In our ever-changing world, we are facing a new data-driven era where the capability to efficiently combine and analyse large data collections is essential for informed decision making in business and government, and for scientific research. Data science is an emerging interdisciplinary field with its focus on high performance computation and quantitative expression of the confidence in conclusions, and the clear communication of those conclusions in different discipline context. This unit is our capstone project that presents the opportunity to create a public data product that can illustrate the concepts and skills you have learnt in this discipline. In this unit, you will have an opportunity to explore deeper disciplinary knowledge; while also meeting and collaborating through project-based learning. The capstone project in this unit will allow you to identify and place the data-driven problem into an analytical framework, solve the problem through computational means, interpret the results and communicate your findings to a diverse audience. All such skills are highly valued by employers. This unit will foster the ability to work in an interdisciplinary team, to translate problem between two or more disciplines and this is essential for both professional and research pathways in the future.

Mathematics and statistics are powerful tools in finance and more generally in the world at large. To really experience the power of mathematics and statistics at work, students need to identify and explore interdisciplinary links. Engagement with other disciplines also provides essential foundational skills for using mathematical and statistical ideas in financial contexts and in the world beyond. In this unit you will commence by working on a group project in an area of financial mathematics or statistics. From this project you will acquire skills of teamwork, research, wring and project management as well as disciplinary knowledge. You will then have the opportunity to apply your disciplinary knowledge in an interdisciplinary team to identify and solve problems and communicate your findings.

The aim of the unit is to expand visual/geometric ways of thinking. The Geometry section is concerned mainly with transformations of the Euclidean plane (that is, bijections from the plane to itself), with a focus on the study of isometries (proving the classification theorem for transformations which preserve distances between points), symmetries (including the classification of frieze groups) and affine transformations (transformations which map lines to lines). The basic approach is via vectors and matrices, emphasising the interplay between geometry and linear algebra. The study of affine transformations is then extended to the study of collineations in the real projective plane, including collineations which map conics to conics. The Topology section considers graphs, surfaces and knots from a combinatorial point of view. Key ideas such as homeomorphism, subdivision, cutting and pasting and the Euler invariant are introduced first for graphs (1-dimensional objects) and then for triangulated surfaces (2-dimensional objects). Topics include the classification of surfaces, map colouring, decomposition of knots and knot invariants.

This unit of study unifies and extends mathematical ideas and techniques that most participants will have met in their first and second years, and will be of general interest to all students of pure and applied mathematics. It combines algebra and logic to present and answer a number of related questions of fundamental importance in the development of mathematics, from ancient to modern times.
The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem the undecidability of First Order Logic, and a discussion of Godel's Incompleteness Theorem.
Classical and novel arithmetics are introduced, unified and described abstractly using field and ring axioms and the language of field extensions. Quotient rings are introduced, which are used to construct different finite and infinite fields. A construction of the real numbers, by factoring out rings of Cauchy sequences of rationals by the ideal of null sequences, is presented. Axiomatics are placed in the context of reasoning within first order logic and set theory.

Mathematics is ubiquitous in the modern world. Mathematical ideas contribute to philosophy, art, music, economics, business, science, history, medicine and engineering. To really see the power and beauty of mathematics at work, students need to identify and explore interdisciplinary links. Engagement with other disciplines also provides essential foundational skills for using mathematics in the world beyond the lecture room. In this unit you will commence by working on a group project in an area of mathematics that interests you. From this you will acquire skills of teamwork, research, writing and project management as well as disciplinary knowledge. You will then have the opportunity to apply your disciplinary knowledge in an interdisciplinary team to identify and solve problems and communicate your findings to a diverse audience.

This unit will introduce you to the mathematical theory of modern finance with the special emphasis on the valuation and hedging of financial derivatives, such as: forward contracts and options of European and American style. You will learn about the concept of arbitrage and how to model risk-free and risky securities. Topics covered by this unit include: the notions of a martingale and a martingale measure, the fundamental theorems of asset pricing, complete and incomplete markets, the binomial options pricing model, discrete random walks and the Brownian motion, the Black-Scholes options pricing model and the valuation and heding of exotic options. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills. Students enrolled in this unit at advanced level will have to undertake more challenging assessment tasks, but lectures in the advanced level are held concurrently with those of the corresponding mainstream unit.

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. After setting up basic elements of stochastic processes, such as time, state, increments, stationarity and Markovian property, this unit develops important properties and limit theorems of discrete-time Markov chain and branching processes. You will then establish key results for the Poisson process and continuous-time Markov chains, such as the memoryless property, super positioning, thinning, Kolmogorov's equations and limiting probabilities. Various illustrative examples are provided throughout the unit to demonstrate how stochastic processes can be applied in modeling and analyzing problems of practical interest. By completing this unit, you will develop the essential basis for further studies, such as stochastic calculus, stochastic differential equations, stochastic control and financial mathematics.

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. After setting up basic elements of stochastic processes, such as time, state, increments, stationarity and Markovian property, this unit develops basic properties and limit theory of discrete-time Markov chains and branching processes. You will then establish key results for the Poisson process and continuous-time Markov chains, stopping times and martingales. Various illustrative examples are provided throughout the unit to demonstrate how stochastic processes can be applied in modelling and analysing problems of practical interest. By completing this unit, you will develop the essential basis for further studies, such as stochastic calculus, stochastic differential equations, stochastic control and financial mathematics. Students who undertake the advanced unit MATH3921 will be expected to have a deeper, more sophisticated understanding of the theory in the unit and to be able to work with more complicated applications than students who complete the regular MATH3021 unit.

Data Science is an emerging and inherently interdisciplinary field. A key set of skills in this area fall under the umbrella of Statistical Machine Learning methods. This unit presents the opportunity to bring together the concepts and skills you have learnt from a Statistics or Data Science major, and apply them to a joint project with NUTM3888 where Statistics and Data Science students will form teams with Nutrition students to solve a real world problem using Statistical Machine Learning methods. The unit will cover a wide breadth of cutting edge supervised and unsupervised learning methods will be covered including principal component analysis, multivariate tests, discrimination analysis, Gaussian graphical models, log-linear models, classification trees, k-nearest neighbors, k-means clustering, hierarchical clustering, and logistic regression. In this unit, you will continue to understand and explore disciplinary knowledge, while also meeting and collaborating through project-based learning; identifying and solving problems, analysing data and communicating your findings to a diverse audience. All such skills are highly valued by employers. This unit will foster the ability to work in an interdisciplinary team, and this is essential for both professional and research pathways in the future.

This unit will introduce the fundamental concepts of analysis of data from both observational studies and experimental designs using classical linear methods, together with concepts of collection of data and design of experiments. You will first consider linear models and regression methods with diagnostics for checking appropriateness of models, looking briefly at robust regression methods. Then you will consider the design and analysis of experiments considering notions of replication, randomization and ideas of factorial designs. Throughout the course you will use the R statistical package to give analyses and graphical displays. This unit is essentially an Advanced version of STAT3012, with additional emphasis on the mathematical techniques underlying applied linear models together with proofs of distribution theory based on vector space methods.

