BSc (Hons), MMath or MPhys — 2025 entry Mathematics and Physics

This module presents a complete introduction to modern astronomy and astrophysics. We start by introducing astronomy as a unique science; measurement in astronomy (position on the sky, velocity and distance); and multi-messenger probes. We then discuss acceleration due to gravity (the dominant force in the Universe) and we show that timescales are typically so long that we must use models to calculate what happened in the past and what will happen in the future. Armed with an understanding of how to make measurements in astronomy and how to model gravity, we then move from the smallest scales to the largest, studying first the interstellar medium from which stars form; stellar structure and evolution; and stellar remnants (white dwarfs, neutron stars and black holes). We compare and contrast the latest models for how planets form and we discuss the prospects for detecting life and intelligent life beyond Earth. Finally, we discuss the formation and evolution of galaxies in the Universe and the Universe as a whole. We show that the Universe appears to be mostly dark: dark matter (~22%) and dark energy (~74%). Understanding what these mysterious components are is one of the key challenges for physics in the next decade. The module includes either a computer or telescope project. Students enrolled in physics with astronomy with undertake the telescope project where (weather willing) students will gain hands on experience of taking real observational data. They will develop software tools to analyse and interpret these data and write up a final report. Students not enrolled in physics with astronomy will undertake a computational project of similar scope and length, also culminating in a final report.

The Linear PDEs Module introduces students to linear partial differential equations, mainly in one and two space dimensions. The classical linear PDEs such as the heat, wave and Laplace equation will be analyzed in detail. Questions of existence and uniqueness of solutions will be addressed and their physical and mathematical meaning emphasized.

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.

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.

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 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.

In this module, students will learn key methods adopted in astrophysics to carry out advanced research: scientific computing, statistics, data analysis, machine learning. Much of the course develops highly transferrable skills that apply to science research in general. The goal is to ensure that students are well-prepared for either their research year or their future careers.

This module introduces six advanced topics in physics. Students will be assessed on only four of these topics, with individuals self-selecting the contact sessions, coursework options and exam questions that reflect their preference. An indicative list of these topics include: biological physics, special relativity, particle physics, cosmology, nuclear astrophysics, and quantum computing.

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.

Quantum computers are built around a central processing unit which is a physical device operating according to the rules of quantum mechanics. This module teaches the principles of how quantum processors are built and how they function. The module follows the superconducting architecture, which currently has a significant role in the industry and has been widely studied and implemented. The module will start with reviewing the basic physics required to understand superconducting qubits and the theory of superconducting circuits. It will then continue to the operation of such qubits to perform gates, storage, and readout. After studying the basic building blocks and operations, the module will discuss early demonstrations of important algorithms on superconducting processors. These will lead us naturally to the important topics of improving performance through protection of coherence, advanced control, and validation. The last part of the module will focus on state-of-the-art superconducting processors, focusing on the various challenges in scaling up and the strategies that the industry is pursuing to overcome them.

The module will introduce students to research level equipment and techniques that are used within the research groups of Physics via extended projects. The four-week projects will cover astrophysics, experimental and theoretical nuclear physics, experimental and theoretical soft matter physics, and quantum technologies. Students will gain experience using state-of-the-art equipment and software, analysing and working with large data sets and in problem solving. The module builds upon experience gained during first- and second-year laboratory and computing classes with project-based work that is typically more open ended and less structured. Students are expected to take more responsibility for the planning and direction of work than in previous years. The goal is to help prepare students for independent research within a team and for future project work (e.g. Final Year Projects and MPhys research years). Numbers will be limited on certain projects.

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.

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.

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.   

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.

This 30 credit project module is designed to give students the opportunity to explore an area of interest in physics in some depth, either through experimental, theoretical or computational means, or in the form of a literature survey.  The module also develops generic professional skills such as teamwork, scientific writing and professional ethics.  The module involves writing a dissertation and an oral assessment  

The course will follow the historical development of the main medical imaging techniques.The first part will consider, from a theoretical perspective, the fundamentals of X-ray image formation both in the planar modality and in the Computed Tomography modality. Elements of image processing and image reconstruction will be addressed.The second will look at the physical principles and methods of Nuclear Medicine.The third will look at the principles underlying the application of diagnostic ultrasound in medicine.The fourth will consider Magnetic Resonance Imaging (MRI), one of the most important techniques of medical imaging used in hospitals today.In parallel to the related theoretical classes, students will do X-ray imaging modelling sessions and ultrasound practicals.

