BSc (Hons) or MMath — 2025 entry Mathematics

This module introduces a variety of commonly used techniques from Operations Research. The module leads to a deeper understanding of linear programming problems and the theory that underpins their solving. Tools such as the Simplex Method are presented and an introduction to nonlinear optimisation methods is also provided. This module supports and complements other modules where optimisation and constrained optimisation are considered.

Statistics is the science of collecting and analysing data from random events and modelling them using random variables. This module first recaps the properties of a single random variable as seen in Level 4 Probability and Statistics (MAT1033), then builds on this by considering multiple random variables simultaneously. This module also covers the transformation of random variables, as well as applications such as tail probabilities and limit theorems. 

The module provides an introduction to the theory and properties of curves and surfaces in Euclidean space. Basic concepts and tools from a branch of mathematics known as differential geometry are used to study curves and surfaces, and to understand their geometric properties. This module builds on material from MAT1043 Multivariable Calculus and MAT1034 Linear Algebra. The module is recommended for students intending to select the Year 3 modules MAT3009 Manifolds and Topology, and MAT3044 Riemannian Geometry, although it is not a pre-requisite for either module.

Stochastic processes are a series of random variables. Students will be introduced to both Markov Chains and continuous Markov stochastic processes, where the distributions of future random variables are determined by the value of the most recent random variable. These models are important for modelling things which change over time, such as voting intention or population size. 

This module provides an introduction to abstract algebra, focusing on the theory of algebraic structures called groups and rings. The module builds on preliminary material on groups introduced in MAT1031 Algebra. This module forms the starting point for subsequent algebraic modules, such as MAT3032 Advanced Algebra, MATM035 Representation Theory and MATM011 Lie Algebras.

This module introduces students to inviscid fluid flows including surface waves. By the end of the module, students should be able to recognise dominant features of fluid motion, and to derive some simple solutions of the equations of motion. Students should also have an appreciation of the force balances that produce various classes of flows.

Statistical modelling provides a means of extracting information from data, enabling informed decisions to be made. It is a versatile and powerful tool with widespread applications contributing to advancements in research, business, technology, and society. Students will be introduced to the basic concepts of statistical modelling via linear regression models, which are fundamental in statistics and serve as the basis for more complex modelling techniques. Model fitting, selection and evaluation are covered for: Simple Linear Regression; polynomial regression and multiple regression models. The module concludes with the introduction of simple models in the Design of Experiments, covering the use of  such experiments and the analysis of data arising from them. The statistical software R is fully integrated with the module. Students will reinforce their understanding of concepts of statistical modelling and develop their digital capabilities by using R to conduct analyses of real data sets from business, science and industry.

The module introduces the subject of (infinite dimensional) function spaces and shows how they are structured by metric, norm or inner product. The course naturally extends ideas contained in Real Analysis 1 and 2, and it sets in a wider context the orthogonal decompositions seen in the Fourier analysis part of MAT2011: Linear PDEs.

This module introduces fundamental concepts in analytical dynamics and illustrates their application to real-world problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations and symplectic manifolds. The module also leads to a deeper understanding of the role of symmetries and conservation laws in dynamical systems. This module builds on material from MAT1036 Classical Dynamics and lays the foundations for MAT3039 Quantum Mechanics.

This module introduces students to topological spaces and manifolds. Topology is the study of properties of spaces which are invariant under continuous transformations, and forms one of the cornerstones of pure mathematics. Study of topology leads to a deeper understanding of the concepts of continuity, connectedness and compactness. In the second part smooth manifolds are introduced, together with different construction methods, concepts of orientability and transversality. Finally, in the third part key elements of analysis on manifolds are discussed, such as tensors, differential forms, and the Stokes theorem in full generality.MAT2047 Curves and Surfaces is a recommended prior module, but is not pre-requisite. Students will also find the material in MAT3044 Riemannian Geometry complementary.

This module extends the abstract algebra introduced in MAT2048 Groups & Rings. A deeper insight into groups and rings is developed, and further algebraic structures are introduced. This module complements the module material in MATM035 Representation Theory and MATM011 Lie Algebras

This module provides an introduction to the applications of ordinary, delay and partial differential equations to ecology and epidemiology.

Under the guidance of an academic supervisor, the student will investigate a mathematical topic of interest to them in some depth. They will compose a written report on their studies and give an oral presentation on their work.

Bayesian Statistics is the branch of statistics that relies on subjective probability to create a wide range of statistical models. This module introduces Bayesian methodology and guides students to use prior to posterior analysis for modelling realistic problems. This module then tackles more difficult topics such as Bayesian point estimates, model selection and linear regression. 

Understanding the motions in our atmosphere and oceans is key to understanding, and hence solving, the global climate crisis. By applying a range of mathematical techniques to the governing fluid flow equations we are able to consider simplified models of a hugely complex system which will allow us to gain a deeper understanding about the transports of energy and heat on our planet.

The first half of this module covers various applications of statistical physics to model share prices and financial markets. This mathematics is then applied to calculating prices for some examples of financial derivatives. The second half of the module then focusses on optimization problems with examples including logistics, aerospace, traffic control and finance (which includes pricing, risk managements and portfolio optimizations in financial markets). There is a particular focus on the role of quantum optimization and the use of quantum computer algorithms in finance.

This module introduces fundamental concepts in Quantum Mechanics and its applications to real-world quantum problems. The module covers the mathematics of Hilbert spaces and Dirac notation, the postulates of Quantum Mechanics, the uncertainty principle, the Schroedinger equation with one-dimensional applications to a particle in a potential well and the quantum harmonic oscillator, and angular momentum and spin. This module utilises material from MAT1034 Linear Algebra and MAT2007 Ordinary Differential Equations. The module also builds on material from MAT1036 Classical Dynamics and MAT3008 Lagrangian & Hamiltonian Dynamics, although these modules are not pre-requisite.

