Mathematics and Statistics - MMathStat
Canterbury
This stimulating four year programme provides opportunities for students to develop and demonstrate key mathematical and statistical knowledge and understanding and culminates with Master's level topics that draw upon the School's highly rated research expertise.
Entrants to the programme who meet the AAA entry requirement will be entitled to an annual bursary of £3,000, comprising of £2,000 from the University and £1,000 from the School.
Overview
Mathematics is important to the modern world. All quantitative science, including both physical and social sciences, is based on it. It provides the theoretical framework for physical science, statistics and data analysis as well as computer science. Our programmes reflect this diversity and the excitement generated by new discoveries within mathematics that affect not only the technicalities of science but also our general understanding of the world in which we live.
The programmes share a common core of Mathematics at Stage 1, and then move on to cover abstract, analytical and computational techniques that give you the opportunity to specialise in areas such as non-linear differential equations, computational algebra and geometry, financial mathematics, forecasting, design and analysis of experiments, inference and stochastic processes.
The MMathStat course is aimed at students who have a strong interest in pursuing a deeper study of mathematics and statistics with the flexibility to choose a wide range of optional modules to allow those who wish to specialise in a particular area the opportunity to do so.
The programme provides opportunities for students to develop and demonstrate key mathematical and statistical knowledge and understanding and will prepare successful students with the depth of mathematical and statistical knowledge to enter postgraduate studies at the doctorate level in mathematics, statistics and other closely related subjects. A year of Master's level study in Stage 4 gives students the opportunity to explore more advanced topics, which draws on the School's highly rated research expertise.
Course structure
The following modules are indicative of those offered on this programme. This listing is based on the current curriculum and may change year to year in response to new curriculum developments and innovation.
On most programmes, you study a combination of compulsory and optional modules. You may also be able to take 'wild' modules from other programmes so you can customise your programme and explore other subjects that interest you.
Stage 1
Possible modules may include:
MA306 - Statistics (15 credits)
Increasingly data are collected to inform future decisions, varying from which websites people access on a regular basis to how patients respond to new drugs, to how the stock market responds to global events, or to how animals move around their local habitat. Therefore, most professionals will need to extract useful information from data and to manage and present data in their working lives. This module explores some of the basic concepts of statistics, from data summarisation to the main methods of statistical inference. The techniques that are discussed can be used in their own right for simple statistical analyses, but serve as an important foundation for later, more advanced, modules. The statistical computing package R is used throughout the module for data analysis. The syllabus includes: an introduction to R and investigating data sets, sampling and sampling distributions, point and interval estimation, hypothesis testing, association between variables.
Credits: 15 credits (7.5 ECTS credits).
MA343 - Algebraic Methods (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA344 - Application of Mathematics (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA346 - Linear Algebra (15 credits)
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:
1 Vector Spaces: definition, examples, linearly independent and spanning sets, bases, dimension, subspaces.
2 Linear transformations: definition, examples, matrix of a linear transformation, change of basis, similar matrices.
3 Determinant of a linear transformation.
4 Eigenvalues/eigenvectors and diagonalisation: characteristic polynomial, invariant subspaces and upper triangular forms. Cayley-Hamilton Theorem.
5 Bilinear forms: inner products, norms, Cauchy-Schwarz inequality.
6 Orthonormal systems, the Gram-Schmidt process.
7 Symmetric Matrices. Every real symmetric matrix is diagonalisable.
8 Quadratic forms: Sylvester's Law of Inertia; signature of a quadratic form; application to conics (and quadrics if time permits).
Credits: 15 credits (7.5 ECTS credits).
MA348 - Mathematical Methods 1 (15 credits)
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
Credits: 15 credits (7.5 ECTS credits).
MA349 - Mathematical Methods 2 (15 credits)
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)
Credits: 15 credits (7.5 ECTS credits).
MA351 - Probability (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA352 - Real Analysis 1 (15 credits)
Topics covered will include:
Real Numbers: Rational and real numbers, absolute value and metric structure on the real numbers, induction, countability and uncountability, 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, O and o notation, limits using Taylor series.
Credits: 15 credits (7.5 ECTS credits).