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 aims to provide an advanced level understanding of the physics of stars and nuclear astrophysics. In particular, the course will provide an understanding of advanced nucleosynthetic pathways, an analytical underpinning of resonant reaction rates, together with the experimental techniques involved in their determination.

This module covers circuit-based quantum computers and some of the algorithms that can be implemented using them. The idea of a classical computational algorithm and its complexity is introduced, and used as a language to discuss quantum computational algorithms. Components of quantum circuits are discussed individually, and then built up to show how algorithms such as the quantum Fourier transform and Grover's search algorithm can be implemented, and how such algorithms can be used in applications such as factorizing integers using Shor's algorithm. A suitable computational framework is introduced and used as a tool to implement the algorithms on simulated quantum computers.

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.

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.

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 presents a complete introduction to modern astronomy and astrophysics. We start by introducing astronomy as a unique science; measurement in astronomy (position on the sky, velocity and distance); and multi-messenger probes. We then discuss acceleration due to gravity (the dominant force in the Universe) and we show that timescales are typically so long that we must use models to calculate what happened in the past and what will happen in the future. Armed with an understanding of how to make measurements in astronomy and how to model gravity, we then move from the smallest scales to the largest, studying first the interstellar medium from which stars form; stellar structure and evolution; and stellar remnants (white dwarfs, neutron stars and black holes). We compare and contrast the latest models for how planets form and we discuss the prospects for detecting life and intelligent life beyond Earth. Finally, we discuss the formation and evolution of galaxies in the Universe and the Universe as a whole. We show that the Universe appears to be mostly dark: dark matter (~22%) and dark energy (~74%). Understanding what these mysterious components are is one of the key challenges for physics in the next decade. The module includes either a computer or telescope project. Students enrolled in physics with astronomy with undertake the telescope project where (weather willing) students will gain hands on experience of taking real observational data. They will develop software tools to analyse and interpret these data and write up a final report. Students not enrolled in physics with astronomy will undertake a computational project of similar scope and length, also culminating in a final report.

The Linear PDEs Module introduces students to linear partial differential equations, mainly in one and two space dimensions. The classical linear PDEs such as the heat, wave and Laplace equation will be analyzed in detail. Questions of existence and uniqueness of solutions will be addressed and their physical and mathematical meaning emphasized.

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.

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.

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 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.

In this module, students will learn key methods adopted in astrophysics to carry out advanced research: scientific computing, statistics, data analysis, machine learning. Much of the course develops highly transferrable skills that apply to science research in general. The goal is to ensure that students are well-prepared for either their research year or their future careers.

This module introduces six advanced topics in physics. Students will be assessed on only four of these topics, with individuals self-selecting the contact sessions, coursework options and exam questions that reflect their preference. An indicative list of these topics include: biological physics, special relativity, particle physics, cosmology, nuclear astrophysics, and quantum computing.

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.

Quantum computers are built around a central processing unit which is a physical device operating according to the rules of quantum mechanics. This module teaches the principles of how quantum processors are built and how they function. The module follows the superconducting architecture, which currently has a significant role in the industry and has been widely studied and implemented. The module will start with reviewing the basic physics required to understand superconducting qubits and the theory of superconducting circuits. It will then continue to the operation of such qubits to perform gates, storage, and readout. After studying the basic building blocks and operations, the module will discuss early demonstrations of important algorithms on superconducting processors. These will lead us naturally to the important topics of improving performance through protection of coherence, advanced control, and validation. The last part of the module will focus on state-of-the-art superconducting processors, focusing on the various challenges in scaling up and the strategies that the industry is pursuing to overcome them.

The module will introduce students to research level equipment and techniques that are used within the research groups of Physics via extended projects. The four-week projects will cover astrophysics, experimental and theoretical nuclear physics, experimental and theoretical soft matter physics, and quantum technologies. Students will gain experience using state-of-the-art equipment and software, analysing and working with large data sets and in problem solving. The module builds upon experience gained during first- and second-year laboratory and computing classes with project-based work that is typically more open ended and less structured. Students are expected to take more responsibility for the planning and direction of work than in previous years. The goal is to help prepare students for independent research within a team and for future project work (e.g. Final Year Projects and MPhys research years). Numbers will be limited on certain projects.

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.

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.

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.   

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.

