Master of Science (Mathematics and Statistics)

This subject develops the mathematical methods of applied mathematics and mathematical physics with an emphasis on integral transform and related techniques. An introduction is given to the calculus of variations and the Euler-Lagrange equation. Advanced complex contour integration techniques are used to evaluate and invert Fourier and Laplace transforms. The general theory includes convolutions, Green’s functions and generalized functions. The methods of Laplace, stationary phase, steepest descents and Watson’s lemma are used to asymptotically approximate integrals. Throughout, the theory is set in the context of examples from applied mathematics and mathematical physics such as the brachistochrone problem, Fraunhofer diffraction, Dirac delta function, heat equation and diffusion.

This subject develops the mathematical methods of applied mathematics and mathematical physics with an emphasis on ordinary differential equations. Both analytical and approximate techniques are used to determine solutions of ordinary differential equations. Exact solutions by localised series expansion techniques of second-order linear ordinary differential equations and Sturm-Liouville boundary value problems are explored. Special functions are introduced here. Regular and singular perturbation expansion techniques, asymptotic series solutions, dominant balance, and WKB theory are used to determine approximate solutions of linear and nonlinear differential equations. Throughout, the theory is set in the context of examples from applied mathematics and mathematical physics such as nonlinear oscillators, boundary layers and dispersive phenomena.

Random matrix theory is a diverse topic in mathematics. It draws together ideas from linear algebra, multivariate calculus, analysis, probability theory and mathematical physics, amongst other topics. It also enjoys a wide number of applications, ranging from wireless communication in engineering, to quantum chaos in physics, to the Reimann zeta function zeros in pure mathematics. A self contained development of random matrix theory will be undertaken in this course from a mathematical physics viewpoint. Topics to be covered include Jacobians for matrix transformation, matrix ensembles and their eigenvalue probability density functions, equilibrium measures, global and local statistical quantities, determinantal point processes, products of random matrices and Dyson Brownian motion.

Many processes in the natural sciences, engineering and finance are described mathematically using ordinary or partial differential equations. Only the simplest or those with special structure can be solved exactly. This subject discusses common techniques for computing numerical solutions to differential equations and introduces the major themes of accuracy, stability and efficiency. Understanding these basic properties of scientific computing algorithms should prevent the unwary from using software packages inappropriately or uncritically, and provide a foundation for devising methods for nonstandard problems. We cover both time-independent problems, in one and higher space dimensions, and evolution equations of hyperbolic or parabolic type.

Modern techniques have revolutionised biology and medicine, but interpretative and predictive tools are needed. Mathematical modelling is such a tool, providing explanations for counter-intuitive results and predictions leading to new experimental directions. The broad flavour of the area and the modelling process will be discussed. Applications will be drawn from many areas including population growth, epidemic modelling, biological invasion, pattern formation, tumour modelling, developmental biology and tissue engineering. A large range of mathematical techniques will be discussed, for example discrete time models, ordinary differential equations, partial differential equations, stochastic models and cellular automata.

The first half of this subject is an introduction to two-dimensional conformal field theory with emphasis on the operator formalism and explicit calculations. The second half is an introduction to string theory based on the first half. For concreteness, the representation theory of Virasoro algebra and bosonic strings will be emphasized.

The goal of statistical mechanics is to describe the behaviour of bulk matter starting from a physical description of the interactions between its microscopic constituents. This subject introduces the Gibbs probability distributions of classical statistical mechanics, the relations to thermodynamics and the modern theory of phase transitions and critical phenomena. The central concepts of critical exponents, universality and scaling are emphasized throughout. Applications include the ideal gases, magnets, fluids, one-dimensional Ising and Potts lattice spin models, random walks and percolation as well as approximate methods of solution.

In mathematical physics, a wealth of information comes from the exact, non-perturbative, solution of quantum models in one-dimension and classical models in two-dimensions. This subject is an introduction to this beautiful and deep subject. Yang-Baxter equations, Bethe ansatz and matrix product techniques are developed in the context of the critical two-dimensional Ising model, dimers, free fermions, the 6-vertex model, percolation, quantum spin chains and the stochastic asymmetric simple exclusion model. The algebraic setting incorporates the quantum groups, and the Temperley-Lieb and braid-monoid algebras.