In today's data-rich world more and more people from diverse fields are needing to perform statistical analyses and indeed more and more tools for doing so are becoming available; it is relatively easy to point and click and obtain some statistical analysis of your data. But how do you know if any particular analysis is indeed appropriate? Is there another procedure or workflow which would be more suitable? Is there such thing as a best possible approach in a given situation? All of these questions (and more) are addressed in this unit. You will study the foundational core of modern statistical inference, including classical and cutting-edge theory and methods of mathematical statistics with a particular focus on various notions of optimality. The first part of the unit covers various aspects of distribution theory which are necessary for the second part which deals with optimal procedures in estimation and testing. The framework of statistical decision theory is used to unify many of the concepts. You will rigorously prove key results and apply these to real-world problems in laboratory sessions. By completing this unit you will develop the necessary skills to confidently choose the best statistical analysis to use in many situations.

At the end of this unit you will have received a broad introduction and gained a variety of tools to apply them within your further mathematical studies and/or in other disciplines.

This unit of study lies at the heart of modern algebra. In the unit we investigate the mathematical theory that was originally developed for the purpose of studying polynomial equations. In a nutshell, the philosophy is that it should be possible to completely factorise any polynomial into a product of linear factors by working over a large enough field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory. The basic theoretical tool needed for this program is the concept of a ring, which generalises the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions. Along the way you will see some beautiful gems of mathematics, including Fermat's Theorem on primes expressible as a sum of two squares, solutions to the ancient Greek problems of trisecting the angle, squaring the circle, and doubling the cube, and the crown of the course: Galois' proof that there is no analogue of the quadratic formula for the general quintic equation. On completing this unit of study you will have obtained a deep understanding of modern abstract algebra.

The theory of ordinary differential equations is a classical topic going back to Newton and Leibniz. It comprises a vast number of ideas and methods. The theory has many applications and stimulates new developments in almost all areas of mathematics. The emphasis is on qualitative analysis including phase-plane methods, bifurcation theory and the study of limit cycles. The more theoretical part includes existence and uniqueness theorems, linearisation, and analysis of asymptotic behaviour. The applications in this unit will be drawn from predator-prey systems, population models, chemical reactions, and other equations and systems from mathematical biology. You will learn how to use ordinary differential equations to model biological, chemical, physical and/or economic systems and how to use different methods from dynamical systems theory and the theory of nonlinear ordinary differential equations to find the qualitative outcome of the models. By doing this unit you will develop skills in using and analyzing nonlinear differential equations which will prepare you for further studies in mathematics, systems biology or physics or for careers in mathematical modelling.

This unit is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, we develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. For students, this provides the first taste of the investigation on the deep relation between geometry and topology of mathematical objects, highlighted in the classic Gauss-Bonnet Theorem. Differential geometry also plays an important part in both classical and modern theoretical physics. The unit aims to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface. A second aim is to remind the students about all the content covered in the mathematical units for previous years, most importantly the key ideas in vector calculus, along with some applications. It also helps to prepare the students for honours courses like Riemannian Geometry. By doing this unit you will further appreciate the beauty of mathematics which originated from the need to solve practical problems, develop skills in understanding the geometry of the surrounding environment, prepare yourself for future study or the workplace by developing advanced critical thinking skills and gain a deep understanding of the underlying rules of the Universe.

Measure theory is the study of fundamental ideas as length, area, volume, arc length and surface area. It is the basis for Lebesgue integration theory used in advanced mathematics ever since its development in about 1900. Measure theory is also a key foundation for modern probability theory. The course starts by establishing the basics of measure theory and the theory of Lebesgue integration, including important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. These ideas are applied to Fourier Analysis which leads to results such as the Inversion Formula and Plancherel's Theorem. The Radon-Nikodyn Theorem provides a representation of measures in terms of a density. Key ideas of this theory are applied in detail to probability theory to provide a rigorous framework for probability which takes in and generalizes familiar ideas such as distributions and conditional expectation. When you complete this unit you will have acquired a new generalized way of thinking about key mathematical concepts such as length, area, integration and probability. This will give you a powerful set of intellectual tools and equip you for further study in mathematics and probability.

The questions how to maximise your gain (or to minimise the cost) and how to determine the optimal strategy/policy are fundamental for an engineer, an economist, a doctor designing a cancer therapy, or a government planning some social policies. Many problems in mechanics, physics, neuroscience and biology can be formulated as optimisation problems. Therefore, optimisation theory is an indispensable tool for an applied mathematician. Optimisation theory has many diverse applications and requires a wide range of tools but there are only a few ideas underpinning all this diversity of methods and applications. This course will focus on two of them. We will learn how the concept of convexity and the concept of dynamic programming provide a unified approach to a large number of seemingly unrelated problems. By completing this unit you will learn how to formulate optimisation problems that arise in science, economics and engineering and to use the concepts of convexity and the dynamic programming principle to solve straightforward examples of such problems. You will also learn about important classes of optimisation problems arising in finance, economics, engineering and insurance.

Fluid Dynamics is the study of systems which allow for a macroscopic description in some continuum limit. It is not limited to the study of liquids such as water but includes our atmosphere and even car traffic. Whether a system can be treated as a fluid, depends on the spatial scales involved. Fluid dynamics presents a cornerstone of applied mathematics and comprises a whole gamut of different mathematical techniques, depending on the question we ask of the system under consideration. The course will discuss applications from engineering, physics and mathematics: How and in what situations a system which is not necessarily liquid can be described as a fluid? The link between an Eulerian description of a fluid and a Lagrangian description of a fluid, the basic variables used to describe flows, the need for continuity, momentum and energy equations, simple forms of these equations, geometric and physical simplifying assumptions, streamlines and stream functions, incompressibility and irrotationality and simple examples of irrotational flows. By the end of this unit, students will have received a basic understanding into fluid mechanics and have acquired general methodology which they can apply in their further studies in mathematics and/or in their chosen discipline.

Sophisticated mathematics and numerical programming underlie many computer applications, including weather forecasting, computer security, video games, and computer aided design. This unit of study provides a strong foundational introduction to modern interactive programming, computational algorithms, and numerical analysis. Topics covered include: (I) basics ingredients of programming languages such as syntax, data structures, control structures, memory management and visualisation; (II) basic algorithmic concepts including binary and decimal representations, iteration, linear operations, sources of error, divide-and-concur, algorithmic complexity; and (III) basic numerical schemes for rootfinding, integration/differentiation, differential equations, fast Fourier transforms, Monte Carlo methods, data fitting, discrete and continuous optimisation. You will also learn about the philosophical underpinning of computational mathematics including the emergence of complex behaviour from simple rules, undecidability, modelling the physical world, and the joys of experimental mathematics. When you complete this unit you will have a clear and comprehensive understanding of the building blocks of modern computational methods and the ability to start combining them together in different ways. Mathematics and computing are like cooking. Fundamentally, all you have is sugar, fat, salt, heat, stirring, chopping. But becoming a good chef requires knowing just how to put things together in creative ways that work. In previous study, you should have learned to cook. Now you're going to learn how to make something someone else might want to pay for more than one time.

