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.

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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.

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Random matrix theory is a diverse mathematical tool. It draws together ideas from linear algebra, multivariate calculus, analysis, probability theory, group and representation theory, differential geometry, combinatorics and mathematical physics. It also enjoys a wide number of applications, ranging from wireless communication in engineering, to time series analysis in statistics, quantum chaos and quantum field theory in physics, to the Riemann zeta function zeros and prime numbers in number theory. A self contained development of random matrix theory will be undertaken in this course from various viewpoints.

Topics to be covered include:

• Gaussian random matrix models and their application in likelihood analysis and modelling covariance matrices in time series analysis;
• eigenvalue densities and the concept of eigenvalue repulsion;
• classification of random matrices ensembles;
• derivation of Jacobians for matrix transformations such as diagonalisations;
• joint eigenvalue densities and correlation functions;
• orthogonal polynomials and the concept of determinantal point processes;
• supersymmetry and non-linear sigma-models;
• the log-gas picture;
• free probability theory and its application to matrix sums and products.

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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, prior choice, comparisons of means and proportions, multi-model inference and model selection. The second part of the subject will cover practical implementations of Bayesian methods through Markov Chain Monte Carlo computing and real data applications, focusing on (generalised) linear models and concluding by exploring machine learning techniques such as Gaussian processes.

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This subject uses mathematical modelling, analysis and simulation to provide insight into complex biological phenomena. With a focus on mechanistic modelling and viewing biological systems as dynamic in nature, you will learn how to develop and implement “real-world” models, applicable to current open problems in mathematical biology.

To give you a taste of cutting-edge research in mathematical biology, this subject will present methods from recent research applying mathematics to biology. This will include topics from the following list:

• Dynamical Mmodelling; and
• Chemical Reaction Network Theory.

The subject will include the use of software languages and packages for modelling and simulation. Motivating problems will be drawn from across the spectrum of biology from genetics and molecular biology to ecology.

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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.

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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.

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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 and quantum 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.

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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.

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This subject introduces the fundamental mathematical models used to study infectious diseases at both the epidemiological and within-host scale. The emphasis is on: 1) how models are developed, from conceptualisation through to implementation in software; and 2) how to apply models to questions of epidemiological, public health and biological importance. Statistical techniques for the model-based analysis of relevant data resources will be introduced.

• Epidemiology: epidemic/endemic behaviour and intervention strategies to reduce transmission, the SIR model, including demography, threshold behaviour, phase-plane analysis;
• Viral dynamics: host-pathogen interactions, the mediating influences of immunomodulatory agents and antimicrobials, the TIV model, including the immune response, pharmacokinetic-pharmacodynamic models;
• Model sensitivity and uncertainty analysis, scenario analysis, parameter estimation, model comparison

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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 and quantum 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.

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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.

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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.

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Random matrix theory is a diverse mathematical tool. It draws together ideas from linear algebra, multivariate calculus, analysis, probability theory, group and representation theory, differential geometry, combinatorics and mathematical physics. It also enjoys a wide number of applications, ranging from wireless communication in engineering, to time series analysis in statistics, quantum chaos and quantum field theory in physics, to the Riemann zeta function zeros and prime numbers in number theory. A self contained development of random matrix theory will be undertaken in this course from various viewpoints.

Topics to be covered include:

• Gaussian random matrix models and their application in likelihood analysis and modelling covariance matrices in time series analysis;
• eigenvalue densities and the concept of eigenvalue repulsion;
• classification of random matrices ensembles;
• derivation of Jacobians for matrix transformations such as diagonalisations;
• joint eigenvalue densities and correlation functions;
• orthogonal polynomials and the concept of determinantal point processes;
• supersymmetry and non-linear sigma-models;
• the log-gas picture;
• free probability theory and its application to matrix sums and products.

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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.