Mathematical modelling can give deep insight into many complex systems that arise in nature and technology. It is also used to describe and predict new phenomena, test hypotheses and investigate novel avenues for experiments. This subject presents a series of advanced case studies that demonstrate the utility of mathematical modelling and develop the student's ability to tackle real-world problems arising in scientific, medical or industrial contexts. Mathematical approaches will include discrete, computational and asymptotic methods. The use of appropriate approximations and the interpretation of solutions in the context of the original problem will be emphasised.

This subject develops mathematical methods for the study of the mechanics of fluids and solids and illustrates their use in several contexts. Topics covered include Newtonian fluids at low and at high Reynolds number and the linear theory of elasticity. Applications may be drawn from biological, earth sciences, engineering or physical contexts.

The physical, social and engineering sciences make widespread use of numerical simulations and graphical representations that link underlying their theoretical foundations with experimental or empirical data. These approaches are routinely designed and conducted by researchers with little or no formal training in computation, assembling instead the necessary skills from a variety of sources. There is an art to assembling computational tools that both achieve their goals and make good effective use of the available computational resources.

This subject introduces students to a wide range of skills that are commonly encountered in the design and construction of computational tools in research applications:

• Formulation of the task as a sequence of operations or procedures that express the context of the assigned problem in a form accessible to digital computing (Mathematica).

• Implementation of this formulation using computer languages appropriate for numerically intensive computation (C, C++, Fortran)

• Modularization of computationally intensive tasks, either as user-written procedures or existing libraries (for example BLAS, lapack)

• Documentation of the code to explain both its design, operation and limitations (LaTeX)

• Instrumentation of the code to verify its correct operation and monitor its performance (gprof)

• Optimization of the code, including the use of parallelization (OpenMPI)

• Visualization of data using graphical packages or rendering engines (Geomview, OpenGL)

• Interaction with the code through a graphical user interface (Python, Matlab)

These skills are introduced to the student by undertaking a short project that is selected in consultation with the Subject Coordinator.

This subject will introduce students to the world of quantum information technology, focusing on the fast developing area of quantum computing. The subject will cover basic principles of quantum logic operations in both digital and analogue approaches to quantum processors, through to quantum error correction and the implementation of quantum algorithms for real-world problems. In lab-based classes students will learn to use state-of-the-art quantum computer programing and simulation environments to complete a range of projects.

The subject consists of three main topics. The bijective principle with applications to maps, permutations, lattice paths, trees and partitions. Algebraic combinatorics with applications rings, symmetric functions and tableaux. Ordered sets with applications to generating functions and the structure of combinatorial objects.

The use of mathematical optimisation is widespread in business, where it is a key analytical tool for managing and planning business operations. It is also required in many industrial processes and is useful to government and community organizations. This subject will expose students to operations research techniques as used in industry. A heavy emphasis will be placed on the modelling process that turns an industrial problem into a mathematical formulation. The focus will then be on how to solve the resulting mathematical problem with mixed-integer programming techniques.

Many discrete optimisation problems encountered in practice are too difficult to solve exactly in a reasonable time frame. Approximation algorithms and heuristics are the most widely used approaches for obtaining reasonably accurate solutions to such hard problems. This subject introduces the basic concepts and techniques underlying these “inexact” approaches. We will address the following fundamental questions in the subject: How difficult is the problem under consideration? How closely can an optimal solution be approximated? And how can we go about finding near-optimal solutions in an efficient way? We will discuss methodologies for analysing the complexity and approximability of some important optimisation problems, including the travelling salesman problem, knapsack problem, bin packing, scheduling, network design, covering problems and facility location. We will also learn about various metaheuristics (simulated annealing, Tabu search, GRASP, genetic algorithms) and matheuristics (relax-and-fix, fix-and-optimise, local branching) that are widely used in solving real-world optimisation problems.

Scheduling is critical to manufacturing, mining, and logistics, and is of increasing importance in healthcare and service industries. Most automated systems, ranging from elevators to industrial robots, embed some kind of scheduling algorithms. Building on the Optimisation background provided in Optimisation for Industry, this subject teaches students how to solve more advanced problems. A particular focus will be scheduling problems, but other more general assignment problems will be discussed.