This 30 credit project module is designed to give students the opportunity to explore an area of interest in physics in some depth, either through experimental, theoretical or computational means, or in the form of a literature survey.  The module also develops generic professional skills such as teamwork, scientific writing and professional ethics.  The module involves writing a dissertation and an oral assessment  

The course will follow the historical development of the main medical imaging techniques.The first part will consider, from a theoretical perspective, the fundamentals of X-ray image formation both in the planar modality and in the Computed Tomography modality. Elements of image processing and image reconstruction will be addressed.The second will look at the physical principles and methods of Nuclear Medicine.The third will look at the principles underlying the application of diagnostic ultrasound in medicine.The fourth will consider Magnetic Resonance Imaging (MRI), one of the most important techniques of medical imaging used in hospitals today.In parallel to the related theoretical classes, students will do X-ray imaging modelling sessions and ultrasound practicals.

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 aims to provide an advanced level understanding of the physics of stars and nuclear astrophysics. In particular, the course will provide an understanding of advanced nucleosynthetic pathways, an analytical underpinning of resonant reaction rates, together with the experimental techniques involved in their determination.

This module covers circuit-based quantum computers and some of the algorithms that can be implemented using them. The idea of a classical computational algorithm and its complexity is introduced, and used as a language to discuss quantum computational algorithms. Components of quantum circuits are discussed individually, and then built up to show how algorithms such as the quantum Fourier transform and Grover's search algorithm can be implemented, and how such algorithms can be used in applications such as factorizing integers using Shor's algorithm. A suitable computational framework is introduced and used as a tool to implement the algorithms on simulated quantum computers.

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.

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.

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 presents a complete introduction to modern astronomy and astrophysics. We start by introducing astronomy as a unique science; measurement in astronomy (position on the sky, velocity and distance); and multi-messenger probes. We then discuss acceleration due to gravity (the dominant force in the Universe) and we show that timescales are typically so long that we must use models to calculate what happened in the past and what will happen in the future. Armed with an understanding of how to make measurements in astronomy and how to model gravity, we then move from the smallest scales to the largest, studying first the interstellar medium from which stars form; stellar structure and evolution; and stellar remnants (white dwarfs, neutron stars and black holes). We compare and contrast the latest models for how planets form and we discuss the prospects for detecting life and intelligent life beyond Earth. Finally, we discuss the formation and evolution of galaxies in the Universe and the Universe as a whole. We show that the Universe appears to be mostly dark: dark matter (~22%) and dark energy (~74%). Understanding what these mysterious components are is one of the key challenges for physics in the next decade. The module includes either a computer or telescope project. Students enrolled in physics with astronomy with undertake the telescope project where (weather willing) students will gain hands on experience of taking real observational data. They will develop software tools to analyse and interpret these data and write up a final report. Students not enrolled in physics with astronomy will undertake a computational project of similar scope and length, also culminating in a final report.

The Linear PDEs Module introduces students to linear partial differential equations, mainly in one and two space dimensions. The classical linear PDEs such as the heat, wave and Laplace equation will be analyzed in detail. Questions of existence and uniqueness of solutions will be addressed and their physical and mathematical meaning emphasized.

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.

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 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.

In this module, students will learn key methods adopted in astrophysics to carry out advanced research: scientific computing, statistics, data analysis, machine learning. Much of the course develops highly transferrable skills that apply to science research in general. The goal is to ensure that students are well-prepared for either their research year or their future careers.

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 six advanced topics in physics. Students will be assessed on only four of these topics, with individuals self-selecting the contact sessions, coursework options and exam questions that reflect their preference. An indicative list of these topics include: biological physics, special relativity, particle physics, cosmology, nuclear astrophysics, and quantum computing.

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.

Quantum computers are built around a central processing unit which is a physical device operating according to the rules of quantum mechanics. This module teaches the principles of how quantum processors are built and how they function. The module follows the superconducting architecture, which currently has a significant role in the industry and has been widely studied and implemented. The module will start with reviewing the basic physics required to understand superconducting qubits and the theory of superconducting circuits. It will then continue to the operation of such qubits to perform gates, storage, and readout. After studying the basic building blocks and operations, the module will discuss early demonstrations of important algorithms on superconducting processors. These will lead us naturally to the important topics of improving performance through protection of coherence, advanced control, and validation. The last part of the module will focus on state-of-the-art superconducting processors, focusing on the various challenges in scaling up and the strategies that the industry is pursuing to overcome them.

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.