Lagrangian and Hamiltonian dynamics are a reformulation of classical Newtonian mechanics into a mathematically sophisticated framework that can be applied in many different coordinate systems. This formulation generalises elegantly to modern theories of relativity and quantum mechanics. The unit develops dynamics from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to make apparently complicated dynamical problems appear simpler. In this unit you will also explore connections between geometry and different physical theories beyond classical mechanics. You will be expected to solve fully dynamical systems of some complexity including planetary motion and to investigate stability using perturbation analysis. You will use Hamilton-Jacobi theory to solve problems ranging from geodesic motion (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes. This unit is a useful preparation for units in dynamical systems and chaos, and complements units in differential equations, quantum theory and general relativity.

The aim of this unit is to introduce some fundamental concepts of the theory of partial differential equations (PDEs) arising in Physics, Chemistry, Biology and Mathematical Finance. The focus is mainly on linear equations but some important examples of nonlinear equations and related phenomena re introduced as well. After an introductory lecture, we proceed with first-order PDEs and the method of characteristics. Here, we also nonlinear transport equations and shock waves are discussed. Then the theory of the elliptic equations is presented with an emphasis on eigenvalue problems and their application to solve parabolic and hyperbolic initial boundary-value problems. The Maximum principle and Harnack's inequality will be discussed and the theory of Green's functions.

The unit will begin with a revision of properties of complex numbers and complex functions. This will be followed by material on conformal mappings, Riemann surfaces, complex integration, entire and analytic functions, the Riemann mapping theorem, analytic continuation, and Gamma and Zeta functions. Finally, special topics chosen by the lecturer will be presented, which may include elliptic functions, normal families, Julia sets, functions of several complex variables, or complex manifolds.

The aim of Financial Mathematics is to establish a theoretical background for building models of securities markets and provides computational techniques for pricing financial derivatives and risk assessment and mitigation. Specialists in Financial Mathematics are widely sought after by major investment banks, hedge funds and other, government and private, financial institutions worldwide. This course is foundational for honours and masters programs in Financial Mathematics. Its aim is to introduce the basic concepts and problems of securities markets and to develop theoretical frameworks and computational tools for pricing financial products and hedging the risk associated with them. This unit will focus on two ideas that are fundamental for Financial Mathematics. You will learn how the concept of arbitrage and the concept of martingale measure provide a unified approach to a large variety of seemingly unrelated problems arising in practice. You will also learn how to use the wide range of tools required by Financial Mathematics, including stochastic calculus, partial differential equations, optimisation and statistics. By doing this unit, you will learn how to formulate problems that arise in finance as mathematical problems and how to solve them using the concepts of arbitrage and martingale measure. You will also learn how to choose an appropriate computational method and devise explicit numerical algorithms useful for a practitioner.

Capturing random phenomena is a challenging problem in many disciplines from biology, chemistry and physics through engineering to economics and finance. There is a wide spectrum of problems in these fields, which are described using random processes that evolve with time. Hence it is of crucial importance that applied mathematicians are equipped with tools used to analyse and quantify random phenomena. This unit will introduce an important class of stochastic processes, using the theory of martingales. You will study concepts such as the Ito stochastic integral with respect to a continuous martingale and related stochastic differential equations. Special attention will be given to the classical notion of the Brownian motion, which is the most celebrated and widely used example of a continuous martingale. By completing this unit, you will learn how to rigorously describe and tackle the evolution of random phenomena using continuous time stochastic processes. You will also gain a deep knowledge about stochastic integration, which is an indispensable tool to study problems arising, for example, in Financial Mathematics.

Securities and derivatives are the foundation of modern financial markets. The fixed-income market, for example, is the dominant sector of the global financial market where various interest-rate linked securities are traded, such as zero-coupon and coupon bonds, interest rate swaps and swaptions. This unit will investigate short-term interest rate models, the Heath-Jarrow-Morton approach to instantaneous forward rates and recently developed models of forward London Interbank Offered Rates (LIBORs) and forward swap rates. You will learn about pricing and hedging of credit derivatives, another challenging and practically important problem and become familiar with stochastic models for credit events, dependent default times and credit ratings. You will learn how to value and hedge single-name and multi-name credit derivatives such as vulnerable options, corporate bonds, credit default swaps and collateralized debt obligations. You will also learn about the most recent developments in Financial Mathematics, such as robust pricing and nonlinear evaluations. By doing this unit, you will get a solid grasp of mathematical tools used in valuation and hedging of fixed income securities, develop a broad knowledge of advanced quantitative methods related to interest rates and credit risk and you will learn to use powerful mathematical tools to address important real-world quantitative problems in the finance industry.

One of the most important aims of algebraic topology is to distinguish or classify topological spaces and maps between them up to homeomorphism. Invariants and obstructions are key to achieve this aim. A familiar invariant is the Euler characteristic of a topological space, which was initially discovered via combinatorial methods and has been rediscovered in many different guises. Modern algebraic topology allows the solution of complicated geometric problems with algebraic methods. Imagine a closed loop of string that looks knotted in space. How would you tell if you can wiggle it about to form an unknotted loop without cutting the string? The space of all deformations of the loop is an intractable set. The key idea is to associate algebraic structures, such as groups or vector spaces, with topological objects such as knots, in such a way that complicated topological questions can be phrased as simpler questions about the algebraic structures. In particular, this turns questions about an intractable set into a conceptual or finite, computational framework that allows us to answer these questions with certainty. In this unit you will learn about fundamental group and covering spaces, homology and cohomology theory. These form the basis for applications in other domains within mathematics and other disciplines, such as physics or biology. At the end of this unit you will have a broad and coherent knowledge of Algebraic Topology, and you will have developed the skills to determine whether seemingly intractable problems can be solved with topological methods.

Commutative Algebra provides the foundation to study modern uses of Algebra in a wide array of settings, from within Mathematics and beyond. The techniques of Commutative Algebra underpin some of the most important advances of mathematics in the last century, most notably in Algebraic Geometry and Algebraic Topology. This unit will teach students the core ideas, theorems, and techniques from Commutative Algebra, and provide examples of their basic applications. Topics covered include affine varieties, Noetherian rings, Hilbert basis theorem, localisation, the Nullstellansatz, ring specta, homological algebra, and dimension theory. Applications may include topics in scheme theory, intersection theory, and algebraic number theory. On completion of this unit students will be thoroughly prepared to undertake further study in algebraic geometry, algebraic number theory, and other areas of mathematics. Students will also gain facility with important examples of abstract ideas with far-reaching consequences.