Riemannian geometry is the study of geometric properties of spaces, called manifolds, typically in higher dimensions, which are described smoothly by sets of well-defined co-ordinates. The emphasis is on how mathematically to understand the notion of distance via a metric on such spaces, and how this is used to investigate key associated geometric concepts. Students are introduced to the key concepts of manifolds and metrics, with illustrative examples such as spheres, hyperbolic spaces and Lie groups. The module continues with geodesics, isometries, covariant derivatives, the Levi Civita connection, and concludes with curvature and its properties. Modules in Curves and Surfaces and in Manifolds and Topology contain related material.

Under the guidance of an academic supervisor, the student will investigate a mathematical topic of interest to them in some depth. They will compose a written report on their studies, and give an oral presentation on their work.   

Data science is the study of data to extract meaningful and actionable insights at all levels of society such as dynamical systems and social media networks. This module introduces the role of data in society and provides students with the underpinning mathematics that drives data methodology and algorithms. This module then covers wide-ranging topics with a focus on the Surrey brand of data, as research into data is part of the department research agenda. 

The module will introduce students to key concepts in education as well as the tools and strategies for effective public engagement in their discipline areas and for different audiences. Students will gain practical experience by planning, implementing, and evaluating an education or public engagement-based activity. Opportunities will also exist for classroom (e.g. school; early-year University) teaching experience and / or involvement in external exhibitions. The projects will be related to current Faculty, department and / or outreach needs thus leading to informed student partnership activities.

This module introduces the ideas of viscous fluid flows which build on the ideas some students would have seen in MAT2050: Inviscid Fluid Dynamics. By the end of the module, students should be familiar with the Navier-Stokes equations, and should be able to solve these equations in various simplified situations as well as in a variety of geometries.

Graph theory is an aesthetically appealing branch of pure mathematics with strong links to other areas of mathematics (combinatorics, algebra, topology, probability, optimisation and numerics) and well-developed applications to a wide range of other disciplines (including operations research, data science, chemistry, systems biology, statistical mechanics and quantum field theory). This module provides an introduction to graph theory. There is an emphasis on theorems and proofs.

Time series are a collection of observations taken over time. This covers a great deal of situations such as stock markets, rainfall or even goals scored by a sports team. Features of the time series will lead to an appropriate choice of model. These models will be validated, and then can be used to forecast the future. Despite the modest pre-requisites of Level 4 Probability and Statistics (MAT1033), students will gain resourcefulness and resilience through learning mathematical proofs as well as gain digital capabilities through using R to conduct analyses of data sets and writing a report.

Under the guidance of an academic supervisor, the student will investigate a mathematical topic of interest to them in some depth. They will compose a written report on their studies and give oral presentations on their work.

This module introduces several principles and processes which underpin most physical science and engineering disciplines, which you are likely to study beyond the Foundation Year. Specifically, you will study topics that include S.I. units and measurement theory, electric and magnetic fields and their interactions, the properties of ideal gases, heat transfer and thermodynamics, fluid statics and dynamics, and engineering instrumentation and measurement. You will attend several lectures and a tutorial each teaching week alongside guided independent study opportunities to develop your understanding of topics more deeply, supported by the use of the university’s virtual learning platform.

The module aims to introduce the students to the core elements of microeconomic theory. The module will begin with a discussion of Economics as a science and its basic principles and concepts. The focus will then move onto the market equilibrium, i.e., the supply and demand model and the impact of government intervention in the market outcomes. Consumer theory and the theory of the firm will be studied to understand the background to the supply and demand model before turning to the welfare implications of different market structures.

The module covers the principles of chemistry relevant to degree-level study in disciplines requiring a strong background in this subject, (e.g. the BEng in both the Chemical and Civil Engineering programmes at the University of Surrey). There will be a strong focus placed on the fundamental principles of physical chemistry, with a basic introduction to organic and analytical chemistry techniques.  Learning will include examples of industrial processes and case studies and there will be an overarching theme of sustainability running through the module linked to several topics (in particular, fuels, combustion and polymers).  Module content will be delivered via weekly lectures, interspersed with opportunities for you to reflect on what you have just learned. Additional support is provided in weekly tutorials. There are guided independent study opportunities to develop your understanding of topics more deeply, supported by the use of the university’s virtual learning platform.

A foundation level physics module designed to reinforce and broaden basic A-Level Physics material in electricity and electronics, nuclear physics, develop practical skills, and prepare students for the more advanced concepts and applications in the first year of their Engineering or Physical Sciences degree. You will attend several lectures and a tutorial each teaching week alongside guided independent study opportunities to develop your understanding of topics more deeply, supported using the university’s virtual learning platform.

In this module the students will learn about the macroeconomic environment and the theoretical and conceptual frameworks which underpin it. It is designed to prepare the students for the more advanced level macroeconomics in the first year of their Economics programme. The module will begin with the introduction of key macroeconomic topics, i.e., economic growth and business cycles, unemployment, inflation and open economy. The focus will then move onto developing a theoretical model to study and analyse the short-run macroeconomic equilibrium. The role of fiscal and monetary policy in shaping economic outcomes will also be discussed. The methodological approach will include the use of mathematical tools in solving and analysing the theoretical models. 

This module introduces a variety of commonly used techniques from Operations Research. The module leads to a deeper understanding of linear programming problems and the theory that underpins their solving. Tools such as the Simplex Method are presented and an introduction to nonlinear optimisation methods is also provided. This module supports and complements other modules where optimisation and constrained optimisation are considered.

Statistics is the science of collecting and analysing data from random events and modelling them using random variables. This module first recaps the properties of a single random variable as seen in Level 4 Probability and Statistics (MAT1033), then builds on this by considering multiple random variables simultaneously. This module also covers the transformation of random variables, as well as applications such as tail probabilities and limit theorems. 

The module provides an introduction to the theory and properties of curves and surfaces in Euclidean space. Basic concepts and tools from a branch of mathematics known as differential geometry are used to study curves and surfaces, and to understand their geometric properties. This module builds on material from MAT1043 Multivariable Calculus and MAT1034 Linear Algebra. The module is recommended for students intending to select the Year 3 modules MAT3009 Manifolds and Topology, and MAT3044 Riemannian Geometry, although it is not a pre-requisite for either module.