Many practical problems in management, operations research, telecommunication and computer networking can be modelled as optimisation problems on networks. Here the underlying structure is a graph. This subject is an introduction to optimisation problems on networks with a focus on theoretical results and efficient algorithms. It covers classical problems that can be solved in polynomial time, such as shortest paths, maximum matchings, maximum flows, and minimum cost flows. Other topics include complexity and NP-completeness, matroids and greedy algorithms, approximation algorithms, multicommodity flows, and network design. This course is beneficial for all students of discrete mathematics, operations research, and computer science.

The subject is about the use of generating functions for enumeration of combinatorial structures, including partitions of numbers, partitions of sets, permutations with restricted cycle structure, connected graphs, and other types of graph. The subject covers the solution of recurrence relations, methods of asymptotic enumeration, and some applications in statistical mechanics. The methods covered have widespread applicability, including in areas of pure and applied mathematics and computer science.

Modern computers have developed far beyond being great devices for numerical simulations or tedious but straightforward algebra; and in 1990 the first mathematical research paper was published whose sole author was a thinking machine known as Shalosh B Ekhad. This course will discuss some of the great advances made in using computers to purely algorithmically discover (and prove!) nontrivial mathematical theorems in for example Number Theory and Algebraic Combinatorics. Topics include: Automated hypergeometric summation, Groebner basis, Chaos theory, Number guessing, Recurrence relations, BBP formulas.

The physical, social and engineering sciences make widespread use of numerical simulations and graphical representations that link underlying their theoretical foundations with experimental or empirical data. These approaches are routinely designed and conducted by researchers with little or no formal training in computation, assembling instead the necessary skills from a variety of sources. There is an art to assembling computational tools that both achieve their goals and make good effective use of the available computational resources.

This subject introduces students to a wide range of skills that are commonly encountered in the design and construction of computational tools in research applications:

• Formulation of the task as a sequence of operations or procedures that express the context of the assigned problem in a form accessible to digital computing (Mathematica).

• Implementation of this formulation using computer languages appropriate for numerically intensive computation (C, C++, Fortran)

• Modularization of computationally intensive tasks, either as user-written procedures or existing libraries (for example BLAS, lapack)

• Documentation of the code to explain both its design, operation and limitations (LaTeX)

• Instrumentation of the code to verify its correct operation and monitor its performance (gprof)

• Optimization of the code, including the use of parallelization (OpenMPI)

• Visualization of data using graphical packages or rendering engines (Geomview, OpenGL)

• Interaction with the code through a graphical user interface (Python, Matlab)

These skills are introduced to the student by undertaking a short project that is selected in consultation with the Subject Coordinator.

This subject will introduce students to the world of quantum information technology, focusing on the fast developing area of quantum computing. The subject will cover basic principles of quantum logic operations in both digital and analogue approaches to quantum processors, through to quantum error correction and the implementation of quantum algorithms for real-world problems. In lab-based classes students will learn to use state-of-the-art quantum computer programing and simulation environments to complete a range of projects.

Measure Theory formalises and generalises the notion of integration. It is fundamental to many areas of mathematics and probability and has applications in other fields such as physics and economics. Students will be introduced to Lebesgue measure and integration, signed measures, the Hahn-Jordan decomposition, the Radon-Nikodym derivative, conditional expectation, Borel sets and standard Borel spaces, product measures, and the Riesz representation theorem.

This subject studies topological spaces and continuous maps between them. It demonstrates the power of topological methods in dealing with problems involving shape and position of objects and continuous mappings, and shows how topology can be applied to many areas, including geometry, analysis, group theory and physics. The aim is to reduce questions in topology to problems in algebra by introducing algebraic invariants associated to spaces and continuous maps. Important classes of spaces studied are manifolds (locally Euclidean spaces) and CW complexes (built by gluing together cells of various dimensions). Topics include: homotopy of maps and homotopy equivalence of spaces, homotopy groups of spaces, the fundamental group, covering spaces; homology theory, including singular homology theory, the axiomatic approach of Eilenberg and Steenrod, and cellular homology.

This course is an introduction to algebraic geometry. Algebraic geometry is the study of zero sets of polynomials. It exploits the interplay between rings of functions and the underlying geometric objects on which they are defined. It is a fundamental tool in may areas of mathematics, including number theory, physics and differential geometry. The syllabus includes affine and projective varieties, coordinate ring of functions, the Nullstellensatz, Zariski topology, regular morphisms, dimension, smoothness and singularities, sheaves, schemes.