Stage 2
Possible modules may include:
MA5501 - Applied Statistical Modelling 1 (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA5503 - Groups and Symmetries (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA5505 - Linear Partial Differential Equations (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA5507 - Mathematical Statistics (15 credits)
This module is a pre-requisite for many of the other statistics modules at Stages 2, 3 and 4, but it can equally well be studied as a module in its own right, extending the ideas of probability and statistics met at Stage 1 and providing practice with the mathematical skills learned in MA348 and MA349. It starts by revising the idea of a probability distribution for one or more random variables and looks at different methods to derive the distribution of a function of random variables. These techniques are then used to prove some of the results underpinning the hypothesis test and confidence interval calculations met at Stage 1, such as for the t-test or the F-test. With these tools to hand, the module moves on to look at how to fit models (probability distributions) to sets of data. A standard technique, the method of maximum likelihood, is used to fit the model to the data to obtain point estimates of the model parameters and to construct hypothesis tests and confidence intervals for these parameters. Outline Syllabus includes: Joint, marginal and conditional distributions of discrete and continuous random variables; Transformations of random variables; Sampling distributions; Point and interval estimation; Properties of estimators; Maximum likelihood; Hypothesis testing; Neyman-Pearson lemma; Maximum likelihood ratio test.
Credits: 15 credits (7.5 ECTS credits).
MA5514 - Rings and Fields (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA5512 - Ordinary Differential Equations (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA5504 - Lagrangian and Hamiltonian Dynamics (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA5502 - Curves and Surfaces (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA501 - Statistics for Insurance (15 credits)
This module covers aspects of Statistics which are particularly relevant to insurance. Some topics (such as risk theory and credibility theory) have been developed specifically for actuarial use. Other areas (such as Bayesian Statistics) have been developed in other contexts but now find applications in actuarial fields. Stochastic processes of events such as accidents, together with the financial flow of their payouts underpin much of the work. Since the earliest games of chance, the probability of ruin has been a topic of interest. Outline Syllabus includes: Decision Theory; Bayesian Statistics; Loss Distributions; Reinsurance; Credibility Theory; Empirical Bayes Credibility theory; Risk Models; Ruin Theory; Generalised Linear Models; Run-off Triangles.
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.
Credits: 15 credits (7.5 ECTS credits).
AC523 - Principles of Finance (30 credits)
This module is concerned with the principles which underlie the investment and financing decision making process. Before a rational decision can be made objectives need to be considered and models need to be built. Short-term decisions are dealt with first, together with relevant costs. One such cost is the time value of money. This leads to long term investment decisions which are examined using the economic theory of choice, first assuming perfect capital markets and certainty. These assumptions are then relaxed so that such problems as incorporating capital rationing and risk into the investment decision are fully considered. The module proceeds by looking at the financing decision. The financial system within which business organisations operate is examined, followed by the specific sources and costs of long and short-term capital, including the management of fixed and working capital
Credits: 30 credits (15 ECTS credits).
Stage 3
Possible modules may include:
MA569 - Starting Research in the Mathematical Sciences (15 credits)
This module has two components: (i) A series of workshops on key skills: The interactive workshops will cover general research methods possibly including library & information systems, mathematical reading, referencing conventions, coherent mathematical writing, typesetting using LaTeX and presentation skills. (ii) Independent project work: A list of possible topics will be offered. The description of each project will include a reference to a single journal article. The student will carry out independent study on one topic based on the journal article listed.
Credits: 15 credits (7.5 ECTS credits).
MA594 - Advanced Regression Modelling with R (15 credits)
R Package: This part will include a general introduction to the package and its components covering: linear models in R, writing your own functions in R, generalized linear models in R.
Further linear regression: Model selection, collinearity, outliers and influential observations, polynomial regression.
Generalized Linear Model: Exponential family; Discrete data distributions: definition, estimation and testing; GLMs: estimation, model selection and model checking; Examples of GLMs: logistic regression and Poisson regression; Overdispersion;
Credits: 15 credits (7.5 ECTS credits).
MA636 - Stochastic Processes (15 credits)
A stochastic process is a process developing in time according to probability rules, for example, models for reserves in insurance companies, queue formation, the behaviour of a population of bacteria, and the persistence (or otherwise) of an unusual surname through successive generations.The syllabus will include coverage of a wide variety of stochastic processes and their applications: Markov chains; processes in continuous-time such as the Poisson process, the birth and death process and queues.
