Mathematics and Statistics

Complex analysis is a core subject in pure and applied mathematics, as well as the physical and engineering sciences. While it is true that physical phenomena are given in terms of real numbers and real variables, it is often too difficult and sometimes not possible, to solve the algebraic and differential equations used to model these phenomena without introducing complex numbers and complex variables and applying the powerful techniques of complex analysis.

Topics include:the topology of the complex plane; convergence of complex sequences and series; holomorphic functions, the Cauchy-Riemann equations, harmonic functions and applications; contour integrals and the Cauchy Integral Theorem; singularities, Laurent series, the Residue Theorem, evaluation of integrals using contour integration, conformal mapping; and aspects of the gamma function.

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This subject provides a basis for further studies in modern analysis, geometry, topology, differential equations and quantum mechanics.It introduces the idea of a metric space with a general distance function, and the resulting concepts of convergence, continuity, completeness, compactness and connectedness. The subject also introduces Hilbert spaces: infinite dimensional vector spaces (typically function spaces) equipped with an inner product that allows geometric ideas to be used to study these spaces and linear maps between them.

Topics include: metric and normed spaces, limits of sequences, open and closed sets, continuity, topological properties, compactness, connectedness; Cauchy sequences, completeness, contraction mapping theorem; Hilbert spaces, orthonormal systems, bounded linear operators and functionals, applications.

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Algebra has a long history of important applications throughout mathematics, science and engineering, and is also studied for its intrinsic beauty. In this subject we study the algebraic laws satisfied by familiar objects such as integers, polynomials and matrices. This abstraction simplifies and unifies our understanding of these structures and enables us to apply our results to interesting new cases. Students will gain further experience with abstract algebraic concepts and methods. General structural results are proved and algorithms developed to determine the invariants they describe.

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Graphs model networks of all types such as telecommunication, transport, computer and social networks. They also model physical structures such as crystals and abstract structures within computer algorithms.

This subject is an introduction to the modern field of graph theory. It emphasises the relationship between proving theorems in mathematics and the construction of algorithms to find the solutions of mathematical problems within the context of graph theory. The subject provides material that supplements other areas of study such as operations research, computer science and discrete mathematics

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This subject is concerned with the study of objects, which are finite in number and typically computable. At a computational level one seeks efficient algorithms and methods for construction and counting of the objects.

The main topics to be covered are: enumeration, permutations, designs, finite geometry, words, Ramsey theory and physical combinatorics. Designs are relevant to statistics, Ramsey theory to computer science, and physical combinatorics to mathematical physics. Words are useful for representing and constructing objects and relating combinatorial objects to algebraic structures.

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This subject introduces three areas of geometry that play a key role in many branches of mathematics and physics. In differential geometry, calculus and the concept of curvature will be used to study the shape of curves and surfaces. In topology, geometric properties that are unchanged by continuous deformations will be studied to find a topological classification of surfaces. In algebraic geometry, curves defined by polynomial equations will be explored. Remarkable connections between these areas will be discussed.

Topics include: Topological classification of surfaces, Euler characteristic, orientability.Introduction to the differential geometry of surfaces in Euclidean space:smooth surfaces, tangent planes, length of curves, Riemannian metrics, Gaussian curvature, minimal surfaces, Gauss-Bonnet theorem.Complex algebraic curves, including conics and cubics, genus.

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Complex analysis is a core subject in pure and applied mathematics, as well as the physical and engineering sciences. While it is true that physical phenomena are given in terms of real numbers and real variables, it is often too difficult and sometimes not possible, to solve the algebraic and differential equations used to model these phenomena without introducing complex numbers and complex variables and applying the powerful techniques of complex analysis.

Topics include:the topology of the complex plane; convergence of complex sequences and series; holomorphic functions, the Cauchy-Riemann equations, harmonic functions and applications; contour integrals and the Cauchy Integral Theorem; singularities, Laurent series, the Residue Theorem, evaluation of integrals using contour integration, conformal mapping; and aspects of the gamma function.

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Most mathematical problems arising from the physical sciences, engineering, life sciences and finance are sufficiently complicated to require computational methods for their solution. This subject introduces students to the process of numerical approximation and computer simulation, applied to simple and commonly encountered stochastic or deterministic models. An emphasis is on the development and implementation of algorithms for the solution of continuous problems including aspects of their efficiency, accuracy and stability. Topics covered will include simple stochastic simulation, direct methods for linear systems, data fitting of linear and nonlinear models, and time-stepping methods for initial value problems.