Functional analysis is one of the major areas of modern mathematics. It can be thought of as an infinite-dimensional generalisation of linear algebra and involves the study of various properties of linear continuous transformations on normed infinite-dimensional spaces. Functional analysis plays a fundamental role in the theory of differential equations, particularly partial differential equations, representation theory, and probability. In this unit you will cover topics that include normed vector spaces, completions and Banach spaces; linear operators and operator norms; Hilbert spaces and the Stone-Weierstrass theorem; uniform boundedness and the open mapping theorem; dual spaces and the Hahn-Banach theorem; and spectral theory of compact self-adjoint operators. A thorough mechanistic grounding in these topics will lead to the development of your compositional skills in the formulation of solutions to multifaceted problems. By completing this unit you will become proficient in using a set of standard tools that are foundational in modern mathematics and will be equipped to proceed to research projects in PDEs, applied dynamics, representation theory, probability, and ergodic theory.

Representation theory is the abstract study of the possible types of symmetry in all dimensions. It is a fundamental area of algebra with applications throughout mathematics and physics: the methods of representation theory lead to conceptual and practical simplification of any problem in linear algebra where symmetry is present. This unit will introduce you to the basic notions of modules over associative algebras and representations of groups, and the ways in which these objects can be classified. You will learn the special properties that distinguish the representation theory of finite groups over the complex numbers, and also the unifying principles which are common to the representation theory of a wider range of algebraic structures. By learning the key concepts of representation theory you will also start to appreciate the power of category-theoretic approaches to mathematics. The mental framework you will acquire from this unit of study will enable you both to solve computational problems in linear algebra and to create new mathematical theory.

Variational and spectral methods are foundational in mathematical models that govern the configurations of many physical systems. They have wide-ranging applications in areas such as physics, engineering, economics, differential geometry, optimal control and numerical analysis. In addition they provide the framework for many important questions in modern geometric analysis. This unit will introduce you to a suite of methods and techniques that have been developed to handle these problems. You will learn the important theoretical advances, along with their applications to areas of contemporary research. Special emphasis will be placed on Sobolev spaces and their embedding theorems, which lie at the heart of the modern theory of partial differential equations. Besides engaging with functional analytic methods such as energy methods on Hilbert spaces, you will also develop a broad knowledge of other variational and spectral approaches. These will be selected from areas such as phase space methods, minimax theorems, the Mountain Pass theorem or other tools in the critical point theory. This unit will equip you with a powerful arsenal of methods applicable to many linear and nonlinear problems, setting a strong foundation for understanding the equilibrium or steady state solutions for fundamental models of applied mathematics.

Computational mathematics fulfils two distinct purposes within Mathematics. On the one hand the computer is a mathematician's laboratory in which to model problems too hard for analytical treatment and to test existing theories; on the other hand, computational needs both require and inspire the development of new mathematics. Computational methods are an essential part of the tool box of any mathematician. This unit will introduce you to a suite of computational methods and highlight the fruitful interplay between analytical understanding and computational practice. In particular, you will learn both the theory and use of numerical methods to simulate partial differential equations, how numerical schemes determine the stability of your method and how to assure stability when simulating Hamiltonian systems, how to simulate stochastic differential equations, as well as modern approaches to distilling relevant information from data using machine learning. By doing this unit you will develop a broad knowledge of advanced methods and techniques in computational applied mathematics and know how to use these in practice. This will provide a strong foundation for research or further study.

Mathematical approaches to many real-world problems are underpinned by powerful and wide ranging mathematical methods and techniques that have become standard in the field and should be in the toolbag of all applied mathematicians. This unit will introduce you to a suite of those methods and give you the opportunity to engage with applications of these methods to well-known problems. In particular, you will learn both the theory and use of asymptotic methods which are ubiquitous in applications requiring differential equations or other continuous models. You will also engage with methods for probabilistic models including information theory and stochastic models. By doing this unit you will develop a broad knowledge of advanced methods and techniques in applied mathematics and know how to use these in practice. This will provide a strong foundation for using mathematics in a broad sweep of practical applications in research, in industry or in further study.

Applied Mathematics harnesses the power of mathematics to give insight into phenomena in the wider world and to solve practical problems. Modelling is the key process that translates a scientific or other phenomenon into a mathematical framework through applying suitable assumptions, identifying important variables and deriving a well-defined mathematical problem. Mathematicians then use this model to explore the real-world phenomenon, including making predictions. Good mathematical modelling is something of an art and is best learnt by example and by writing, refining and analysing your own models. This unit will introduce you to some classic mathematical models and give you the opportunity to analyse, explore and extend these models to make predictions and gain insights into the underlying phenomena. You will also engage with modelling in depth in at least one area of application. By doing this unit you will develop a broad knowledge of advanced mathematical modelling methods and techniques and know how to use these in practice. This will provide a strong foundation for applying mathematics and modelling to many diverse applications and for research or further study.

In applied mathematics, dynamical systems are systems whose state is changing with time. Examples include the motion of a pendulum, the change in the population of insects in a field or fluid flow in a river. These systems are typically represented mathematically by differential equations or difference equations. Dynamical systems theory reveals universal mechanisms behind disparate natural phenomena. This area of mathematics brings together sophisticated theory from many areas of pure and applied mathematics to create powerful methods that are used to understand and control the dynamical building blocks which make up physical, biological, chemical, engineered and even sociological systems. By doing this unit you will develop a broad knowledge of methods and techniques in dynamical systems, and know how to use these to analyse systems in nature and in technology. This will provide a strong foundation for using mathematics in a broad sweep of applications and for research or further study.

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. In this unit you will rigorously establish the basic properties and limit theory of discrete-time Markov chains and branching processes and then, building on this foundation, derive key results for the Poisson process and continuous-time Markov chains, stopping times and martingales. You will learn about various illustrative examples throughout the unit to demonstrate how stochastic processes can be applied in modeling and analysing problems of practical interest, such as queuing, inventory, population, financial asset price dynamics and image processing. By completing this unit, you will develop a solid mathematical foundation in stochastic processes which will become the platform for further studies in advanced areas such as stochastic analysis, stochastic differential equations, stochastic control and financial mathematics.

Classical linear models are widely used in science, business, economics and technology. This unit will introduce the fundamental concepts of analysis of data from both observational studies and experimental designs using linear methods, together with concepts of collection of data and design of experiments. You will first consider linear models and regression methods with diagnostics for checking appropriateness of models, looking briefly at robust regression methods. Then you will consider the design and analysis of experiments considering notions of replication, randomization and ideas of factorial designs. Throughout the course you will use the R statistical package to give analyses and graphical displays. This unit includes material in STAT3022, but has an additional component on the mathematical techniques underlying applied linear models together with proofs of distribution theory based on vector space methods.