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.
Credits: 15 credits (7.5 ECTS credits).
MA639 - Time Series Modelling and Simulation (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA772 - Analysis of Variance (15 credits)
Analysis of variance is a fundamentally important method for the statistical analysis of data. It is used widely in biological, medical, psychological, sociological and industrial research when we wish to compare more than two treatments at once. In analysing experimental data, the appropriate form of analysis of variance is determined by the design of the experiment, and we shall therefore discuss some aspects of experimental design in this module. Lectures are supplemented by computing classes which explore the analysis of variance facilities of the statistical package R. Syllabus: One-way ANOVA (fixed effects model); alternative models; least squares estimation; expectations of mean squares; distributional results; ANOVA table; follow-up analysis; multiple comparisons; least significant difference; confidence intervals; contrasts; orthogonal polynomials; checking assumptions; residual plots; Bartlett's test; transformations; one-way ANOVA (random effects model); types of experiment; experimental and observational units; treatment structure; randomisation; replication; blocking; the size of an experiment; two-way ANOVA; the randomised complete block design; two-way layout with interaction; the general linear model; matrix formulation; models of full rank; constraints; motivations for using least squares; properties of estimators; model partitions; extra sum of squares principle; orthogonality; multiple regression; polynomial regression; comparison of regression lines; analysis of covariance; balanced incomplete block designs; Latin square designs; Youden rectangles; factorial experiments; main effects and interactions.
Credits: 15 credits (7.5 ECTS credits).
MA595 - Graphs and Combinatorics (15 credits)
Combinatorics is a field in mathematics that studies discrete, usually finite, structures, such as graphs. It not only plays an important role in numerous parts of mathematics, but also has real world applications. In particular, it underpins a variety of computational processes used in digital technologies and the design of computing hardware.
Among other things, this module provides an introduction to graph theory. Graphs are discrete objects consisting of vertices that are connected by edges. We will discuss a variety of concepts and results in graph theory, and some fundamental graph algorithms. Topics may include, but are not restricted to: trees, shortest paths problems, walks on graphs, graph colourings and embeddings, flows and matchings, and matrices and graphs.
In addition to graphs, the module may cover other topics in combinatorics such as: problems in extremal set theory, enumerative problems, Principle of Inclusion and Exclusion, and, for M-level students, Ramsey theory, computational complexity and the P versus NP problem.
Credits: 15 credits (7.5 ECTS credits).
MA605 - Symmetries, Groups and Invariants (15 credits)
In this module we will study certain configurations with symmetries as they arise in real world applications. Examples include knots described by "admissible diagrams" or chemical structures described by colouring patterns. Different diagrams and patterns can describe essentially the same structure, so the problem of classification up to equivalence arises. This will be solved by attaching invariants which are then put in normal form to distinguish them. The syllabus will be as follows: (a) Review of basic methods from linear algebra, group theory and discrete mathematics; (b) Permutation groups, transitivity, primitivity, Burnside formula; (c) Finitely generated Abelian groups; (d) Applications to knot theory, Reidemeister moves, the Abelian knot group; (e) Examples, observations, generalizations and proofs.
Credits: 15 credits (7.5 ECTS credits).
MA617 - Asymptotics and Perturbation Methods (15 credits)
The lectures will introduce students to asymptotic and perturbation methods for the approximate evaluation of integrals and to obtain approximations for solutions of ordinary differential equations. These methods are widely used in the study of physically significant differential equations which arise in Applied Mathematics, Physics and Engineering. The material is chosen so as to demonstrate a range of mathematical techniques available and to illustrate some different applications which are amenable to such analysis.
Credits: 15 credits (7.5 ECTS credits).
MA572 - Complex Analysis (15 credits)
This module is concerned with complex functions, that is functions which are both defined for and assume complex values. Their theory follows a quite different development from that of real functions, is remarkable in its directness and elegance, and leads to many useful applications.Topics covered will include: Complex numbers. Domains and simple connectivity. Cauchy-Riemann equations. Integration and Cauchy's theorem. Singularities and residues. Applications.