Stochastic processes are a series of random variables. Students will be introduced to both Markov Chains and continuous Markov stochastic processes, where the distributions of future random variables are determined by the value of the most recent random variable. These models are important for modelling things which change over time, such as voting intention or population size. 

This module provides an introduction to abstract algebra, focusing on the theory of algebraic structures called groups and rings. The module builds on preliminary material on groups introduced in MAT1031 Algebra. This module forms the starting point for subsequent algebraic modules, such as MAT3032 Advanced Algebra, MATM035 Representation Theory and MATM011 Lie Algebras.

This module introduces students to inviscid fluid flows including surface waves. By the end of the module, students should be able to recognise dominant features of fluid motion, and to derive some simple solutions of the equations of motion. Students should also have an appreciation of the force balances that produce various classes of flows.

Statistical modelling provides a means of extracting information from data, enabling informed decisions to be made. It is a versatile and powerful tool with widespread applications contributing to advancements in research, business, technology, and society. Students will be introduced to the basic concepts of statistical modelling via linear regression models, which are fundamental in statistics and serve as the basis for more complex modelling techniques. Model fitting, selection and evaluation are covered for: Simple Linear Regression; polynomial regression and multiple regression models. The module concludes with the introduction of simple models in the Design of Experiments, covering the use of  such experiments and the analysis of data arising from them. The statistical software R is fully integrated with the module. Students will reinforce their understanding of concepts of statistical modelling and develop their digital capabilities by using R to conduct analyses of real data sets from business, science and industry.

The module introduces the subject of (infinite dimensional) function spaces and shows how they are structured by metric, norm or inner product. The course naturally extends ideas contained in Real Analysis 1 and 2, and it sets in a wider context the orthogonal decompositions seen in the Fourier analysis part of MAT2011: Linear PDEs.

This module introduces fundamental concepts in analytical dynamics and illustrates their application to real-world problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations and symplectic manifolds. The module also leads to a deeper understanding of the role of symmetries and conservation laws in dynamical systems. This module builds on material from MAT1036 Classical Dynamics and lays the foundations for MAT3039 Quantum Mechanics.

This module introduces students to topological spaces and manifolds. Topology is the study of properties of spaces which are invariant under continuous transformations, and forms one of the cornerstones of pure mathematics. Study of topology leads to a deeper understanding of the concepts of continuity, connectedness and compactness. In the second part smooth manifolds are introduced, together with different construction methods, concepts of orientability and transversality. Finally, in the third part key elements of analysis on manifolds are discussed, such as tensors, differential forms, and the Stokes theorem in full generality.MAT2047 Curves and Surfaces is a recommended prior module, but is not pre-requisite. Students will also find the material in MAT3044 Riemannian Geometry complementary.

This module extends the abstract algebra introduced in MAT2048 Groups & Rings. A deeper insight into groups and rings is developed, and further algebraic structures are introduced. This module complements the module material in MATM035 Representation Theory and MATM011 Lie Algebras

This module provides an introduction to the applications of ordinary, delay and partial differential equations to ecology and epidemiology.

Under the guidance of an academic supervisor, the student will investigate a mathematical topic of interest to them in some depth. They will compose a written report on their studies and give an oral presentation on their work.

Bayesian Statistics is the branch of statistics that relies on subjective probability to create a wide range of statistical models. This module introduces Bayesian methodology and guides students to use prior to posterior analysis for modelling realistic problems. This module then tackles more difficult topics such as Bayesian point estimates, model selection and linear regression. 

Understanding the motions in our atmosphere and oceans is key to understanding, and hence solving, the global climate crisis. By applying a range of mathematical techniques to the governing fluid flow equations we are able to consider simplified models of a hugely complex system which will allow us to gain a deeper understanding about the transports of energy and heat on our planet.

The first half of this module covers various applications of statistical physics to model share prices and financial markets. This mathematics is then applied to calculating prices for some examples of financial derivatives. The second half of the module then focusses on optimization problems with examples including logistics, aerospace, traffic control and finance (which includes pricing, risk managements and portfolio optimizations in financial markets). There is a particular focus on the role of quantum optimization and the use of quantum computer algorithms in finance.

Under the guidance of an academic supervisor, the student will investigate a mathematical topic of interest to them in some depth. They will compose a written report on their studies, and give an oral presentation on their work.   

This module introduces fundamental concepts in Quantum Mechanics and its applications to real-world quantum problems. The module covers the mathematics of Hilbert spaces and Dirac notation, the postulates of Quantum Mechanics, the uncertainty principle, the Schroedinger equation with one-dimensional applications to a particle in a potential well and the quantum harmonic oscillator, and angular momentum and spin. This module utilises material from MAT1034 Linear Algebra and MAT2007 Ordinary Differential Equations. The module also builds on material from MAT1036 Classical Dynamics and MAT3008 Lagrangian & Hamiltonian Dynamics, although these modules are not pre-requisite.

Riemannian geometry is the study of geometric properties of spaces, called manifolds, typically in higher dimensions, which are described smoothly by sets of well-defined co-ordinates. The emphasis is on how mathematically to understand the notion of distance via a metric on such spaces, and how this is used to investigate key associated geometric concepts. Students are introduced to the key concepts of manifolds and metrics, with illustrative examples such as spheres, hyperbolic spaces and Lie groups. The module continues with geodesics, isometries, covariant derivatives, the Levi Civita connection, and concludes with curvature and its properties. Modules in Curves and Surfaces and in Manifolds and Topology contain related material.

Data science is the study of data to extract meaningful and actionable insights at all levels of society such as dynamical systems and social media networks. This module introduces the role of data in society and provides students with the underpinning mathematics that drives data methodology and algorithms. This module then covers wide-ranging topics with a focus on the Surrey brand of data, as research into data is part of the department research agenda. 