In today's data-rich world, more and more people from diverse fields need to perform statistical analyses, and indeed there are more and more tools to do this becoming available. It is relatively easy to "point and click" and obtain some statistical analysis of your data. But how do you know if any particular analysis is indeed appropriate? Is there another procedure or workflow which would be more suitable? Is there such a thing as a "best possible" approach in a given situation? All of these questions (and more) are addressed in this unit. You will study the foundational core of modern statistical inference, including classical and cutting-edge theory and methods of mathematical statistics with a particular focus on various notions of optimality. The first part of the unit covers aspects of distribution theory which are applied in the second part which deals with optimal procedures in estimation and testing. The framework of statistical decision theory is used to unify many of the concepts that are introduced in this unit. You will rigorously prove key results and apply these to real-world problems in laboratory sessions. By completing this unit, you will develop the necessary skills to confidently choose the best statistical analysis to use in many situations.

This unit will study basic concepts and methods of time series analysis applicable in many real world problems in numerous fields, including economics, finance, insurance, physics, ecology, chemistry, computer science and engineering. This unit will investigate the basic methods of modelling and analyzing of time series data (ie. data containing serially dependence structure). This can be achieved through learning standard time series procedures on identification of components, autocorrelations, partial autocorrelations and their sampling properties. After setting up these basics, students will learn the theory of stationary univariate time series models including ARMA, ARIMA and SARIMA and their properties. Then the identification, estimation, diagnostic model checking, decision making and forecasting methods based on these models will be developed with applications. The spectral theory of time series, estimation of spectra using periodogram and consistent estimation of spectra using lag-windows will be studied in detail. Further, the methods of analyzing long memory and time series and heteroscedastic time series models including ARCH, GARCH, ACD, SCD and SV models from financial econometrics and the analysis of vector ARIMA models will be developed with applications. By completing this unit, students will develop the essential basis for further studies, such as financial econometrics and financial time series. The skills gained through this unit of study will form a strong foundation to work in a financial industry or in a related research organization.

In our ever-changing world, we are facing a new data-driven era where the capability to efficiently combine and analyse large data collections is essential for informed decision making in business and government, and for scientific research. Statistics and data analytics consulting provide an important framework for many individuals to seek assistance with statistics and data-driven problems. This unit of study will provide students with an opportunity to gain real-life experience in statistical consulting or work with collaborative (interdisciplinary) research. In this unit, you will have an opportunity to have practical experience in a consultation setting with real clients. You will also apply your statistical knowledge in a diverse collection of consulting projects while learning project and time management skills. In this unit you will need to identify and place the client's problem into an analytical framework, provide a solution within a given time frame and communicate your findings back to the client. All such skills are highly valued by employers. This unit will foster the expertise needed to work in a statistical consulting firm or data analytical team which will be essential for data-driven professional and research pathways in the future.

Applied Statistics fundamentally brings statistical learning to the wider world. Some data sets are complex due to the nature of their responses or predictors or have high dimensionality. These types of data pose theoretical, methodological and computational challenges that require knowledge of advanced modelling techniques, estimation methodologies and model selection skills. In this unit you will investigate contemporary model building, estimation and selection approaches for linear and generalised linear regression models. You will learn about two scenarios in model building: when an extensive search of the model space is possible; and when the dimension is large and either stepwise algorithms or regularisation techniques have to be employed to identify good models. These particular data analysis skills have been foundational in developing modern ideas about science, medicine, economics and society and in the development of new technology and should be in the toolkit of all applied statisticians. This unit will provide you with a strong foundation of critical thinking about statistical modelling and technology and give you the opportunity to engage with applications of these methods across a wide scope of applications and for research or further study.

Probability Theory lays the theoretical foundations that underpin the models we use when analysing phenomena that involve chance. This unit introduces the students to modern probability theory and applies it to problems in mathematical statistics. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that bring together both the theory of discrete random variables and the theory of continuous random variables that were introduce to earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and also verify important results in Mathematical Statistics. These involve exponential families, efficient estimation, large-sample testing and Bayesian methods. Finally you will verify important convergence properties of the expectation-maximisation (EM) algorithm. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability or mathematical statistics.

Probability Theory lays the theoretical foundations that underpin the models we use when analysing phenomena that involve chance. This unit introduces the students to modern probability theory (based on measure theory) that was developed by Andrey Kolmogorov. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that brings together both the theory of discrete random variables and the theory of continuous random variables that were introduced earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, conditional expectation, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and to thoroughly study discrete-time martingales. Originally used to model betting strategies, martingales are a powerful generalisation of random walks that allow us to prove fundamental results such as the Strong Law of Large Numbers or analyse problems such as the gambler's ruin. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability, stochastic systems or financial mathematics.

In our interconnected world, networks are an increasingly important representation of datasets and systems. This unit will investigate how this network approach to problems can be pursued through the combination of mathematical models and datasets. You will learn different mathematical models of networks and understand how these models explain non-intuitive phenomena, such as the small world phenomenon (short paths between nodes despite clustering), the friendship paradox (our friends typically have more friends than we have), and the sudden appearance of epidemic-like processes spreading through networks. You will learn computational techniques needed to infer information about the mathematical models from data and, finally, you will learn how to combine mathematical models, computational techniques, and real-world data to draw conclusions about problems. More generally, network data is a paradigm for high-dimensional interdependent data, the typical problem in data science. By doing this unit you will develop computational and mathematical skills of wide applicability in studies of networks, data science, complex systems, and statistical physics.

With explosions in availability of computing power and facilities for gathering data in recent times, a key skill of any graduate is the ability to work with increasingly complex datasets. There may include, for example, data sets with multiple levels of observations gathered from diverse sources using a variety of methods. Being able to apply computational skills to implement appropriate software, as well as bringing to bear statistical expertise in the design of the accompanying algorithms are both vital when facing the challenge of analysing complicated data. This unit is made up of three distinct modules, each focusing on a different aspect of applications of statistical methods to complex data. These include (but are not restricted to) the development of a data product that interrogate large and complicated data structures; using sophisticated statistical methods to improve computational efficiency for large data sets or computationally intensive statistical methods; and the analysis of categorical ordinal data. Across all modules you will develop expertise in areas of statistical methodology, statistical analysis as well as computational statistics. Additional modules may be delivered, depending on the areas of expertise of available staff and distinguished visitors.

Increased computing power has meant that many Bayesian methods can now be easily implemented and provide solutions to problems that have previously been intractable. Bayesian methods allow researchers to incorporate prior knowledge into their statistical models. This unit is made up of three distinct modules, each focusing on a different niche in the application of Bayesian statistical methods to complex data in, for example, geophysics, ecology and hydrology. These include (but are not restricted to) Bayesian methods and models; statistical inversion; approximate Bayesian inference for semiparametric regression. Across all modules you will develop expertise in Bayesian computational statistics. On completion of this unit you will be able to apply appropriate Bayesian methods to a variety of applications in science, and other data-heavy disciplines to develop a better understanding of the information inherent in complex datasets.