Credits: 15 credits (7.5 ECTS credits).
MA574 - Polynomials in Several Variables (15 credits)
Systems of polynomial equations arise naturally in many applications of mathematics. This module focuses on methods for solving such systems and understanding the solutions sets. The key abstract concept is an ideal in a commutative ring and the fundamental computational concept is Buchberger's algorithm for computing a Groebner basis for an ideal in a polynomial ring. The syllabus includes: multivariate polynomials, Hilbert's Basis Theorem, monomial orders, division algorithms, Groebner bases, Hilbert's Nullstellensatz, elimination theory, linear equations over systems of polynomials, and syzygies.
Credits: 15 credits (7.5 ECTS credits).
MA587 - Numerical Solution of Differential Equations (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA567 - Topology (15 credits)
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.
Credits: 15 credits (7.5 ECTS credits).
MA568 - Orthogonal Polynomials and Special Functions (15 credits)
This module provides an introduction to the study of orthogonal polynomials and special functions. They are essentially useful mathematical functions with remarkable properties and applications in mathematical physics and other branches of mathematics. Closely related to many branches of analysis, orthogonal polynomials and special functions are related to important problems in approximation theory of functions, the theory of differential, difference and integral equations, whilst having important applications to recent problems in quantum mechanics, mathematical statistics, combinatorics and number theory. The emphasis will be on developing an understanding of the structural, analytical and geometrical properties of orthogonal polynomials and special functions. The module will utilise physical, combinatorial and number theory problems to illustrate the theory and give an insight into a plank of applications, whilst including some recent developments in this field. The development will bring aspects of mathematics as well as computation through the use of MAPLE. The topics covered will include: The hypergeometric functions, the parabolic cylinder functions, the confluent hypergeometric functions (Kummer and Whittaker) explored from their series expansions, analytical and geometrical properties, functional and differential equations; sequences of orthogonal polynomials and their weight functions; study of the classical polynomials and their applications as well as other hypergeometric type polynomials.
Credits: 15 credits (7.5 ECTS credits).
CB600 - Games and Networks (15 credits)
The module is divided into three main topics, namely Combinatorial Optimisation, Dynamic Programming and Game Theory. A more detailed listing of content is given below.
Combinatorial Optimisation:
The Shortest Path Problem
The Minimal Spanning Tree Problem
Flows in Networks
Scheduling Theory
Computational Complexity
Theory of Games:
Matrix Games Pure Strategies
Matrix Games Mixed Strategies
Bimatrix Games
N-person Games
Multi-criteria Decision Theory
Credits: 15 credits (7.5 ECTS credits).
MA549 - Discrete Mathematics (15 credits)
Recently some quite novel applications have been found for "Discrete Mathematics", as opposed to the Continuous Mathematics based on the Differential and Integral Calculus. Thus methods for the encoding of information in order to safeguard against eavesdropping or distortion by noise, for example in online banking and digital television, have involved using some basic results from abstract algebra. This module will provide a self-contained introduction to this general area and will cover most of the following topics: (a) Modular arithmetic, polynomials and finite fields: Applications to orthogonal Latin squares, cryptography, coin-tossing over a telephone, linear feedback shift registers and m-sequences. (b) Error correcting codes: Binary block, linear and cyclic codes including repetition, parity-check, Hamming, simplex, Reed-Muller, BCH, Golay codes; channel capacity; Maximum likelihood, nearest neighbour, syndrome and algebraic decoding.
Credits: 15 credits (7.5 ECTS credits).
Stage 4
Possible modules may include:
MA858 - Computational Statistics (15 credits)
This applied statistics module focusses on problems that occur in the fields of ecology, biology, genetics and psychology. Motivated by real examples, you will learn how to define and fit stochastic models to the data. In more complex situations this will mean using optimisation routines in MATLAB to obtain maximum likelihood estimates for the parameters. You will also learn how construct, fit and evaluate such stochastic models. Outline Syllabus includes: Function optimisation. Basic likelihood tools. Fundamental features of modelling. Model selection. The EM algorithm. Simulation techniques. Generalised linear models.
Credits: 15 credits (7.5 ECTS credits).