The module will introduce students to key concepts in education as well as the tools and strategies for effective public engagement in their discipline areas and for different audiences. Students will gain practical experience by planning, implementing, and evaluating an education or public engagement-based activity. Opportunities will also exist for classroom (e.g. school; early-year University) teaching experience and / or involvement in external exhibitions. The projects will be related to current Faculty, department and / or outreach needs thus leading to informed student partnership activities.

This module introduces the ideas of viscous fluid flows which build on the ideas some students would have seen in MAT2050: Inviscid Fluid Dynamics. By the end of the module, students should be familiar with the Navier-Stokes equations, and should be able to solve these equations in various simplified situations as well as in a variety of geometries.

Graph theory is an aesthetically appealing branch of pure mathematics with strong links to other areas of mathematics (combinatorics, algebra, topology, probability, optimisation and numerics) and well-developed applications to a wide range of other disciplines (including operations research, data science, chemistry, systems biology, statistical mechanics and quantum field theory). This module provides an introduction to graph theory. There is an emphasis on theorems and proofs.

Time series are a collection of observations taken over time. This covers a great deal of situations such as stock markets, rainfall or even goals scored by a sports team. Features of the time series will lead to an appropriate choice of model. These models will be validated, and then can be used to forecast the future. Despite the modest pre-requisites of Level 4 Probability and Statistics (MAT1033), students will gain resourcefulness and resilience through learning mathematical proofs as well as gain digital capabilities through using R to conduct analyses of data sets and writing a report.

Under the guidance of an academic supervisor, the student will investigate a mathematical topic of interest to them in some depth. They will compose a written report on their studies and give oral presentations on their work.

This module introduces several principles and processes which underpin most physical science and engineering disciplines, which you are likely to study beyond the Foundation Year. Specifically, you will study topics that include S.I. units and measurement theory, electric and magnetic fields and their interactions, the properties of ideal gases, heat transfer and thermodynamics, fluid statics and dynamics, and engineering instrumentation and measurement. You will attend several lectures and a tutorial each teaching week alongside guided independent study opportunities to develop your understanding of topics more deeply, supported by the use of the university’s virtual learning platform.

The module aims to introduce the students to the core elements of microeconomic theory. The module will begin with a discussion of Economics as a science and its basic principles and concepts. The focus will then move onto the market equilibrium, i.e., the supply and demand model and the impact of government intervention in the market outcomes. Consumer theory and the theory of the firm will be studied to understand the background to the supply and demand model before turning to the welfare implications of different market structures.

The module covers the principles of chemistry relevant to degree-level study in disciplines requiring a strong background in this subject, (e.g. the BEng in both the Chemical and Civil Engineering programmes at the University of Surrey). There will be a strong focus placed on the fundamental principles of physical chemistry, with a basic introduction to organic and analytical chemistry techniques.  Learning will include examples of industrial processes and case studies and there will be an overarching theme of sustainability running through the module linked to several topics (in particular, fuels, combustion and polymers).  Module content will be delivered via weekly lectures, interspersed with opportunities for you to reflect on what you have just learned. Additional support is provided in weekly tutorials. There are guided independent study opportunities to develop your understanding of topics more deeply, supported by the use of the university’s virtual learning platform.

A foundation level physics module designed to reinforce and broaden basic A-Level Physics material in electricity and electronics, nuclear physics, develop practical skills, and prepare students for the more advanced concepts and applications in the first year of their Engineering or Physical Sciences degree. You will attend several lectures and a tutorial each teaching week alongside guided independent study opportunities to develop your understanding of topics more deeply, supported using the university’s virtual learning platform.

In this module the students will learn about the macroeconomic environment and the theoretical and conceptual frameworks which underpin it. It is designed to prepare the students for the more advanced level macroeconomics in the first year of their Economics programme. The module will begin with the introduction of key macroeconomic topics, i.e., economic growth and business cycles, unemployment, inflation and open economy. The focus will then move onto developing a theoretical model to study and analyse the short-run macroeconomic equilibrium. The role of fiscal and monetary policy in shaping economic outcomes will also be discussed. The methodological approach will include the use of mathematical tools in solving and analysing the theoretical models. 

This module introduces a variety of commonly used techniques from Operations Research. The module leads to a deeper understanding of linear programming problems and the theory that underpins their solving. Tools such as the Simplex Method are presented and an introduction to nonlinear optimisation methods is also provided. This module supports and complements other modules where optimisation and constrained optimisation are considered.

Statistics is the science of collecting and analysing data from random events and modelling them using random variables. This module first recaps the properties of a single random variable as seen in Level 4 Probability and Statistics (MAT1033), then builds on this by considering multiple random variables simultaneously. This module also covers the transformation of random variables, as well as applications such as tail probabilities and limit theorems. 

The module provides an introduction to the theory and properties of curves and surfaces in Euclidean space. Basic concepts and tools from a branch of mathematics known as differential geometry are used to study curves and surfaces, and to understand their geometric properties. This module builds on material from MAT1043 Multivariable Calculus and MAT1034 Linear Algebra. The module is recommended for students intending to select the Year 3 modules MAT3009 Manifolds and Topology, and MAT3044 Riemannian Geometry, although it is not a pre-requisite for either module.

Stochastic processes are a series of random variables. Students will be introduced to both Markov Chains and continuous Markov stochastic processes, where the distributions of future random variables are determined by the value of the most recent random variable. These models are important for modelling things which change over time, such as voting intention or population size. 

This module provides an introduction to abstract algebra, focusing on the theory of algebraic structures called groups and rings. The module builds on preliminary material on groups introduced in MAT1031 Algebra. This module forms the starting point for subsequent algebraic modules, such as MAT3032 Advanced Algebra, MATM035 Representation Theory and MATM011 Lie Algebras.

This module introduces students to inviscid fluid flows including surface waves. By the end of the module, students should be able to recognise dominant features of fluid motion, and to derive some simple solutions of the equations of motion. Students should also have an appreciation of the force balances that produce various classes of flows.