The subject covers aspects of multilinear and commutative algebra as well as two substantial applications. Within multilinear algebra this includes bilinear forms and `multilinear products’ of vector spaces, such as tensor products. Commutative algebra concerns itself with properties of commutative rings, such as polynomial rings and their quotients and to modules over such rings. Both topics have wide application, both to other parts of mathematics and to physics. Much of this theory was developed for applications in geometry and in number theory, and the theorems can be used to cast substantial light on problems from geometry and number theory.

As well as being beautiful in its own right, algebra is used in many areas of mathematics, computer science and physics. This subject provides a grounding in several fundamental areas of modern advanced algebra including Lie groups, combinatorial group theory, category theory and homological algebra.
The material complements that covered in the subject Commutative and Mutlilinear Algebra without assuming it as prerequisite.

Functional analysis is a fundamental area of pure mathematics, with countless applications to the theory of differential equations, engineering, and physics.
The students will be exposed to the theory of Banach spaces, the concept of dual spaces, the weak-star topology, the Hahn-Banach theorem, the axiom of choice and Zorn's lemma, Krein-Milman, operators on Hilbert space, the Peter-Weyl theorem for compact topological groups, the spectral theorem for infinite dimensional normal operators, and connections with harmonic analysis.

Symmetries arise in mathematics as groups and Representation Theory is the study of groups via their actions on vector spaces. It has important applications in many fields: physics, chemistry, economics, biology and others. This subject will provide the basic tools for studying actions on vector spaces. The course will focus on teaching the basics of representation theory via favourite examples: symmetric groups, diagram algebras, matrix groups, reflection groups. In each case the irreducible characters and irreducible modules for the group (or algebra) will be analysed, developing more and more powerful tools as the course proceeds. Examples that will form the core of the material for the course include SL2, cyclic and dihedral groups, diagram algebras: Temperley-Lieb, symmetric group and Hecke algebras, Brauer and BMW algebras, compact Lie groups. Among the tools and motivation that will play a role in the study are characters and character formulas, induction, restriction and tensor products, and connections to statistical mechanics, mathematical physics and geometry.
If time permits, there may be some treatment of loop groups, affine Lie algebras and Dynkin diagrams.

This subject extends the methods of calculus and linear algebra to study the geometry and topology of higher dimensional spaces. The ideas introduced are of great importance throughout mathematics, physics and engineering. This subject will cover basic material on the differential topology of manifolds including integration on manifolds, and give an introduction to Riemannian geometry. Topics include: Differential Topology: smooth manifolds, tangent spaces, inverse and implicit function theorems, differential forms, bundles, transversality, integration on manifolds, de Rham cohomology; Riemanian Geometry: connections, geodesics, and curvature of Riemannian metrics; examples coming from Lie groups, hyperbolic geometry, and other homogeneous spaces.

Riemann surfaces arise from complex analysis. They are central in mathematics, appearing in seemingly diverse areas such as differential and algebraic geometry, number theory, integrable systems, statistical mechanics and string theory.

The first part of the subject studies complex analysis. It assumes students have completed a first course in complex analysis so begins with a quick review of analytic functions and Cauchy's theorem, emphasising topological aspects such as the argument principle and Rouche's theorem.

Topics also include: Schwarz's lemma; limits of analytic functions, normal families, Riemann mapping theorem; multiple-valued functions, differential equations and Riemann surfaces. The second part of the subject studies Riemann surfaces and natural objects on them such as holomorphic differentials and quadratic differentials.

Topics may also include: divisors, Riemann-Roch theorem; the moduli space of Riemann surfaces, Teichmueller space; integrable systems.

This subject will present an introduction to an advanced topic in geometry or topology serving to prepare students for a PhD in mathematics. The specific content will vary depending on the subject coordinator. 

The physical, social and engineering sciences make widespread use of numerical simulations and graphical representations that link underlying their theoretical foundations with experimental or empirical data. These approaches are routinely designed and conducted by researchers with little or no formal training in computation, assembling instead the necessary skills from a variety of sources. There is an art to assembling computational tools that both achieve their goals and make good effective use of the available computational resources.