MA881 - Probability and Classical Inference (15 credits)
This module begins by introducing probability, primarily as a tool that underlies the subsequent material on statistical inference. This includes, for example, various notions of convergence for random variables. Classical statistical inference assumes that data follow a probability model with some unknown parameters, and the main aims are to estimate these parameters and to test hypotheses about them. The focus of the module is to develop general methods of statistical inference that can be applied to a wide range of problems. Outline syllabus includes: probability axioms; marginal, joint and conditional distributions; Bayes theorem; important distributions; convergence of random variables; sampling distributions; likelihood; point estimation; interval estimation; likelihood-ratio, Wald and score tests; estimating equations.
Credits: 15 credits (7.5 ECTS credits).
MA883 - Bayesian Statistics (15 credits)
The origins of Bayesian inference lie in Bayes' Theorem for density functions; the likelihood function and the prior distribution combine to provide a posterior distribution which reflects beliefs about an unknown parameter based on the data and prior beliefs. Statistical inference is determined solely by the posterior distribution. So, for example, an estimate of the parameter could be the mean value of the posterior distribution. This module will provide a full description of Bayesian analysis and cover popular models, such as the normal distribution. Initially, the flavour will be one of describing the Bayesian counterparts to well known classical procedures such as hypothesis testing and confidence intervals. Current methods for inference involving posterior distributions typically involve sampling strategies. That is, due to the complicated nature of some posterior distributions, analytic methods fail to provide meaningful summaries. Hence, sampling from the posterior has become popular. A full description of sampling techniques, starting from rejection sampling, will be given. Outline Syllabus includes: Conjugate models (prior and posterior belong to the same family of parametric models). Predictive distributions; Bayes estimates; Sampling density functions; Gibbs and Metropolis-Hastings samplers; Winbugs; Bayesian regression and hierarchical models; Bayesian model choice; Decision theory; Objective priors; Exchangeability.
Credits: 15 credits (7.5 ECTS credits).
MA889 - Analysis of Large Data Sets (15 credits)
This module considers statistical analysis when we observe multiple characteristics on an experimental unit. For example, a sample of students' marks on several exams or the genders, ages and blood pressures of a group of patients. We are particularly interested in understanding the relationships between the characteristics and differences between experimental units. Regression methods can be used if one characteristic can be treated as a response variable and the others as explanatory variables. Variable selection on the explanatory variables can be daunting if the number of characteristics is large and suitable methods will be investigated. Outline Syllabus includes: measure of dependence, principal component analysis, factor analysis, canonical correlation analysis, hypothesis testing, discriminant analysis, clustering, scaling, information criterion methods for variable selection, false discovery rate, penalised maximum likelihood.
Credits: 15 credits (7.5 ECTS credits).
MA962 - Geometric Integration (15 credits)
The equations studied in this module will be ordinary differential systems, especially Hamiltonian systems. The aim of this subject area is to obtain and study numerical solutions of these systems that preserve specific qualitative and geometric properties. For certain differential equations, these geometric methods can be far superior to standard numerical methods. The syllabus includes: A review of basic numerical methods, variational methods and Hamiltonian mechanics; Properties that numerical methods can preserve (first integrals, symplecticity, time reversibility); Geometric numerical methods (modified Euler and Runge-Kutta methods, splitting methods); Use and misuse of the various notions of error.
Credits: 15 credits (7.5 ECTS credits).
MA964 - Applied Algebraic Topology (15 credits)
There is growing interest in applying the methods of algebraic topology to data analysis, sensor networks, robotics, etc. The module will develop the necessary elements of algebra and topology, and investigate how these techniques are used in various applications. The syllabus will include: an introduction to manifolds, CW complexes and simplicial complexes; an investigation of the elements of homotopy theory; an exploration of homological and computational methods; applications such as homological sensor networks and topological data analysis.
Credits: 15 credits (7.5 ECTS credits).
MA965 - Symmetries, Groups and Invariants (15 credits)
In this module we will study certain configurations with symmetries as they arise in real world applications. Examples include knots described by "admissible diagrams" or chemical structures described by "colouring patterns". Different diagrams and patterns can describe essentially the same structure, so the problem of classification up to equivalence arises. This will be solved by attaching "invariants" which are then put in "normal form" to distinguish them. The syllabus will be as follows: (a) Review of basic methods from linear algebra, group theory and discrete mathematics; (b) Permutation groups, transitivity, primitivity, Burnside formula; (c) Finitely generated Abelian groups; (d) Applications to knot theory, Reidemeister moves, the Abelian knot group; (e) Examples, observations, generalizations and proofs; (f) General Polya-enumeration (as an extension of the Burnside formula).