Statistical modelling provides a means of extracting information from data, enabling informed decisions to be made. It is a versatile and powerful tool with widespread applications contributing to advancements in research, business, technology, and society. Students will be introduced to the basic concepts of statistical modelling via linear regression models, which are fundamental in statistics and serve as the basis for more complex modelling techniques. Model fitting, selection and evaluation are covered for: Simple Linear Regression; polynomial regression and multiple regression models. The module concludes with the introduction of simple models in the Design of Experiments, covering the use of  such experiments and the analysis of data arising from them. The statistical software R is fully integrated with the module. Students will reinforce their understanding of concepts of statistical modelling and develop their digital capabilities by using R to conduct analyses of real data sets from business, science and industry.

The module introduces the subject of (infinite dimensional) function spaces and shows how they are structured by metric, norm or inner product. The course naturally extends ideas contained in Real Analysis 1 and 2, and it sets in a wider context the orthogonal decompositions seen in the Fourier analysis part of MAT2011: Linear PDEs.

This module introduces fundamental concepts in analytical dynamics and illustrates their application to real-world problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations and symplectic manifolds. The module also leads to a deeper understanding of the role of symmetries and conservation laws in dynamical systems. This module builds on material from MAT1036 Classical Dynamics and lays the foundations for MAT3039 Quantum Mechanics.

This module introduces students to topological spaces and manifolds. Topology is the study of properties of spaces which are invariant under continuous transformations, and forms one of the cornerstones of pure mathematics. Study of topology leads to a deeper understanding of the concepts of continuity, connectedness and compactness. In the second part smooth manifolds are introduced, together with different construction methods, concepts of orientability and transversality. Finally, in the third part key elements of analysis on manifolds are discussed, such as tensors, differential forms, and the Stokes theorem in full generality.MAT2047 Curves and Surfaces is a recommended prior module, but is not pre-requisite. Students will also find the material in MAT3044 Riemannian Geometry complementary.

This module extends the abstract algebra introduced in MAT2048 Groups & Rings. A deeper insight into groups and rings is developed, and further algebraic structures are introduced. This module complements the module material in MATM035 Representation Theory and MATM011 Lie Algebras

This module provides an introduction to the applications of ordinary, delay and partial differential equations to ecology and epidemiology.

Bayesian Statistics is the branch of statistics that relies on subjective probability to create a wide range of statistical models. This module introduces Bayesian methodology and guides students to use prior to posterior analysis for modelling realistic problems. This module then tackles more difficult topics such as Bayesian point estimates, model selection and linear regression. 

Understanding the motions in our atmosphere and oceans is key to understanding, and hence solving, the global climate crisis. By applying a range of mathematical techniques to the governing fluid flow equations we are able to consider simplified models of a hugely complex system which will allow us to gain a deeper understanding about the transports of energy and heat on our planet.

The first half of this module covers various applications of statistical physics to model share prices and financial markets. This mathematics is then applied to calculating prices for some examples of financial derivatives. The second half of the module then focusses on optimization problems with examples including logistics, aerospace, traffic control and finance (which includes pricing, risk managements and portfolio optimizations in financial markets). There is a particular focus on the role of quantum optimization and the use of quantum computer algorithms in finance.

This module introduces fundamental concepts in Quantum Mechanics and its applications to real-world quantum problems. The module covers the mathematics of Hilbert spaces and Dirac notation, the postulates of Quantum Mechanics, the uncertainty principle, the Schroedinger equation with one-dimensional applications to a particle in a potential well and the quantum harmonic oscillator, and angular momentum and spin. This module utilises material from MAT1034 Linear Algebra and MAT2007 Ordinary Differential Equations. The module also builds on material from MAT1036 Classical Dynamics and MAT3008 Lagrangian & Hamiltonian Dynamics, although these modules are not pre-requisite.

Graph theory is an aesthetically appealing branch of pure mathematics with strong links to other areas of mathematics (combinatorics, algebra, topology, probability, optimisation and numerics) and well-developed applications to a wide range of other disciplines (including operations research, data science, chemistry, systems biology, statistical mechanics and quantum field theory). This module provides an introduction to graph theory. There is an emphasis on theorems and proofs.

Riemannian geometry is the study of geometric properties of spaces, called manifolds, typically in higher dimensions, which are described smoothly by sets of well-defined co-ordinates. The emphasis is on how mathematically to understand the notion of distance via a metric on such spaces, and how this is used to investigate key associated geometric concepts. Students are introduced to the key concepts of manifolds and metrics, with illustrative examples such as spheres, hyperbolic spaces and Lie groups. The module continues with geodesics, isometries, covariant derivatives, the Levi Civita connection, and concludes with curvature and its properties. Modules in Curves and Surfaces and in Manifolds and Topology contain related material.

Data science is the study of data to extract meaningful and actionable insights at all levels of society such as dynamical systems and social media networks. This module introduces the role of data in society and provides students with the underpinning mathematics that drives data methodology and algorithms. This module then covers wide-ranging topics with a focus on the Surrey brand of data, as research into data is part of the department research agenda. 

The module will introduce students to key concepts in education as well as the tools and strategies for effective public engagement in their discipline areas and for different audiences. Students will gain practical experience by planning, implementing, and evaluating an education or public engagement-based activity. Opportunities will also exist for classroom (e.g. school; early-year University) teaching experience and / or involvement in external exhibitions. The projects will be related to current Faculty, department and / or outreach needs thus leading to informed student partnership activities.

Time series are a collection of observations taken over time. This covers a great deal of situations such as stock markets, rainfall or even goals scored by a sports team. Features of the time series will lead to an appropriate choice of model. These models will be validated, and then can be used to forecast the future. Despite the modest pre-requisites of Level 4 Probability and Statistics (MAT1033), students will gain resourcefulness and resilience through learning mathematical proofs as well as gain digital capabilities through using R to conduct analyses of data sets and writing a report.

This module introduces the ideas of viscous fluid flows which build on the ideas some students would have seen in MAT2050: Inviscid Fluid Dynamics. By the end of the module, students should be familiar with the Navier-Stokes equations, and should be able to solve these equations in various simplified situations as well as in a variety of geometries.