This subject introduces students to a wide range of skills that are commonly encountered in the design and construction of computational tools in research applications:

• Formulation of the task as a sequence of operations or procedures that express the context of the assigned problem in a form accessible to digital computing (Mathematica).

• Implementation of this formulation using computer languages appropriate for numerically intensive computation (C, C++, Fortran)

• Modularization of computationally intensive tasks, either as user-written procedures or existing libraries (for example BLAS, lapack)

• Documentation of the code to explain both its design, operation and limitations (LaTeX)

• Instrumentation of the code to verify its correct operation and monitor its performance (gprof)

• Optimization of the code, including the use of parallelization (OpenMPI)

• Visualization of data using graphical packages or rendering engines (Geomview, OpenGL)

• Interaction with the code through a graphical user interface (Python, Matlab)

These skills are introduced to the student by undertaking a short project that is selected in consultation with the Subject Coordinator.

This subject will introduce students to the world of quantum information technology, focusing on the fast developing area of quantum computing. The subject will cover basic principles of quantum logic operations in both digital and analogue approaches to quantum processors, through to quantum error correction and the implementation of quantum algorithms for real-world problems. In lab-based classes students will learn to use state-of-the-art quantum computer programing and simulation environments to complete a range of projects.

The subject covers the key aspects of the theory of stochastic processes that plays the central role in modern probability and has numerous applications in natural sciences and industry. We discuss the following topics: ways to construct and specify random processes, discrete time martingales, Levy processes and more general continuous time Markov processes, point processes. Applications to modelling random phenomena evolving in time are discussed throughout the course.

The theory of statistical inference is important for applied statistics and as a discipline in its own right. After reviewing random samples and related probability techniques including inequalities and convergence concepts the theory of statistical inference is developed. The principles of data reduction are discussed and related to model development. Methods of finding estimators are given, with an emphasis on multi-parameter models, along with the theory of hypothesis testing and interval estimation. Both finite and large sample properties of estimators are considered. Applications may include robust and distribution free methods, quasi-likelihood and generalized estimating equations. It is expected that students completing this course will have the tools to be able to develop inference procedures in novel settings.

This subject provides an introduction to stochastic calculus and mathematics of financial derivatives. Stochastic calculus is essentially a theory of integration of a stochastic process with respect to another stochastic process, created for situations where conventional integration will not be possible. Apart from being an interesting and deep mathematical theory, stochastic calculus has been used with great success in numerous application areas, from engineering and control theory to mathematical biology, theory of cognition and financial mathematics.

Multivariate statistics concerns the analysis of collections of random variables that has general applications across the sciences and more recently in bioinformatics. It overlaps machine learning and data mining, and leads into functional data analysis. Here random vectors and matrices are introduced along with common multivariate distributions. Multivariate techniques for clustering, classification and data reduction are given. These include discriminant analysis and principal components. Classical multi-variate regression and analysis of variance methods are considered. These approaches are then extended to high dimensional data, such as that commonly encountered in bioinformatics, motivating the development of multiple hypothesis testing techniques. Finally, functional data is introduced.

Computing techniques and data mining methods are indispensable in modern statistical research and data science applications, where “Big Data” problems are often involved. This subject will introduce a number of recently developed methods and applications in computational statistics and data science that are scalable to large datasets and high-performance computing. The data mining methods to be introduced include general model diagnostic and assessment techniques, kernel and local polynomial nonparametric regression, basis expansion and nonparametric spline regression, generalised additive models, classification and regression trees, forward stagewise and gradient boosting models. Important statistical computing algorithms and techniques used in data science will be explained in detail. These include the bootstrap resampling and inference, cross-validation, the EM algorithm and Louis method, and Markov chain Monte Carlo methods including adaptive rejection and squeeze sampling, sequential importance sampling, slice sampling, Gibbs sampler and Metropolis-Hastings algorithm.

This subject mostly explores the key concept from Probability Theory: convergence of probability distributions, which is fundamental for Mathematical Statistics and is widely used in other applications. We study in depth the classical method of characteristic functions and discuss alternative approaches to proving limit theorems of Probability Theory.

Statistical models are central to applications of statistics and their development motivates new statistical theories and methodologies. Commencing with a review of linear and generalized linear models, analysis of variance and experimental design, the theory of linear mixed models is developed and model selection techniques are introduced. Approaches to non and semiparametric inference, including generalized additive models, are considered. Specific applications may include longitudinal data, survival analysis and time series modelling.