Credits: 15 credits (7.5 ECTS credits).
MA995 - Graphs and Combinatorics (15 credits)
Combinatorics is a field in mathematics that studies discrete, usually finite, structures, such as graphs. It not only plays an important role in numerous parts of mathematics, but also has real world applications. In particular, it underpins a variety of computational processes used in digital technologies and the design of computing hardware. Among other things, this module provides an introduction to graph theory. Graphs are discrete objects consisting of vertices that are connected by edges. We will discuss a variety of concepts and results in graph theory, and some fundamental graph algorithms. Topics may include, but are not restricted to, trees, shortest paths problems, walks on graphs, graph colourings and embeddings, flows and matchings, and matrices and graphs. In addition to graphs the module may cover other topics in combinatorics such as Ramsey theory, problems in extremal set theory, enumerative problems, Principle of Inclusion and Exclusion, and the P versus NP problem.
Credits: 15 credits (7.5 ECTS credits).
MA884 - Principles of Data Collection (15 credits)
This module will focus on sample surveys, experimental design and clinical trials. It will consider the key principles for designing a survey or an experiment to ensure that any inferences drawn about the population being studied or about the treatments being compared are valid. The discussion of experimental design will be extended into the field of clinical trials (trials conducted on humans) with the added considerations that this introduces. Use will be made of R and Excel for practical examples. Outline Syllabus includes: simple, stratified, cluster and multi-stage sampling; ratio and regression estimators; questionnaire design; completely randomised, randomised block and Latin square designs; factorial designs, fractional replication and confounding; incomplete block designs; analysis of covariance; practical aspects of clinical trials; parallel group trials, sample size; multicentre and crossover trials.
Credits: 15 credits (7.5 ECTS credits).
MA888 - Stochastic Models in Ecology and Medicine (15 credits)
This module considers the development and application of stochastic models in two specific areas. The ecological part is focused on the analysis of data collected on wild animals. Particular attention will be given to estimating how long wild animals live, and also to estimating the sizes of mobile animal populations. The medical part also considers the estimation of survival, but in this case for human beings, with less data loss due to individuals leaving the study than is typical in ecological studies. In survival data it is often known only that individuals survived for a certain period of time, with exact survival time being unknown. This is called censoring and its implications will be discussed in detail. Outline Syllabus includes: Estimating abundance; estimating survival; using covariates; multi-state models; parameter redundancy; human survival data with censoring; the hazard and related functions; parametric and semiparametric survival models.
Credits: 15 credits (7.5 ECTS credits).
MA871 - Asymptotics and Perturbation Methods (15 credits)
The lectures will introduce students to asymptotic and perturbation methods for the approximate evaluation of integrals and to obtaining approximations for solutions of ordinary differential equations. These methods are widely used in the study of physically significant differential equations which arise in Applied Mathematics, Physics and Engineering. The material is chosen so as to demonstrate a range of mathematical techniques available and to illustrate some different applications which are amenable to such analysis.
Asymptotics. Ordering symbols. Asymptotic sequences, expansions and series. Differentiation and integration of asymptotic expansions. Dominant balance. Solution of algebraic and transcendental equations.
Asymptotic evaluation of integrals. Integration by parts. Laplace's method and Watson's lemma. Method of stationary phase.
Approximate solution of linear differential equations. Classification of singular points. Local behaviour at irregular singular points. Asymptotic expansions in the complex plane. Stokes phenomena: Stokes and anti-Stokes lines, dominance and sub-dominance. Connections between sectors of validity. Airy functions.
Matched asymptotic expansions. Regular and singular perturbation problems. Asymptotic matching. Boundary layer theory: inner, outer and intermediate expansions and limits. Uniform approximation.
WKB method. Schrödinger equation and Sturm-Liouville problems. Turning points.