The module will introduce students to research level equipment and techniques that are used within the research groups of Physics via extended projects. The four-week projects will cover astrophysics, experimental and theoretical nuclear physics, experimental and theoretical soft matter physics, and quantum technologies. Students will gain experience using state-of-the-art equipment and software, analysing and working with large data sets and in problem solving. The module builds upon experience gained during first- and second-year laboratory and computing classes with project-based work that is typically more open ended and less structured. Students are expected to take more responsibility for the planning and direction of work than in previous years. The goal is to help prepare students for independent research within a team and for future project work (e.g. Final Year Projects and MPhys research years). Numbers will be limited on certain projects.

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.

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.

The course will follow the historical development of the main medical imaging techniques.The first part will consider, from a theoretical perspective, the fundamentals of X-ray image formation both in the planar modality and in the Computed Tomography modality. Elements of image processing and image reconstruction will be addressed.The second will look at the physical principles and methods of Nuclear Medicine.The third will look at the principles underlying the application of diagnostic ultrasound in medicine.The fourth will consider Magnetic Resonance Imaging (MRI), one of the most important techniques of medical imaging used in hospitals today.In parallel to the related theoretical classes, students will do X-ray imaging modelling sessions and ultrasound practicals.

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 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.

This module covers circuit-based quantum computers and some of the algorithms that can be implemented using them. The idea of a classical computational algorithm and its complexity is introduced, and used as a language to discuss quantum computational algorithms. Components of quantum circuits are discussed individually, and then built up to show how algorithms such as the quantum Fourier transform and Grover's search algorithm can be implemented, and how such algorithms can be used in applications such as factorizing integers using Shor's algorithm. A suitable computational framework is introduced and used as a tool to implement the algorithms on simulated quantum computers.

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.

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.

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.

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

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 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.

This module aims to provide an advanced level understanding of explosive nuclear astrophysics and the physics of stars. In particular, the course will provide an analytical underpinning of resonant reaction rates, together with the experimental techniques involved in their determination, as well as a theoretical treatment of nuclear reactions and celestial objects.

This module will introduce the students to the principles and formalism of General Relativity and its applications to Black Holes and astrophysical phenomena.

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.

Quantum Mechanics topics are essential building blocks for our understanding of many physical systems. The module assumes basic knowledge in quantum mechanics from modules in earlier years, but will provide a review at the beginning. Topics include a review of quantum mechanics, operator methods and applications to the harmonic oscillator, spin & angular momentum, symmetries in quantum mechanics. The second half of the module then moves beyond isolated quantum systems by addressing the topic of Quantum Entanglement and Quantum Coherence. Here the important concepts of entanglement and decoherence will be studied by developing an understanding of open quantum systems and the density matrix formalism.

This module comprises two independent halves, on quantum simulations (Q Sim), and quantum biology (Q Bio).Quantum Simulation. The quantum simulation part of this module introduces students to the use of quantum computers in the simulation of physical systems using mapping of Hamiltonians from standard quantum mechanics to a representation suitable for application on quantum computers, along with a study of wavefunction ansatz design, algorithms, and error mitigation and correction.Quantum biology is the study of how quantum mechanical phenomena, such as quantum superposition, tunnelling, and entanglement, can be exploited by living systems to provide evolutionary and/or biological advantages. This half module will cover a range of biological molecules and processes that may exploit quantum effects, such as magnetoreception, photosynthetic light harvesting, and DNA mutations. The goal of this half module is to develop an understanding of how quantum processes could have an impact in nature and how this knowledge can be further used for applications in health and medical sciences. In addition, using research progress in the field of quantum biology and illustrative examples, this course will help to develop an experimental and theoretical understanding of how quantum processes may play a crucial role in maintaining the non-equilibrium state of the biomolecular systems. Many of the subjects acquired through this half module are likely to be of potential use in future project work,

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.

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.

The course provides an introduction to nuclear energy generation and applications of nuclear science. Nuclear reactors, their physics and operation are described. Nuclear reactor safety case work is also discussed. Future potential energy generation mechanisms such including nuclear fusion will be discussed. The module will also present a range of applications of radioactivity measurement including aspects of Environmental Science and Medical Diagnogstics and treatment therapy. The module will include some aspects of calculus and first order differential equations.

The Linear PDEs Module introduces students to linear partial differential equations, mainly in one and two space dimensions. The classical linear PDEs such as the heat, wave and Laplace equation will be analyzed in detail. Questions of existence and uniqueness of solutions will be addressed and their physical and mathematical meaning emphasized.