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The theory of Lie algebras is fundamental to the study of groups of continuous symmetries acting on vector spaces, with applications to diverse areas including geometry, number theory and the theory of differential equations. Moreover, since quantum mechanical systems are described by Hilbert spaces acted on by continuous symmetries, Lie algebras and their representations are also fundamental to modern mathematical physics. This subject develops the basic theory in a way accessible to both pure mathematics and mathematical physics students, with an emphasis on examples. The main theorems are: the classification of complex semi-simple Lie algebras in terms of Cartan matrices and Dynkin diagrams, and the classification of finite-dimensional representations of these algebras in terms of highest weight theory.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Game theory is a branch of mathematics where the interactions between rational decision makers (players) are modelled and analysed. Game theory can broadly be divided into the study of noncooperative games and cooperative games. For noncooperative games we study two-player games, games in extensive form, games of perfect and imperfect information, games with complete and incomplete information, games with chance moves, repeated games, and Bayesian games. To analyse these games we introduce the concepts of Nash equilibria, evolutionary stable strategies, subgame perfect equilibria, and Harsanyi games. For cooperative games we study coalitional games with transferable utility, and introduce the concepts of coalitions, characteristic functions, the core, the Shapley value, and the nucleolus. We discuss in detail the well known Bonderava-Shapley theorem which gives conditions for the nonemptyness of the core. This subject provides a rigorous mathematical treatment of game theory, and will include applications selected from queueing theory, biology, population dynamics, resource allocation, auction theory, political science, and military applications.

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Many optimisation problems in the real world are inherently nonlinear. A variety of industries, including telecommunications networks, underground mining, microchip design, computer vision, facility location and supply chain management, depend on the efficient solution of nonlinear programs. This subject introduces the foundational mathematical concepts behind nonlinear optimisation. Some of the concepts covered include convex analysis, optimality conditions, conic programming, and duality. Various methods to solve nonlinear programs are covered, including iterative methods such as conjugate gradient methods, barrier methods and subgradient methods. This subject also explores the application of geometric methods such as perturbation and variational approaches to problems in facility location and network design.

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Measure Theory introduces the modern conceptual framework of analysis that has led to a transformation and generalisation of such basic objects as functions, and such notions as continuity, differentiability and integrability.

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 the core topics of Lebesgue's theory of integration, and abstract measure theory, in particular signed measures, the Hahn-Jordan decomposition, the Radon-Nikodym derivative. Additional topics may include rudiments of probability theory (conditional expectation, Borel sets and measures) and geometric analysis (rectifiable curves, Hausdorff measure and dimension).

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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.

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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 will cover affine and projective varieties, the Nullstellensatz, Zariski topology, morphisms, sheaves, schemes and may include additional topics in algebra and algebraic geometry (e.g., dimension, smoothness, singularities).

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Homological algebra is a set of tools designed to linearise problems from geometry and topology, algebra, mathematical physics, and other areas of mathematics. This subject provides an introduction to this fascinating field as well as to basic notions of category theory.

The subject covers categories, functors, limits and colimits, adjoint pairs, and abelian categories; chain complexes and homology, chain maps and chain homotopies, resolutions, and derived functors; the functors Hom and tensor product and their derived functors.

Further topics covered may include the derived category; (co)homology of groups or algebras; simplicial methods; spectral sequences; and other topics of interest.

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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.

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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.

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This subject extends the methods of calculus and linear algebra to study the 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. Topics include: smooth manifolds, tangent spaces, inverse and implicit function theorems; differential forms, integration on manifolds and de Rham cohomology; submersions and fibre bundles; immersions and transversality; examples coming from Lie groups and homogeneous spaces. Additional topics may include: Morse theory; intersection theory; characteristic classes and Chern-Weil theory; the Thom isomorphism; bordism theory.

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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.

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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.

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The theory of Lie algebras is fundamental to the study of groups of continuous symmetries acting on vector spaces, with applications to diverse areas including geometry, number theory and the theory of differential equations. Moreover, since quantum mechanical systems are described by Hilbert spaces acted on by continuous symmetries, Lie algebras and their representations are also fundamental to modern mathematical physics. This subject develops the basic theory in a way accessible to both pure mathematics and mathematical physics students, with an emphasis on examples. The main theorems are: the classification of complex semi-simple Lie algebras in terms of Cartan matrices and Dynkin diagrams, and the classification of finite-dimensional representations of these algebras in terms of highest weight theory.