This module is an introduction to classical and modern methods in the Calculus of Variations. Variational methods have many applications in Mathematics, Physics and Engineering. Examples will be included in the module to illustrate the theory of the Calculus of Variations in practice. This module complements module material in MAT3004 Introduction to Function Spaces, and MAT3008 Lagrangian & Hamiltonian Dynamics, but these modules are not pre-requisite.

Asset prices in financial markets go up and down in accordance with how markets digest the flow of information. To understand the causal relation between information flow and price movements, it is necessary to model market information and use this to infer the price dynamics. In this way, market dynamics can be replicated artificially on a computer. This module explains the powerful process of artificially generating realistic market models.   The module begins with elements of probability theory. We will then learn the idea of conditional expectation and the Bayes formula, which gives the optimal inference under uncertainty. The meaning of the Bayes formula will be explained, leading to the understanding of what is meant by “intelligence”.   The module then covers the basics of stochastic process (specifically, the Brownian motion) and calculus (specifically, the Ito calculus), sufficient to follow the contents of the module. Then simple models for flows of information in financial markets will be introduced, and by use of the Bayes formula the associated price dynamics will be derived.   The module concludes with a brief application to the asset valuation problems in financial markets (such as options or other derivatives). 

Mathematics underpinning real-world uncertain events has become indispensable in many applications, including in particular financial markets. This module will begin with the introduction to probability theory and stochastic processes, with an emphasis on the Ito calculus for treating functions of Brownian motion. Such functions are commonly used in financial markets to model asset price dynamics, required for the valuation of financial contracts.   The module then discusses structures of financial markets, with an emphasis on the equity market. Several of the standard and exotic contingent claims will be introduced, and the need for mathematical models for the valuation and risk management of these products will be explained.   The pricing of a standard call option will then be worked out in a single-period binomial model, for which option price will be worked out in two ways: first using the portfolio replication and no arbitrage argument, and second using the risk-neutral expectation argument.   The model is then extended into multi-period binomial tree model, leading to the Cox-Ross-Rubinstein option pricing formula.   Finally, a continuous-time geometric Brownian motion model, originally introduced by Samuelson, will be considered, and used to deduce the famous Black-Scholes option pricing formula. This can be applied for the purpose of both pricing, as well as risk-management purposes, which will be demonstrated by working out the hedging strategy. The meaning of the pricing formula, and how it can be used in practical investment banking context, will be explained.

Understanding the motions in our atmosphere and oceans is key to understanding, and hence solving, the global climate crisis. By applying a range of mathematical techniques to the governing fluid flow equations we are able to consider simplified models of a hugely complex system which will allow us to gain a deeper understanding about the transports of energy and heat on our planet.

This module provides an introduction to the applications of ordinary, delay and partial differential equations to ecology and epidemiology.

The module is an introduction to nonlinear partial differential equations (PDEs) with a focus on hyperbolic and dispersive PDEs. The module takes key classes of equations as the organising centre. Each class of PDEs is considered and the properties, analytical techniques, and analysis of each is taken in turn.

Group theory is a branch of mathematics developed to understand symmetries, which are powerful tools for understanding the properties of complex mathematical problems and physical systems. Group representations map abstract group elements to linear transformations on vector spaces. Representation theory allows us to better understand the properties of symmetry groups, and leads to powerful and compact solutions of otherwise difficult and intractable problems. This module builds on groups theory in MAT1031 Algebra and MAT2048 Groups & Rings.

This module introduces programming in Python for data science, with a focus on data pre-processing, data mining and analysis, machine learning and deep learning. Besides the practical hands-on experience with writing code, this course also covers the theoretical background on different data analysis techniques and machine learning approaches. The goal is to develop an understanding of how information can be extracted from data and how this information can be further used to make predictions, but importantly how this is done practically in terms of writing clear and transparent source code. Using real-world data sets and illustrative examples, this course will help to develop a theoretical understanding of data science as well as practical experience by developing useful software tools. Many of the techniques acquired through this module are likely to be of potential use in the dissertation project.

Data science is the study of data to extract meaningful and actionable insights at all levels of society such as dynamical systems and social media networks. This module introduces the role of data in society and provides students with the underpinning mathematics that drives data methodology and algorithms. This module then covers wide-ranging topics with a focus on the Surrey brand of data, as research into data is part of the department research agenda. 

This 15-credit M-Level module introduces important topics and techniques in theoretical physics that have a wide range of applications in many areas physics and engineering and which the students will not have met before. Both the mathematical techniques and their applications are covered at a level appropriate for Masters level students coming to the end of their degree and who should be able to pull many different ideas in theoretical physics together.

The presentations of the module will focus on data-driven methods for the analysis of dynamical systems and time-series data and on related machine learning problems such as dimensionality reduction, manifold learning, regression, and classification.  Python will be used to implement data-driven methods. The methods will then be applied to typical benchmark problems such as chaotic dynamical systems, metastable stochastic systems, and fluid dynamics problems, but also, for instance, to image classification problems to highlight similarities with classical supervised learning applications.

This module introduces a variety of commonly used techniques from Operations Research. The module leads to a deeper understanding of linear programming problems and the theory that underpins their solving. Tools such as the Simplex Method are presented and an introduction to nonlinear optimisation methods is also provided. This module supports and complements other modules where optimisation and constrained optimisation are considered.

Statistics is the science of collecting and analysing data from random events and modelling them using random variables. This module first recaps the properties of a single random variable as seen in Level 4 Probability and Statistics (MAT1033), then builds on this by considering multiple random variables simultaneously. This module also covers the transformation of random variables, as well as applications such as tail probabilities and limit theorems. 

The module provides an introduction to the theory and properties of curves and surfaces in Euclidean space. Basic concepts and tools from a branch of mathematics known as differential geometry are used to study curves and surfaces, and to understand their geometric properties. This module builds on material from MAT1043 Multivariable Calculus and MAT1034 Linear Algebra. The module is recommended for students intending to select the Year 3 modules MAT3009 Manifolds and Topology, and MAT3044 Riemannian Geometry, although it is not a pre-requisite for either module.