Algebra is one of the broadest fields of mathematics, underlying most aspects of mathematics. It is sometimes considered "the mathematics of symmetry" or the "language of mathematics". In its most general description, algebra includes number theory, algebraic geometry and the classical study of algebraic structures such as rings and groups as well as their representations. Advanced algebra intersects other fields of modern mathematics, for instance via algebraic topology, homological algebra and categorical representation theory; and modern physics, via Lie groups and Lie algebras. You will learn about fundamental concepts of a branch of advanced algebra and its role in modern mathematics and its applications. You will develop problem-solving skills using algebraic techniques applied to diverse situations. Learning an area of pure mathematics means building a mental framework of theoretical concepts, stocking that framework with plentiful examples with which to develop an intuition of what statements are likely to be true, testing the framework with specific calculations, and finally gaining the deep understanding required to create technically sophisticated proofs of general results. The selection of topics is guided by their relevance for current research. Having gained an abstract understanding of symmetry, you will discover the manifestation of algebraic structures everywhere!

Algebra is one of the broadest fields of mathematics, underlying most aspects of mathematics. It is sometimes considered "the mathematics of symmetry" or the "language of mathematics". In its most general description, algebra includes number theory, algebraic geometry and the classical study of algebraic structures such as rings and groups as well as their representations. Advanced algebra intersects other fields of modern mathematics, for instance via algebraic topology, homological algebra and categorical representation theory; and modern physics, via Lie groups and Lie algebras. You will learn about fundamental concepts of a branch of advanced algebra and its role in modern mathematics and its applications. You will develop problem-solving skills using algebraic techniques applied to diverse situations. Learning an area of pure mathematics means building a mental framework of theoretical concepts, stocking that framework with plentiful examples with which to develop an intuition of what statements are likely to be true, testing the framework with specific calculations, and finally gaining the deep understanding required to create technically sophisticated proofs of general results. The selection of topics is guided by their relevance for current research. Having gained an abstract understanding of symmetry, you will discover the manifestation of algebraic structures everywhere!

Differential equations and the notion of convergence are ubiquitous within the natural sciences, engineering and mathematics. Analysis has, thus, far reaching applications, and it is a major discipline in its own right. The origins of many major areas such as topology, functional and harmonic analysis have their roots in real and complex analysis. Analysis makes unexpected appearances in other areas such as number theory, where it played a key role in a recent breakthrough on arithmetic progression of prime numbers by Fields medalist Terrence Tao. Analysis deals with any kind of limit process, notions of distance, measure, continuity or differentiability. It makes up a crucial part of diverse areas in mathematics. The fields of application of analysis that you will encounter in this unit may include partial differential equations, differential geometry, harmonic analysis, topological groups, optimal control, scattering theory, ergodic theory, differential topology or mathematical physics. The selection of topics in this unit is guided by their relevance for applications and current research. In this unit, you will gain an understanding of the systematic, abstract foundations of a branch of analysis and develop tools needed to get to the present frontiers.

Differential equations and the notion of convergence are ubiquitous within the natural sciences, engineering and mathematics. Analysis has, thus, far reaching applications, and it is a major discipline in its own right. The origins of many major areas such as topology, functional and harmonic analysis have their roots in real and complex analysis. Analysis makes unexpected appearances in other areas such as number theory, where it played a key role in a recent breakthrough on arithmetic progression of prime numbers by Fields medalist Terrence Tao. Analysis deals with any kind of limit process, notions of distance, measure, continuity or differentiability. It makes up a crucial part of diverse areas in mathematics. The fields of application of analysis that you will encounter in this unit may include partial differential equations, differential geometry, harmonic analysis, topological groups, optimal control, scattering theory, ergodic theory, differential topology or mathematical physics. The selection of topics in this unit is guided by their relevance for applications and current research. In this unit, you will gain an understanding of the systematic, abstract foundations of a branch of analysis and develop tools needed to get to the present frontiers.

Geometry, as one of the most ancient branches of pure mathematics, arose from the necessity and desire to describe and thoroughly understand the surrounding world and the universe. The development of geometry substantially contributes to the evolution of mathematics as a whole subject through the concepts and notions of axiom and manifold, which lays the foundation of modern mathematics. Despite the abstract appearance of modern geometry, the objects and problems of modern geometry can usually be traced back to practical situations. A good example is the recent breakthrough in image identification technology, which is rooted in differential geometry. From both a research and an educational perspective, geometry provides perfect opportunities for the implementation and interaction of ideas and techniques from other branches of mathematics like algebra, analysis, topology and probability, and other subjects like chemistry, finance and physics through topics including financial derivatives, Einstein Equations and black holes, which have attracted enormous public attention in recent years. You will learn to approach questions initially through intuition and then make this rigorous using mathematical tools. Through the selection of topics in this unit, you will train your mathematical imagination to discover the geometric framework of a complex problem.

Geometry, as one of the most ancient branches of pure mathematics, arose from the necessity and desire to describe and thoroughly understand the surrounding world and the universe. The development of geometry substantially contributes to the evolution of mathematics as a whole subject through the concepts and notions of axiom and manifold, which lays the foundation of modern mathematics. Despite the abstract appearance of modern geometry, the objects and problems of modern geometry can usually be traced back to practical situations. A good example is the recent breakthrough in image identification technology, which is rooted in differential geometry. From both a research and an educational perspective, geometry provides perfect opportunities for the implementation and interaction of ideas and techniques from other branches of mathematics like algebra, analysis, topology and probability, and other subjects like chemistry, finance and physics through topics including financial derivatives, Einstein Equations and black holes, which have attracted enormous public attention in recent years. You will learn to approach questions initially through intuition and then make this rigorous using mathematical tools. Through the selection of topics in this unit, you will train your mathematical imagination to discover the geometric framework of a complex problem.

Topology is the mathematical theory of the "shape of spaces". It gives a flexible framework in which the fabric of space is like rubber and thus enables the study of the general shape of a space. The spaces often arise indirectly: as the solution space of a set of equations; as the parameter space for a family of objects; as a point cloud from a data set; and so on. This leads to strong interactions between topology and a plethora of mathematical and scientific areas. The love of the study and use of topology is far reaching, including the use of topological techniques in the phases of matter and transition which received the 2016 Nobel Prize in Physics. This unit introduces you to a selection of topics in pure or applied topology. Topology receives strength from its areas of applications and imparts insights in return. A wide spectrum of methods is used, dividing topology into the areas of algebraic, computational, differential, geometric and set-theoretic topology. You will learn the methods, key results, and role in current mathematics of at least one of these areas, and gain an understanding of current research problems and open conjectures in the field.