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This subject offers a wide ranging introduction to the modern theory of partial differential equations (PDEs) in pure mathematics. Thus we will study questions of existence, uniqueness, regularity, and long time behaviour (e.g.\ energy dispersion) for solutions to PDEs. We will discuss these questions first for the classical equations (Laplace's equation, the heat equation, and the wave equation) which will lead us to the broader theory of elliptic, parabolic, and hyperbolic equations. The course covers mostly linear equations, but exposes the student also to some of the most interesting non-linear equations arising in physics and geometry.

Further topics may include: Calculus of variations, Hamilton-Jacobi equations, Systems of Conservation laws; Non-linear elliptic equations, Schauder theory; Quasi-linear hyperbolic equations, propagation of singularities, blow up phenomena.

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This course is an introduction to algebraic number theory. Algebraic number theory studies the structure of the integers and algebraic numbers, combining methods from commutative algebra, complex analysis, and Galois theory. This subject covers the basic theory of number fields, rings of integers and Dedekind domains, zeta functions, decomposition of primes in number fields and ramification, the ideal class group, and local fields. Additional topics may include Dirichlet L-functions and Dirichlet’s theorem; quadratic forms and the theorem of Hasse-Minkowski; local and global class field theory; adeles; and other topics of interest.

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This subject extends notions from calculus, linear algebra and differential equations to study spaces with geometric structures. The concepts introduced are of great importance in mathematics, physics, and all areas in which local properties of spaces are used to model systems.

Topics include: smooth manifolds, vector bundles, multilinear algebra; Frobenius’ theorem, exterior differentiation, Lie differentiation, flows of vector fields; connections and curvature; bilinear forms, metrics, length, volume, Levi-Civita connection; parallel transport, geodesics, holonomy; connections on principal bundles; examples including Lie groups, hyperbolic geometry and homogeneous spaces.

Additional topics may include: second fundamental form and minimal submanifolds; Jacobi fields and applications to topology; constant curvature and Einstein metrics; Hodge star operator, Hodge Laplacian and harmonic forms; Lorentzian geometry and Einstein's equations; Kähler geometry; symplectic geometry; gauge theory.

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This subject explores a range of key concepts in modern Probability Theory that are fundamental for Mathematical Statistics and are widely used in other applications. We study measurable space, product measure, Fubini's theorem, conditional expectation and conditional probability, construction of i.i.d. and beyond, discrete-time martingales.

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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.

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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.

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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, and generalised additive models. Important statistical computing algorithms and techniques used in data science will be explained in detail. These include unsupervised learning of meaningful components, bootstrap resampling and inference, cross-validation, the Expectation-Maximisation (EM) algorithm and variational approximation, and Markov chain Monte Carlo methods including adaptive rejection and squeeze sampling, sequential importance sampling, slice sampling, Gibbs samplers and the Metropolis--Hastings algorithm.

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The subject covers some key aspects of the theory of stochastic processes that plays a 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, functional central limit theorem, Levy processes, renewal processes and Markov processes (discrete and continuous state space). Applications to modelling random phenomena evolving in time are discussed throughout the course.

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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.

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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.

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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.

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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.

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This subject develops the advanced topics and methods of stochastic processes and discusses possible applications of the models covered in the course. It serves to prepare students for research in Probability Theory. The specific content will vary depending on the subject coordinator.

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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, prior choice, comparisons of means and proportions, multi-model inference and model selection. The second part of the subject will cover practical implementations of Bayesian methods through Markov Chain Monte Carlo computing and real data applications, focusing on (generalised) linear models and concluding by exploring machine learning techniques such as Gaussian processes.

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Modern statistics and data science deals with data having multiple dimensions. Multivariate methods are used to handle these types of data. Approaches to supervised and unsupervised learning with multivariate data are discussed. In particular, methods for classification, clustering, and dimension reduction are introduced, which are particularly suited to high-dimensional data. Both parametric and nonparametric approaches are discussed.

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