Stochastic processes are a series of random variables. Students will be introduced to both Markov Chains and continuous Markov stochastic processes, where the distributions of future random variables are determined by the value of the most recent random variable. These models are important for modelling things which change over time, such as voting intention or population size. 

This module provides an introduction to abstract algebra, focusing on the theory of algebraic structures called groups and rings. The module builds on preliminary material on groups introduced in MAT1031 Algebra. This module forms the starting point for subsequent algebraic modules, such as MAT3032 Advanced Algebra, MATM035 Representation Theory and MATM011 Lie Algebras.

This module introduces students to inviscid fluid flows including surface waves. By the end of the module, students should be able to recognise dominant features of fluid motion, and to derive some simple solutions of the equations of motion. Students should also have an appreciation of the force balances that produce various classes of flows.

Statistical modelling provides a means of extracting information from data, enabling informed decisions to be made. It is a versatile and powerful tool with widespread applications contributing to advancements in research, business, technology, and society. Students will be introduced to the basic concepts of statistical modelling via linear regression models, which are fundamental in statistics and serve as the basis for more complex modelling techniques. Model fitting, selection and evaluation are covered for: Simple Linear Regression; polynomial regression and multiple regression models. The module concludes with the introduction of simple models in the Design of Experiments, covering the use of  such experiments and the analysis of data arising from them. The statistical software R is fully integrated with the module. Students will reinforce their understanding of concepts of statistical modelling and develop their digital capabilities by using R to conduct analyses of real data sets from business, science and industry.

The module introduces the subject of (infinite dimensional) function spaces and shows how they are structured by metric, norm or inner product. The course naturally extends ideas contained in Real Analysis 1 and 2, and it sets in a wider context the orthogonal decompositions seen in the Fourier analysis part of MAT2011: Linear PDEs.

This module introduces fundamental concepts in analytical dynamics and illustrates their application to real-world problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations and symplectic manifolds. The module also leads to a deeper understanding of the role of symmetries and conservation laws in dynamical systems. This module builds on material from MAT1036 Classical Dynamics and lays the foundations for MAT3039 Quantum Mechanics.

This module introduces students to topological spaces and manifolds. Topology is the study of properties of spaces which are invariant under continuous transformations, and forms one of the cornerstones of pure mathematics. Study of topology leads to a deeper understanding of the concepts of continuity, connectedness and compactness. In the second part smooth manifolds are introduced, together with different construction methods, concepts of orientability and transversality. Finally, in the third part key elements of analysis on manifolds are discussed, such as tensors, differential forms, and the Stokes theorem in full generality.MAT2047 Curves and Surfaces is a recommended prior module, but is not pre-requisite. Students will also find the material in MAT3044 Riemannian Geometry complementary.

This module extends the abstract algebra introduced in MAT2048 Groups & Rings. A deeper insight into groups and rings is developed, and further algebraic structures are introduced. This module complements the module material in MATM035 Representation Theory and MATM011 Lie Algebras

This module provides an introduction to the applications of ordinary, delay and partial differential equations to ecology and epidemiology.

Bayesian Statistics is the branch of statistics that relies on subjective probability to create a wide range of statistical models. This module introduces Bayesian methodology and guides students to use prior to posterior analysis for modelling realistic problems. This module then tackles more difficult topics such as Bayesian point estimates, model selection and linear regression. 

Understanding the motions in our atmosphere and oceans is key to understanding, and hence solving, the global climate crisis. By applying a range of mathematical techniques to the governing fluid flow equations we are able to consider simplified models of a hugely complex system which will allow us to gain a deeper understanding about the transports of energy and heat on our planet.

The first half of this module covers various applications of statistical physics to model share prices and financial markets. This mathematics is then applied to calculating prices for some examples of financial derivatives. The second half of the module then focusses on optimization problems with examples including logistics, aerospace, traffic control and finance (which includes pricing, risk managements and portfolio optimizations in financial markets). There is a particular focus on the role of quantum optimization and the use of quantum computer algorithms in finance.

This module introduces fundamental concepts in Quantum Mechanics and its applications to real-world quantum problems. The module covers the mathematics of Hilbert spaces and Dirac notation, the postulates of Quantum Mechanics, the uncertainty principle, the Schroedinger equation with one-dimensional applications to a particle in a potential well and the quantum harmonic oscillator, and angular momentum and spin. This module utilises material from MAT1034 Linear Algebra and MAT2007 Ordinary Differential Equations. The module also builds on material from MAT1036 Classical Dynamics and MAT3008 Lagrangian & Hamiltonian Dynamics, although these modules are not pre-requisite.

Graph theory is an aesthetically appealing branch of pure mathematics with strong links to other areas of mathematics (combinatorics, algebra, topology, probability, optimisation and numerics) and well-developed applications to a wide range of other disciplines (including operations research, data science, chemistry, systems biology, statistical mechanics and quantum field theory). This module provides an introduction to graph theory. There is an emphasis on theorems and proofs.

Riemannian geometry is the study of geometric properties of spaces, called manifolds, typically in higher dimensions, which are described smoothly by sets of well-defined co-ordinates. The emphasis is on how mathematically to understand the notion of distance via a metric on such spaces, and how this is used to investigate key associated geometric concepts. Students are introduced to the key concepts of manifolds and metrics, with illustrative examples such as spheres, hyperbolic spaces and Lie groups. The module continues with geodesics, isometries, covariant derivatives, the Levi Civita connection, and concludes with curvature and its properties. Modules in Curves and Surfaces and in Manifolds and Topology contain related material.

Data science is the study of data to extract meaningful and actionable insights at all levels of society such as dynamical systems and social media networks. This module introduces the role of data in society and provides students with the underpinning mathematics that drives data methodology and algorithms. This module then covers wide-ranging topics with a focus on the Surrey brand of data, as research into data is part of the department research agenda. 