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Stochastic processes occur in finance as models for asset prices, in telecommunications as models for data traffic, in computational biology as hidden Markov models for gene structure, in chemistry as models for reactions, in manufacturing as models for assembly and inventory processes, in biology as models for the growth and dispersion of plant and animal populations, in speech pathology and speech recognition and many other areas.

This course introduces the theory of stochastic processes including Poisson processes, Markov chains in discrete and continuous time, and renewal processes. These processes are illustrated using examples from real-life situations. It then considers in more detail important applications in areas such as queues and networks (the foundation of telecommunication models), finance, and genetics.

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This subject demonstrates how the mathematical modelling process naturally gives rise to certain classes of ordinary and partial differential equations in many contexts, including the infectious diseases, the flow of traffic and the dynamics of particles and of fluids. It advances the student’s knowledge of the modelling process, as well addressing important mathematical ideas in deterministic modelling and the challenges raised by system nonlinearity.

• Infectious disease models and other contexts leading to systems of autonomous first-order differential equations; initial value problem, phase space, critical points, local linearization and stability; qualitative behaviour of plane autonomous systems, structural stability; formulation, interpretation and critique of models.
• Conservation laws and flux functions leading to first-order quasilinear-linear partial differential equations; characteristics, fans, shocks and applications including modelling traffic flow.
• Introduction to continuum mechanics: basic principles; tensor algebra and tensor calculus; the ideal fluid model and potential flow; the Newtonian fluid, Navier-Stokes equations and simple solutions.

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This subject gives an example-oriented overview of various advanced topics that are important for mathematical physics and physics students, as well as being of interest to students of pure and applied mathematics. These topics include:

• Further differential equations: Bessel functions, Legendre polynomials, spherical harmonics and applications such as the Laplace/Schrodinger equation in polar/spherical coordinates;
• Further vector calculus: Differential forms, integration, Stokes’ theorem and applications such as Maxwell’s equations, charge conservation and Dirac monopoles;
• Hilbert spaces: L2 spaces, bounded and unbounded operators, normalisable and non-normalisable eigenfunctions, distributions and applications to quantum theory;
• Group theory: Lie groups and algebras, representations and applications such as quantum spin and particle physics.

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This subject introduces the concepts of mathematical and computational modelling of biological systems, and how they are applied to data in order to study the underlying drivers of observed behaviour. The subject emphasises the role of abstraction and simplification of biological systems, where required details of the underlying biological mechanisms will be provided. Combined with an introduction to Monte Carlo methods for statistical inference, students will learn how to identify common patterns in the rich and diverse nature of biological phenomena and appreciate how the modelling process leads to new insight into biological phenomena.

• Modelling: Deterministic and stochastic population-level dynamic models; and agent-based computational models will be introduced and studied. Indicative examples will be drawn from health (e.g. infectious diseases, cell tumor growth, developmental biology), ecology (e.g. predator-prey systems, sustainable harvesting, environmental decision making) and biotechnology (e.g. biochemical and metabolic models).
• Simulation: Sampling based methods (i.e. Monte Carlo methods) for parameter estimation and hypothesis testing will be introduced, and their importance in modern computational biology discussed.

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This subject is concerned with the study of objects, which are finite in number and typically computable. At a computational level one seeks efficient algorithms and methods for construction and counting of the objects.

The main topics to be covered are: enumeration, permutations, designs, finite geometry, words, Ramsey theory and physical combinatorics. Designs are relevant to statistics, Ramsey theory to computer science, and physical combinatorics to mathematical physics. Words are useful for representing and constructing objects and relating combinatorial objects to algebraic structures.

View detailed information in the Handbook

Operations Research aims to develop and apply scientific methods to improve decision-making. It deals with theoretical development of models and methods, as well as their practical applications to find optimal or near-optimal solutions to problems arising in various disciplines, such as engineering, management, economics, and education.

This subject introduces some major techniques and algorithms for solving nonlinear optimisation problems. Unconstrained and constrained systems will be considered, for both convex and non-convex problems. The methods covered include: interval search techniques, Newton and quasi-Newton methods, penalty methods for nonlinear programs, and methods based on duality. The emphasis is both on being able to apply and implement the techniques discussed, and on understanding the underlying mathematical principles. Examples involve the formulation of operations research models for linear regression, multi-facility location analysis and network flow optimisation.