Topology is the mathematical theory of the "shape of spaces". It gives a flexible framework in which the fabric of space is like rubber and thus enables the study of the general shape of a space. The spaces often arise indirectly: as the solution space of a set of equations; as the parameter space for a family of objects; as a point cloud from a data set; and so on. This leads to strong interactions between topology and a plethora of mathematical and scientific areas. The love of the study and use of topology is far reaching, including the use of topological techniques in the phases of matter and transition which received the 2016 Nobel Prize in Physics. This unit introduces you to a selection of topics in pure or applied topology. Topology receives strength from its areas of applications and imparts insights in return. A wide spectrum of methods is used, dividing topology into the areas of algebraic, computational, differential, geometric and set-theoretic topology. You will learn the methods, key results, and role in current mathematics of at least one of these areas, and gain an understanding of current research problems and open conjectures in the field.

In his book on Applied Mathematics, Alain Goriely states "There is great beauty in mathematics and great beauty in the world around us. Applied Mathematics brings the two together in a way that is not always beautiful, but is always interesting and exciting. "In this unit you will explore classic problems in Applied Mathematics and their solutions or investigate an area of Applied Mathematics that is currently the focus of active research. You will delve deeply into powerful mathematical methods and use this mathematics to investigate and resolve problems in the real world, whether that is in computation, the social sciences or the natural sciences. You will learn how the synergies between mathematics and real world problems that are found throughout Applied Mathematics both drive the creation of new mathematical methods and theory, and give powerful insights into the underlying problems, resulting in new ways of seeing the world and new types of technology. By doing this unit you will grow in your appreciation of the links between mathematical theory and its practical outcomes in other disciplines and learn to use mathematics in deeply profound ways in one or more areas of application.

In his book on Applied Mathematics, Alain Goriely states "There is great beauty in mathematics and great beauty in the world around us. Applied Mathematics brings the two together in a way that is not always beautiful, but is always interesting and exciting. "In this unit you will explore classic problems in Applied Mathematics and their solutions or investigate an area of Applied Mathematics that is currently the focus of active research. You will delve deeply into powerful mathematical methods and use this mathematics to investigate and resolve problems in the real world, whether than is in computation, the social sciences or the natural sciences. You will learn how the synergies between mathematics and real world problems that are found throughout Applied Mathematics both drive the creation of new mathematical methods and theory, and give powerful insights into the underlying problems, resulting in new ways of seeing the world and new types of technology. By doing this unit you will grow in your appreciation of the links between mathematical theory and its practical outcomes in other disciplines and learn to use mathematics in deeply profound ways in one or more areas of application.

Deterministic and stochastic systems lie at the heart of applied mathematics. They are dynamical models of the real world, whose reach is widespread and growing rapidly. Interest in such models grew from the discovery of chaos in simple models of atmospheric circulation, at almost the same time as astonishingly well-ordered and predictable behaviour was observed in models of particle physics. These starting points led to the development of new tools in applied mathematics, which turned out to be profoundly effective at describing emergent behaviours and change. The Economist magazine has stated that "The equations of a good theory are taken to represent physical reality because they can be used to make predictions". This unit will present a toolbox for describing and predicting outcomes. The tools also allow for methods of checking how parameters in a model could be changed to compare predictions to observations. You will learn how profound mathematical theory is applied to produce tools that are universal, adaptable and far-reaching. You will adapt and apply this fundamental theory to these to explore classical and current applications of mathematics to real world problems. You will use methods, developed to study classical areas, as springboards for new tools for innovative applications such as artificial intelligence and machine learning.

Deterministic and stochastic systems lie at the heart of applied mathematics. They are dynamical models of the real world; whose reach is widespread and growing rapidly. Interest in such models grew from the discovery of chaos in simple models of atmospheric circulation, at almost the same time as astonishingly well-ordered and predictable behaviour was observed in models of particle physics. These starting points led to the development of new tools in applied mathematics, which turned out to be profoundly effective at describing emergent behaviours and change. The Economist magazine has stated that "The equations of a good theory are taken to represent physical reality because they can be used to make predictions". This unit will present a toolbox for describing and predicting outcomes. The tools also allow for methods of checking how parameters in a model could be changed to comparepredictions to observations. You will learn how profound mathematical theory is applied to produce tools that are universal, adaptable and far-reaching. You will adapt and apply this fundamental theory to these to explore classical and current applications of mathematics to real world problems. You will use methods, developed to study classical areas, as springboards for new tools for innovative applications such as artificial intelligence and machine learning.

"Mathematical modelling applies mathematical frameworks, such as ordinary and partial differential equations, to capture the dynamics of natural phenomena, including fluid dynamics, Newtonian and relativistic mechanics, climate, ecology, and physiology. Modelling often falls into two styles, mechanistic and phenomenological. Mechanistic modelling seeks to understand how large-scale phenomena are driven by simple, local dynamics usually governed by physical or biological laws or properties. On the other hand, phenomenological modelling seeks to capture large-scale trends of a system, such as growth, decay, and oscillations, without necessarily accounting for smaller-scale dynamics. In practice, most models combine elements of both styles. In this unit you will learn about how these mathematical frameworks are constructed and applied for particular types of phenomena which may include mathematical oncology, high Reynolds number fluid flow, stellar atmosphere, terrestrial climates, populations of cells or organisms or other areas of mathematical interest. You will analyse both classical and new models and critique their applicability and use their predictions to explore aspects of the natural world. Inspired by these ideas, you will have the opportunity to create new models in tutorials and assignments and to use them to solve complex mathematical and scientific problems. By doing this unit, you will learn how mathematics is applied in both simple and complicated models and explore the ways that mathematical analysis creates insight into natural phenomena. "

"Mathematical modelling applies mathematical frameworks, such as ordinary and partial differential equations, to capture the dynamics of natural phenomena, including fluid dynamics, Newtonian and relativistic mechanics, climate, ecology, and physiology. Modelling often falls into two styles, mechanistic and phenomenological. Mechanistic modelling seeks to understand how large-scale phenomena are driven by simple, local dynamics usually governed by physical or biological laws or properties. On the other hand, phenomenological modelling seeks to capture large-scale trends of a system, such as growth, decay, and oscillations, without necessarily accounting for smaller-scale dynamics. In practice, most models combine elements of both styles. In this unit you will learn about how these mathematical frameworks are constructed and applied for particular types of phenomena which may include mathematical oncology, high Reynolds number fluid flow, stellar atmosphere, terrestrial climates, populations of cells or organisms or other areas of mathematical interest. You will analyse both classical and new models and critique their applicability and use their predictions to explore aspects of the natural world. Inspired by these ideas, you will have the opportunity to create new models in tutorials and assignments and to use them to solve complex mathematical and scientific problems. By doing this unit, you will learn how mathematics is applied in both simple and complicated models and explore the ways that mathematical analysis creates insight into natural phenomena. "

How to maximise gains (or to minimise costs) and how to determine optimal strategies or policies are fundamental questions for engineers, economists, doctors designing a cancer therapy, fund managers or a government agency planning social policies. Several problems in Science (e. g. mechanics, physics, neuroscience or biology) can be also formulated as optimisation problems in random environments. The theory of stochastic optimal control and games is an indispensable tool in many areas of applied mathematics. In the first part of this unit, you will be familiarised with the dynamic programming principle and learn how to show that it provides a unified approach to a large number of seemingly unrelated problems. The second part is devoted to backward stochastic differential equations and their applications to stochastic optimal control and game theory. You will learn how to solve continuous time problems based either on the Wiener process or more general classes of stochastic processes. After completing this unit, you will be able to formulate a diverse suite of problems arising in finance, applied sciences, engineering and medicine as stochastic optimal control problems and solve them using the concepts of the Bellman principle, Hamilton-Jacobi-Bellman equation and backward stochastic differential equations.