This subject builds on methods and techniques learned in theoretical subjects by studying the application of statistics in real contexts. Emphasis is on the skills needed for a practising statistician, including the development of mature statistical thinking, organizing the structure of a statistical problem, the contribution to the design of research from a statistical point of view, measurement issues and data processing. The subject deals with thinking about data in a broad context, and skills required in statistical consulting.

Mathematical modelling of various types of risk has become an important component of the modern financial industry. The subject discusses the key aspects of the mathematics of market risk. Main concepts include loss distributions, risk and dependence measures, copulas, risk aggregation and allocation principles, elements of extreme value theory. The main theme is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers.

Modern data sets are growing in size and complexity due to the astonishing development of data acquisition and storage capabilities. This subject focuses on developing rigorous statistical learning methods that are needed to extract relevant features from large data sets, assess the reliability of the selected features, and obtain accurate inferences and predictions. This subject covers recent methodological developments in this area such as inference for high-dimensional inference regression, empirical Bayes methods, model selection and model combining methods, and post-selection inference methods.

Complex data consisting of dependent measurements collected at different times and locations are increasingly important in a wide range of disciplines, including environmental sciences, biomedical sciences, engineering and economics. This subject will introduce you to advanced statistical methods and probability models that have been developed to address complex data structures, such as functional data, geo-statistical data, lattice data, and point process data. A unifying theme of this subject will be the development of inference, classification and prediction methods able to cope with the dependencies that often arise in these data.

This subject develops the advanced methods of stochastic processes and discusses possible applications of the models discussed in the course. The topics will include: Levy processes, large deviation theory, point processes and stochastic networks.

Modern data collection technologies are creating unprecedented challenges in statistics and data science related to the analysis and interpretation of massive data sets where observations exhibit patterns through time and space. This subject introduces probability models and advanced statistical inference methods for the analysis of temporal and spatio-temporal data. The subject balances rigorous theoretical development of the methods and their properties with real-data applications. Topics include inference methods for univariate and multivariate time series models, spatial models, lattice models, and inference methods for spatio-temporal processes. The subject will also address aspects related to computational and statistical trade-offs, and the use of statistical software.

This is an advanced course that prepares students for research careers in statistics, which nowadays typically require a combination of applied, methodological and theoretical skills. By considering a variety of statistical topics in depth it introduces students to the technical skills and methods of proof needed to conduct research in modern statistics. Topics covered may include U-statistics, asymptotic distributions of statistics, inference in parametric and nonparametric models, curve estimation, Edgeworth expansions and the bootstrap and sequential methods.

Bayesian inference treats all unknowns as random variables, and the core task is to update the probability distribution for each unknown as new data is observed. After introducing Bayes’ Theorem to transform prior probabilities into posterior probabilities, the first part of this subject introduces theory and methodological aspects underlying Bayesian statistical learning including credible regions, comparisons of means and proportions, multi-model inference and model selection. The second part of the subject focuses on advanced supervised and unsupervised Bayesian machine learning methods in the context of Gaussian processes and Dirichlet processes. The subject will also cover practical implementations of Bayesian methods through Markov Chain Monte Carlo computing and real data applications.

The physical, social and engineering sciences make widespread use of numerical simulations and graphical representations that link underlying their theoretical foundations with experimental or empirical data. These approaches are routinely designed and conducted by researchers with little or no formal training in computation, assembling instead the necessary skills from a variety of sources. There is an art to assembling computational tools that both achieve their goals and make good effective use of the available computational resources.

This subject introduces students to a wide range of skills that are commonly encountered in the design and construction of computational tools in research applications:

• Formulation of the task as a sequence of operations or procedures that express the context of the assigned problem in a form accessible to digital computing (Mathematica).

• Implementation of this formulation using computer languages appropriate for numerically intensive computation (C, C++, Fortran)

• Modularization of computationally intensive tasks, either as user-written procedures or existing libraries (for example BLAS, lapack)

• Documentation of the code to explain both its design, operation and limitations (LaTeX)

• Instrumentation of the code to verify its correct operation and monitor its performance (gprof)

• Optimization of the code, including the use of parallelization (OpenMPI)

• Visualization of data using graphical packages or rendering engines (Geomview, OpenGL)

• Interaction with the code through a graphical user interface (Python, Matlab)

These skills are introduced to the student by undertaking a short project that is selected in consultation with the Subject Coordinator.