Multiple scales analysis and related methods. Secular terms. Multiple scales method. Method of strained coordinates (Lindstedt-Poincaré method).
Credits: 15 credits (7.5 ECTS credits).
MA561 - Introduction to Lie Groups and Algebras (15 credits)
Lie groups and their associated Lie algebras are studied by both pure and applied mathematicians and by physicists; this is a topic renowned for both its mathematical beauty and its immense utility. Lie groups include translation, rotation and scaling groups as well as unitary, symplectic and special linear matrix groups. We will study in detail the lower dimensional groups that arise in many applications, and more general theory such as the structure of their associated Lie algebras. Special topics include a look at the lowest dimensional exceptional Lie group G2, and Lie group actions and their invariants.
Credits: 15 credits (7.5 ECTS credits).
Teaching & Assessment
Teaching: use of relevant statistical packages; computer laboratory classes; dissertation module; lectures and extended example classes; supervised example classes.
Assessment: coursework involving: complex theoretical questions, analysis of real-world data using appropriate computing packages, projects and dissertation, written unseen examinations.
Programme aims
- provide an excellent quality of education in mathematics and statistics, informed by research and scholarship
- equip students with a broad base of knowledge and skills to analyse and solve mathematical and statistical problems, and to pursue statistics to an advanced level, including development of a systematic understanding of theoretical and practical statistics
- develop in students appreciation of recent developments in statistics, and of the links between the theory of statistics and their practical application
- develop students' capacity for rigorous reasoning and precise expression and to communicate their conclusions effectively and precisely
- develop in students the ability to work independently with a minimum amount of supervision within agreed guidelines
- ensure that students are competent in the use of information technology and can use appropriate software to solve problems
- provide successful students with the depth of knowledge of statistics sufficient to enter a career as a professional statistician or to begin a PhD in Statistics
- provide successful students with eligibility for the Royal Statistical Society's award of GradStat, a recognised professional qualification in statistics that is a stepping stone to the Society's Chartered Statistician (CStat) award (subject to successful accreditation of the programme by the Royal Statistical Society)
Learning outcomes
Knowledge and understanding
- The fundamental concepts and techniques of calculus, algebra, analysis, numerical mathematics and linear algebra (SB3.12).
- Statistical understanding in the subjects of probability and inference and their application to a range of problems (SB3.9, SB3.14, SB3.18).
- Information technology skills relevant to mathematicians and statisticians and understanding of the role of statistical programming.
- Methods, models and techniques appropriate to mathematics and statistics and appreciation of the links between different concepts and methods.
- The role of logical mathematical argument and deductive reasoning, including the formal process of mathematical proof (SB3.14).
- Project work on an advanced statistical topic based on substantial independent work.
Intellectual skills
You develop your intellectual skills in the following areas:
- Ability to demonstrate a reasonable understanding of the main body of knowledge for mathematics and statistics.
- Ability to demonstrate skill in calculation and manipulation of mathematical and statistical material.
- Ability to apply a range of concepts and principles in various contexts relevant to mathematics and statistics.
- Ability to construct and apply logical arguments in mathematics and statistics.
- Ability to demonstrate skill in solving problems in mathematics and statistics using appropriate methods.
- Ability in relevant computer skills and usage.
- Ability to work with relatively little guidance.
- Ability to demonstrate understanding of some advanced statistical theory and techniques and, where appropriate, to demonstrate ability to apply techniques appropriately.
Subject-specific skills
You gain subject-skills in the following areas:
- Ability to demonstrate knowledge of core and advanced mathematical and statistical concepts and topics, both explicitly and by applying them to the solution of problems
- Ability to comprehend problems and abstract the essentials of problems so as to facilitate modelling, mathematical and statistical analysis and interpretation
- Ability to use computational and more general IT facilities in mathematical and statistical work
- Ability to present mathematical and statistical arguments and the conclusions from them with clarity and accuracy.
- Ability to apply some advanced statistical techniques, including model fitting, and interpret the results appropriately.