Stochastics examines phenomena in which chance plays a central role. The theory of stochastic phenomena has applications in engineering systems, the physical and life sciences and economics, to give just a few examples. Applications of stochastic processes arise particularly naturally in finance where there are fluctuations in stock prices and practitioners are required to solve different types of optimisation problems in stochastically driven systems. For this reason, it is particularly important that mathematicians in general and especially mathematicians specialising in problems in the financial industry are equipped with tools to analyse and quantify random phenomena. This unit will expose you to critical topics in the theory and application of stochastic processes and analysis in mathematical finance. You will learn how to identify problems that require the application of stochastic theory, how to rigorously describe such problems using appropriate mathematical frameworks and how to tackle and solve the problem once it has been phrased in terms of stochastic theory. Along the way, you will also gain a deep knowledge about diverse topics in finance and the relevance of mathematical analysis in the financial industry.

The great power of the discipline of Statistics is the possibility to make inferences concerning a large population based on optimally learning from increasingly large and complex data. Critical to successful inference is a deep understanding of the theory when the number of samples and the number of observed features is large and require complex statistical methods to be analysed correctly. In this unit you will learn how to integrate concepts from a diverse suite of specialities in mathematics and statistics such as optimisation, functional approximations and complex analysis to make inferences for highly complicated data. In particular, this unit explores advanced topics in statistical methodology examining both theoretical foundations and details of implementation to applications. The unit is made up of 3 distinct modules. These include (but are not restricted to) Asymptotic theory for statistics and econometrics, Theory and algorithms for statistical learning with big data, and Introduction to optimal semiparametric optimality.

The great power of the discipline of Statistics is the possibility to make inferences concerning a large population based on only observing a relatively small sample from it. Of course, this "magic" does not come without a price, we must construct statistical models to approximate these populations and samples from them, develop mathematical tools using probability theory, appreciate the limitations of our methods and, most importantly, understand what assumptions need to be made for such inferences to be valid, and develop ways to check these assumptions. Implementing these methods to possibly complex data structures is also a challenge that must be overcome. This unit explores advanced topics in statistical methodology examining both theoretical foundations and details of implementation to applications. The unit is made up of 3 distinct modules. These include (but are not restricted to) Advanced Survival Analysis, Extreme Value Theory and Statistical Methods in Bioinformatics.

Independent research can be a life changing experience. In this unit you will complete a research project in a discipline of the Mathematical Sciences, such as Pure Mathematics, Applied Mathematics, Statistics, Data Science, or Financial Mathematics and Statistics. Together with your supervisor, you will identify a suitable research problem and develop a strategy to address it. This may include synthesizing and generalising results from the mathematical literature, generating new examples, proving new theorems, writing and applying new mathematical models or devising new statistical tests. Working on a research project will also provide broader opportunities such as being part of a research group, interacting with both your peers and discipIine experts, attending research seminars and workshops. Students in some disciplines, such as Applied Mathematics and Statistics, may also have the opportunity to interact with interdisciplinary teams as part of their research project. You will communicate the research plan and findings via an oral presentation and a 40 to 60 page thesis. Successful completion of your Master of Mathematical Sciences degree will clearly demonstrate that you have mastered significant research and professional skills for either undertaking a PhD or any variety of future careers.

Independent research can be a life changing experience. In this unit you will complete a research project in a discipline of the Mathematical Sciences, such as Pure Mathematics, Applied Mathematics, Statistics, Data Science, or Financial Mathematics and Statistics. Together with your supervisor, you will identify a suitable research problem and develop a strategy to address it. This may include synthesizing and generalising results from the mathematical literature, generating new examples, proving new theorems, writing and applying new mathematical models or devising new statistical tests. Working on a research project will also provide broader opportunities such as being part of a research group, interacting with both your peers and discipIine experts, attending research seminars and workshops. Students in some disciplines, such as Applied Mathematics and Statistics, may also have the opportunity to interact with interdisciplinary teams as part of their research project. You will communicate the research plan and findings via an oral presentation and a 40 to 60 page thesis. Successful completion of your Master of Mathematical Sciences degree will clearly demonstrate that you have mastered significant research and professional skills for either undertaking a PhD or any variety of future careers.

Independent research can be a life changing experience. In this unit you will complete a research project in a discipline of the Mathematical Sciences, such as Pure Mathematics, Applied Mathematics, Statistics, Data Science, or Financial Mathematics and Statistics. Together with your supervisor, you will identify a suitable research problem and develop a strategy to address it. This may include synthesizing and generalising results from the mathematical literature, generating new examples, proving new theorems, writing and applying new mathematical models or devising new statistical tests. Working on a research project will also provide broader opportunities such as being part of a research group, interacting with both your peers and discipIine experts, attending research seminars and workshops. Students in some disciplines, such as Applied Mathematics and Statistics, may also have the opportunity to interact with interdisciplinary teams as part of their research project. You will communicate the research plan and findings via an oral presentation and a 40 to 60 page thesis. Successful completion of your Master of Mathematical Sciences degree will clearly demonstrate that you have mastered significant research and professional skills for either undertaking a PhD or any variety of future careers.

Independent research can be a life changing experience. In this unit you will complete a research project in a discipline of the Mathematical Sciences, such as Pure Mathematics, Applied Mathematics, Statistics, Data Science, or Financial Mathematics and Statistics. Together with your supervisor, you will identify a suitable research problem and develop a strategy to address it. This may include synthesizing and generalising results from the mathematical literature, generating new examples, proving new theorems, writing and applying new mathematical models or devising new statistical tests. Working on a research project will also provide broader opportunities such as being part of a research group, interacting with both your peers and discipIine experts, attending research seminars and workshops. Students in some disciplines, such as Applied Mathematics and Statistics, may also have the opportunity to interact with interdisciplinary teams as part of their research project. You will communicate the research plan and findings via an oral presentation and a 40 to 60 page thesis. Successful completion of your Master of Mathematical Sciences degree will clearly demonstrate that you have mastered significant research and professional skills for either undertaking a PhD or any variety of future careers.