The module will introduce students to key concepts in education as well as the tools and strategies for effective public engagement in their discipline areas and for different audiences. Students will gain practical experience by planning, implementing, and evaluating an education or public engagement-based activity. Opportunities will also exist for classroom (e.g. school; early-year University) teaching experience and / or involvement in external exhibitions. The projects will be related to current Faculty, department and / or outreach needs thus leading to informed student partnership activities.

Time series are a collection of observations taken over time. This covers a great deal of situations such as stock markets, rainfall or even goals scored by a sports team. Features of the time series will lead to an appropriate choice of model. These models will be validated, and then can be used to forecast the future. Despite the modest pre-requisites of Level 4 Probability and Statistics (MAT1033), students will gain resourcefulness and resilience through learning mathematical proofs as well as gain digital capabilities through using R to conduct analyses of data sets and writing a report.

This module introduces the ideas of viscous fluid flows which build on the ideas some students would have seen in MAT2050: Inviscid Fluid Dynamics. By the end of the module, students should be familiar with the Navier-Stokes equations, and should be able to solve these equations in various simplified situations as well as in a variety of geometries.

This module is an introduction to classical and modern methods in the Calculus of Variations. Variational methods have many applications in Mathematics, Physics and Engineering. Examples will be included in the module to illustrate the theory of the Calculus of Variations in practice. This module complements module material in MAT3004 Introduction to Function Spaces, and MAT3008 Lagrangian & Hamiltonian Dynamics, but these modules are not pre-requisite.

Asset prices in financial markets go up and down in accordance with how markets digest the flow of information. To understand the causal relation between information flow and price movements, it is necessary to model market information and use this to infer the price dynamics. In this way, market dynamics can be replicated artificially on a computer. This module explains the powerful process of artificially generating realistic market models.   The module begins with elements of probability theory. We will then learn the idea of conditional expectation and the Bayes formula, which gives the optimal inference under uncertainty. The meaning of the Bayes formula will be explained, leading to the understanding of what is meant by “intelligence”.   The module then covers the basics of stochastic process (specifically, the Brownian motion) and calculus (specifically, the Ito calculus), sufficient to follow the contents of the module. Then simple models for flows of information in financial markets will be introduced, and by use of the Bayes formula the associated price dynamics will be derived.   The module concludes with a brief application to the asset valuation problems in financial markets (such as options or other derivatives). 

Mathematics underpinning real-world uncertain events has become indispensable in many applications, including in particular financial markets. This module will begin with the introduction to probability theory and stochastic processes, with an emphasis on the Ito calculus for treating functions of Brownian motion. Such functions are commonly used in financial markets to model asset price dynamics, required for the valuation of financial contracts.   The module then discusses structures of financial markets, with an emphasis on the equity market. Several of the standard and exotic contingent claims will be introduced, and the need for mathematical models for the valuation and risk management of these products will be explained.   The pricing of a standard call option will then be worked out in a single-period binomial model, for which option price will be worked out in two ways: first using the portfolio replication and no arbitrage argument, and second using the risk-neutral expectation argument.   The model is then extended into multi-period binomial tree model, leading to the Cox-Ross-Rubinstein option pricing formula.   Finally, a continuous-time geometric Brownian motion model, originally introduced by Samuelson, will be considered, and used to deduce the famous Black-Scholes option pricing formula. This can be applied for the purpose of both pricing, as well as risk-management purposes, which will be demonstrated by working out the hedging strategy. The meaning of the pricing formula, and how it can be used in practical investment banking context, will be explained.

Understanding the motions in our atmosphere and oceans is key to understanding, and hence solving, the global climate crisis. By applying a range of mathematical techniques to the governing fluid flow equations we are able to consider simplified models of a hugely complex system which will allow us to gain a deeper understanding about the transports of energy and heat on our planet.

This module provides an introduction to the applications of ordinary, delay and partial differential equations to ecology and epidemiology.

The module is an introduction to nonlinear partial differential equations (PDEs) with a focus on hyperbolic and dispersive PDEs. The module takes key classes of equations as the organising centre. Each class of PDEs is considered and the properties, analytical techniques, and analysis of each is taken in turn.

Group theory is a branch of mathematics developed to understand symmetries, which are powerful tools for understanding the properties of complex mathematical problems and physical systems. Group representations map abstract group elements to linear transformations on vector spaces. Representation theory allows us to better understand the properties of symmetry groups, and leads to powerful and compact solutions of otherwise difficult and intractable problems. This module builds on groups theory in MAT1031 Algebra and MAT2048 Groups & Rings.

This module introduces programming in Python for data science, with a focus on data pre-processing, data mining and analysis, machine learning and deep learning. Besides the practical hands-on experience with writing code, this course also covers the theoretical background on different data analysis techniques and machine learning approaches. The goal is to develop an understanding of how information can be extracted from data and how this information can be further used to make predictions, but importantly how this is done practically in terms of writing clear and transparent source code. Using real-world data sets and illustrative examples, this course will help to develop a theoretical understanding of data science as well as practical experience by developing useful software tools. Many of the techniques acquired through this module are likely to be of potential use in the dissertation project.

Data science is the study of data to extract meaningful and actionable insights at all levels of society such as dynamical systems and social media networks. This module introduces the role of data in society and provides students with the underpinning mathematics that drives data methodology and algorithms. This module then covers wide-ranging topics with a focus on the Surrey brand of data, as research into data is part of the department research agenda. 

This 15-credit M-Level module introduces important topics and techniques in theoretical physics that have a wide range of applications in many areas physics and engineering and which the students will not have met before. Both the mathematical techniques and their applications are covered at a level appropriate for Masters level students coming to the end of their degree and who should be able to pull many different ideas in theoretical physics together.

The presentations of the module will focus on data-driven methods for the analysis of dynamical systems and time-series data and on related machine learning problems such as dimensionality reduction, manifold learning, regression, and classification.  Python will be used to implement data-driven methods. The methods will then be applied to typical benchmark problems such as chaotic dynamical systems, metastable stochastic systems, and fluid dynamics problems, but also, for instance, to image classification problems to highlight similarities with classical supervised learning applications.