A significant part of the subject is the project, where students work in groups on a practical operations research problem.

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Complex analysis is a core subject in pure and applied mathematics, as well as the physical and engineering sciences. While it is true that physical phenomena are given in terms of real numbers and real variables, it is often too difficult and sometimes not possible, to solve the algebraic and differential equations used to model these phenomena without introducing complex numbers and complex variables and applying the powerful techniques of complex analysis.

Topics include:the topology of the complex plane; convergence of complex sequences and series; holomorphic functions, the Cauchy-Riemann equations, harmonic functions and applications; contour integrals and the Cauchy Integral Theorem; singularities, Laurent series, the Residue Theorem, evaluation of integrals using contour integration, conformal mapping; and aspects of the gamma function.

View detailed information in the Handbook

Stochastic processes occur in finance as models for asset prices, in telecommunications as models for data traffic, in computational biology as hidden Markov models for gene structure, in chemistry as models for reactions, in manufacturing as models for assembly and inventory processes, in biology as models for the growth and dispersion of plant and animal populations, in speech pathology and speech recognition and many other areas.

This course introduces the theory of stochastic processes including Poisson processes, Markov chains in discrete and continuous time, and renewal processes. These processes are illustrated using examples from real-life situations. It then considers in more detail important applications in areas such as queues and networks (the foundation of telecommunication models), finance, and genetics.

View detailed information in the Handbook

Graphs model networks of all types such as telecommunication, transport, computer and social networks. They also model physical structures such as crystals and abstract structures within computer algorithms.

This subject is an introduction to the modern field of graph theory. It emphasises the relationship between proving theorems in mathematics and the construction of algorithms to find the solutions of mathematical problems within the context of graph theory. The subject provides material that supplements other areas of study such as operations research, computer science and discrete mathematics

View detailed information in the Handbook

This subject introduces the essential features of decision-making techniques encountered in operations research, management, industry, business and economics. It shows how to construct formal mathematical models for practical decision-making as encountered in two-person games, multi-objective optimisation problems, stochastic decision problems, group decision and social choice, and decision-making under uncertainty. It shows students further uses of linear programming and introduces dynamic programming techniques.

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Linear models are central to the theory and practice of modern statistics. They are used to model a response as a linear combination of explanatory variables and are the most widely used statistical models in practice. Starting with examples from a range of application areas this subject develops an elegant unified theory that includes the estimation of model parameters, quadratic forms, hypothesis testing using analysis of variance, model selection, diagnostics on model assumptions, and prediction. Both full rank models and models that are not of full rank are considered. The theory is illustrated using common models and experimental designs.

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Linear models are central to the theory and practice of modern statistics. They are used to model a response as a linear combination of explanatory variables and are the most widely used statistical models in practice. Starting with examples from a range of application areas this subject develops an elegant unified theory that includes the estimation of model parameters, quadratic forms, hypothesis testing using analysis of variance, model selection, diagnostics on model assumptions, and prediction. Both full rank models and models that are not of full rank are considered. The theory is illustrated using common models and experimental designs.

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Stochastic processes occur in finance as models for asset prices, in telecommunications as models for data traffic, in computational biology as hidden Markov models for gene structure, in chemistry as models for reactions, in manufacturing as models for assembly and inventory processes, in biology as models for the growth and dispersion of plant and animal populations, in speech pathology and speech recognition and many other areas.

This course introduces the theory of stochastic processes including Poisson processes, Markov chains in discrete and continuous time, and renewal processes. These processes are illustrated using examples from real-life situations. It then considers in more detail important applications in areas such as queues and networks (the foundation of telecommunication models), finance, and genetics.

View detailed information in the Handbook

This subject introduces a measured-theoretic approach to probability theory and presents its fundamentals concepts and results.

Topics covered include: probability spaces and random variables, expectation, conditional expectation and distributions, elements of multivariate distribution theory, modes of convergence in probability theory, characteristics functions and their application in key limit theorems.

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Modern applied statistics combines the power of modern computing and theoretical statistics. This subject considers the computational techniques required for the practical implementation of statistical theory, and includes Bayes and Monte-Carlo methods. The subject focuses on the application of these techniques to generalised linear models, which are commonly used in the analysis of categorical data.

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