This subject will introduce students to the world of quantum information technology, focusing on the fast developing area of quantum computing. The subject will cover basic principles of quantum logic operations in both digital and analogue approaches to quantum processors, through to quantum error correction and the implementation of quantum algorithms for real-world problems. In lab-based classes students will learn to use state-of-the-art quantum computer programing and simulation environments to complete a range of projects.

Quantum Mechanics introduces a dramatically new and rich understanding of the universe. In addition to providing a much deeper insight into the world of atoms and subatomic particles than afforded by classical Newtonian physics, Quantum Mechanics underpins advances in science across all disciplines, from molecular biology to astrophysics. This subject provides a rigorous mathematical formalism for advanced quantum mechanics, laying the foundation for further fundamental theoretical physics and research-level experimental physics in frontier areas such as quantum communication and quantum computation.

The subject describes the Hilbert-space formulation of quantum wave mechanics, including density matrix descriptions for single and joint Hilbert space systems; symmetries and conservation laws including rotations and angular momentum; many-body systems of identical particles; time-dependent perturbation theory, and scattering theory.

This subject introduces quantum field theory, the combination of quantum mechanics and relativity that explains the fundamental structure of matter and the physics of the early universe. The course has an emphasis on quantum electrodynamics. Specific topics will include an introduction to classical field theory, the Euler-Lagrange equations and Noether’s theorem; the Dirac and Klein-Gordon equations; the quantisation of free scalar, Dirac and vector fields; covariant perturbation theory, the Smatrix and Feynman diagrams; the computation of elementary processes in quantum electrodynamics.

This subject provides an advanced introduction to Einstein's theory of general relativity. Specific topics may inlcude special relativity, manifolds and curvature, experimental tests, Einstein's equations, the Schwarzschild solution and black holes, weak fields and gravitational radiation. Examples will be drawn from particle physics, astrophysics and cosmology.

This subject provides an advanced introduction to non-equilibrium statistical mechanics. The subject focuses on collective phenomena in complex many-body systems with an emphasis on diffusive processes, stability and the emergence of long-range order, with examples drawn from physics, chemistry, biology and economics. Specific topics include diffusive stochastic processes (Fokker-Planck equations), birth-death processes (master equations), kinetic transport, and spatio-temporal pattern formation in unstable nonlinear systems (bifurcations, chaos, reaction-diffusion equations). 

This subject provides an advanced introduction to physical cosmology. Specific topics may include the isotropic homogeneous Universe, the Robertson Walker metric, the Friedmann equations, baryogenesis, inflation, big-bang nucleosynthesis, the recombination era, density fluctuations as the origin of galaxies, the cosmic microwave background, linear and non-linear growth of structure, the Press-Schechter mass function, reionization of the IGM and gravitational lensing. Examples are drawn from past and current cosmological observations.

Particle Physics is the study of the elementary constituents of matter, and the fundamental forces of nature. The subject introduces modern elementary particle physics, with an emphasis on the theoretical description of the Standard Model of Particle Physics and its experimental basis. Specific topics may include basic group theory; parity and CP violation; global and local symmetries; non-abelian gauge theory; QCD and the quark model; running coupling constants and asymptotic freedom; spontaneous symmetry breaking and the Higgs mechanism; the complete Standard Model Lagrangian; interactions of particles with matter; accelerators and detectors; deep inelastic scattering and structure functions; flavour mixing and neutrino oscillations.

Modern science and business makes extensive use of computers for simulation, because complex real-world systems often cannot be analysed exactly, but can be simulated. Using simulation we can perform virtual experiments with the system, to see how it responds when we change parameters, which thus allows us to optimise its performance. We use the language R, which is one of the most popular modern languages for data analysis.

In this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the School's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

In this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the School's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

In this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the School's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

In this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the School's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

In this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the School's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

In this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the School's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

In this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the Department's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.

In this subject, students undertake a substantial research program in the area of Mathematics and Statistics. The research will be conducted under the supervision of a member of the Department's academic staff. A list of the research interests of the Department of Mathematics and Statistics is outlined on the website of the Department. The results will be reported in the form of a thesis and an oral presentation.