Transferable skills
You gain transferable skills in the following areas:
- Problem-solving skills; ability to work independently to solve problems involving qualitative and quantitative information
- Communication skills
- Computational skills
- Information-retrieval skills, in relation to primary and secondary information sources, including information retrieval through on-line computer searches
- Information technology skills, including scientific word-processing
- Time-management and organisational skills, as evidenced by the ability to plan and implement efficient and effective modes of working
- Study skills needed for continuing professional development
KIS Course data
Entry requirements
Home/EU students
The University will consider applications from students offering a wide range of qualifications, typical requirements are listed below, students offering alternative qualifications should contact the Admissions Office for further advice. It is not possible to offer places to all students who meet this typical offer/minimum requirement.
Students can also enter the MMathStat programme by transfer from the standard 3-year degree programmes at the end of Stage 2, provided they have passed the core modules and met the average mark threshold of Stage 2 of the MMathStat programme.
Qualification | Typical offer/minimum requirement |
---|---|
A level | AAA including A in Mathematics (not Use of Mathematics). Only one General Studies and Critical Thinking can be accepted against the requirements |
Access to HE Diploma | The University will not necessarily make conditional offers to all Access candidates but will continue to assess them on an individual basis. If we make you an offer, you will need to obtain/pass the overall Access to Higher Education Diploma and may also be required to obtain a proportion of the total level 3 credits and/or credits in particular subjects at merit grade or above |
BTEC Level 3 Extended Diploma (formerly BTEC National Diploma) | The University will consider applicants holding BTEC National Diploma and Extended National Diploma Qualifications (QCF; NQF; OCR) on a case-by-case basis. Please contact us for further advice on your individual circumstances. |
International Baccalaureate | 34 points overall or 17 at HL including Mathematics HL 6 |
International students
The University welcomes applications from international students. Our international recruitment team can guide you on entry requirements. See our International Student website for further information about entry requirements for your country.
If you need to increase your level of qualification ready for undergraduate study, we offer a number of International Foundation Programmes.
Meet our staff in your country
For more advise about applying to Kent, you can meet our staff at a range of international events.
Qualification | Typical offer/minimum requirement |
---|---|
English Language Requirements | International students will need to demonstrate their proficiency in English: Average 6.5 in IELTs test with minimum 6.0 in reading and writing or equivalent. Please see our English language entry requirements web page. |
General entry requirements
Please also see our general entry requirements.
Careers
Students studying this degree programme will develop a broad range of skills and mathematical and statistical understanding that are highly sought after by employers and which open up a wide variety of careers. School of Mathematics, Statistics and Actuarial Science graduates typically find employment in areas involving applications of the subject or they directly enter postgraduate studies at the doctoral level. Recent graduates of the School have gone into careers in medical statistics, the pharmaceutical industry, the aerospace industry, software development, teaching, Civil Service statistics, chartered accountancy, the oil industry and PhD training.
Professional recognition
Successful students will be eligible for the Royal Statistical Society's award of GradStat, a recognised professional qualification in statistics that is a stepping stone to the Society's Chartered Statistician (CStat) award (subject to successful accreditation of the programme by the Royal Statistical Society)
Funding
University funding
Kent offers generous financial support schemes to assist eligible undergraduate students during their studies. See our funding page for more details.
Government funding
You may be eligible for government finance to help pay for the costs of studying. See the Government's student finance website.
The Government has confirmed that EU students applying for university places in the 2017 to 2018 academic year will still have access to student funding support for the duration of their course.
Scholarships
General scholarships
Scholarships are available for excellence in academic performance, sport and music and are awarded on merit. For further information on the range of awards available and to make an application see our scholarships website.
The Kent Scholarship for Academic Excellence
At Kent we recognise, encourage and reward excellence. We have created the Kent Scholarship for Academic Excellence. The scholarship will be awarded to any applicant who achieves a minimum of AAA over three A levels, or the equivalent qualifications (including BTEC and IB) as specified on our scholarships pages.
The scholarship is also extended to those who achieve AAB at A level (or specified equivalents) where one of the subjects is either Mathematics or a Modern Foreign Language. Please review the eligibility criteria.
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Enquiries
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Open days
Our general open days will give you a flavour of what it is like to be an undergraduate, postgraduate or part-time student at Kent. They include a programme of talks for undergraduate students, with subject lectures and demonstrations, plus self-guided walking tours of the campus and accommodation.
Please check which of our locations offers the courses you are interested in before choosing which event to attend.