classification of positions in so called impartial combinatorial games. We then move on to discuss two-player zero-sum games and introduce security levels, pure and mixed strategies, and prove the famous von Neumann Minimax Theorem. We will see how to solve zero-sum two player games using domination and discuss a general method based on linear programming. Subsequently we analyse arbitrary sum two-player games and discuss utility, best responses, Nash equilibria, and the Nash Equilibrium Theorem. The final part of the module is devoted to multi-player games and cooperation; we analyse coalitions, the core of the game, and the Shapley value.

There is no specific mathematical syllabus for this module; students will chose a topic in mathematics, statistics or financial mathematics from a published list on which to base their coursework assessments (different topics for levels 6 and 7). The coursework is supported by a series of workshops covering various forms of written and oral communication. These may include critically evaluating the following: a research article in mathematics, statistics or finance; a survey or magazine article aimed at a scientifically-literate but non-specialist audience; a mathematical biography; a poster presentation of a mathematical topic; a curriculum vitae; an oral presentation with slides or board; a video or podcast on a mathematical topic. Guidance will be given on typesetting mathematics using LaTeX.

There is no specific mathematical syllabus for this module. Students will study a topic in mathematics or statistics, either individually or within a small group, and produce an individual or group project on the topic as well as individual coursework assignments. Projects will be chosen from published lists of individual and of group projects. The coursework and project-work are supported by a series of workshops covering various forms of written and oral communication and by supervision from an academic member of staff.

The workshops may include critically evaluating the following: a research article in mathematics or statistics; a survey or magazine article aimed at a scientifically-literate but non-specialist audience; a mathematical biography; a poster presentation of a mathematical topic; a curriculum vitae; an oral presentation with slides or board; a video or podcast on a mathematical topic. Guidance will be given on typesetting mathematics using LaTeX.

This module is an introduction to point-set topology, a topic that is relevant to many other areas of mathematics. In it, we will be looking at the concept of topological spaces and related constructions. In an Euclidean space, an "open set" is defined as a (possibly infinite) union of open "epsilon-balls". A topological space generalises the notion of "open set" axiomatically, leading to some interesting and sometimes surprising geometric consequences. For example, we will encounter spaces where every sequence of points converges to every point in the space, see why for topologists a doughnut is the same as a coffee cup, and have a look at famous objects such as the Moebius strip or the Klein bottle. 

This module provides a rigorous foundation for the solution of systems of polynomial equations in many variables. In the 1890s, David Hilbert proved four ground-breaking theorems that prepared the way for Emmy Nöther's famous foundational work in the 1920s on ring theory and ideals in abstract algebra. This module will echo that historical progress, developing Hilbert's theorems and the essential canon of ring theory in the context of polynomial rings. It will take a modern perspective on the subject, using the Gröbner bases developed in the 1960s together with ideas of computer algebra pioneered in the 1980s. The syllabus will include

• Multivariate polynomials, monomial orders, division algorithm, Gröbner bases;

• Hilbert's Nullstellensatz and its meaning and consequences for solving polynomials in several variables;

• Elimination theory and applications;

• Linear equations over systems of polynomials, syzygies.

Most differential equations which arise from physical systems cannot be solved explicitly in closed form, and thus numerical solutions are an invaluable way to obtain information about the underlying physical system. The first half of the module is concerned with ordinary differential equations. Several different numerical methods are introduced and error growth is studied. Both initial value and boundary value problems are investigated. The second half of the module deals with the numerical solution of partial differential equations. The syllabus includes: initial value problems for ordinary differential equations; Taylor methods; Runge-Kutta methods; multistep methods; error bounds and stability; boundary value problems for ordinary differential equations; finite difference schemes; difference schemes for partial differential equations; iterative methods; stability analysis.