This module presents a complete introduction to modern astronomy and astrophysics. We start by introducing astronomy as a unique science; measurement in astronomy (position on the sky, velocity and distance); and multi-messenger probes. We then discuss acceleration due to gravity (the dominant force in the Universe) and we show that timescales are typically so long that we must use models to calculate what happened in the past and what will happen in the future. Armed with an understanding of how to make measurements in astronomy and how to model gravity, we then move from the smallest scales to the largest, studying first the interstellar medium from which stars form; stellar structure and evolution; and stellar remnants (white dwarfs, neutron stars and black holes). We compare and contrast the latest models for how planets form and we discuss the prospects for detecting life and intelligent life beyond Earth. Finally, we discuss the formation and evolution of galaxies in the Universe and the Universe as a whole. We show that the Universe appears to be mostly dark: dark matter (~22%) and dark energy (~74%). Understanding what these mysterious components are is one of the key challenges for physics in the next decade. The module includes either a computer or telescope project. Students enrolled in physics with astronomy with undertake the telescope project where (weather willing) students will gain hands on experience of taking real observational data. They will develop software tools to analyse and interpret these data and write up a final report. Students not enrolled in physics with astronomy will undertake a computational project of similar scope and length, also culminating in a final report.

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.

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 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

In this module, students will learn key methods adopted in astrophysics to carry out advanced research: scientific computing, statistics, data analysis, machine learning. Much of the course develops highly transferrable skills that apply to science research in general. The goal is to ensure that students are well-prepared for either their research year or their future careers.

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

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 six advanced topics in physics. Students will be assessed on only four of these topics, with individuals self-selecting the contact sessions, coursework options and exam questions that reflect their preference. An indicative list of these topics include: biological physics, special relativity, particle physics, cosmology, nuclear astrophysics, and quantum computing.

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.

Quantum computers are built around a central processing unit which is a physical device operating according to the rules of quantum mechanics. This module teaches the principles of how quantum processors are built and how they function. The module follows the superconducting architecture, which currently has a significant role in the industry and has been widely studied and implemented. The module will start with reviewing the basic physics required to understand superconducting qubits and the theory of superconducting circuits. It will then continue to the operation of such qubits to perform gates, storage, and readout. After studying the basic building blocks and operations, the module will discuss early demonstrations of important algorithms on superconducting processors. These will lead us naturally to the important topics of improving performance through protection of coherence, advanced control, and validation. The last part of the module will focus on state-of-the-art superconducting processors, focusing on the various challenges in scaling up and the strategies that the industry is pursuing to overcome them.

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.

The module will introduce students to research level equipment and techniques that are used within the research groups of Physics via extended projects. The four-week projects will cover astrophysics, experimental and theoretical nuclear physics, experimental and theoretical soft matter physics, and quantum technologies. Students will gain experience using state-of-the-art equipment and software, analysing and working with large data sets and in problem solving. The module builds upon experience gained during first- and second-year laboratory and computing classes with project-based work that is typically more open ended and less structured. Students are expected to take more responsibility for the planning and direction of work than in previous years. The goal is to help prepare students for independent research within a team and for future project work (e.g. Final Year Projects and MPhys research years). Numbers will be limited on certain projects.

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.

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.

The course will follow the historical development of the main medical imaging techniques.The first part will consider, from a theoretical perspective, the fundamentals of X-ray image formation both in the planar modality and in the Computed Tomography modality. Elements of image processing and image reconstruction will be addressed.The second will look at the physical principles and methods of Nuclear Medicine.The third will look at the principles underlying the application of diagnostic ultrasound in medicine.The fourth will consider Magnetic Resonance Imaging (MRI), one of the most important techniques of medical imaging used in hospitals today.In parallel to the related theoretical classes, students will do X-ray imaging modelling sessions and ultrasound practicals.

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 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.

This module covers circuit-based quantum computers and some of the algorithms that can be implemented using them. The idea of a classical computational algorithm and its complexity is introduced, and used as a language to discuss quantum computational algorithms. Components of quantum circuits are discussed individually, and then built up to show how algorithms such as the quantum Fourier transform and Grover's search algorithm can be implemented, and how such algorithms can be used in applications such as factorizing integers using Shor's algorithm. A suitable computational framework is introduced and used as a tool to implement the algorithms on simulated quantum computers.

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.

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.

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.

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

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 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.

This module aims to provide an advanced level understanding of explosive nuclear astrophysics and the physics of stars. In particular, the course will provide an analytical underpinning of resonant reaction rates, together with the experimental techniques involved in their determination, as well as a theoretical treatment of nuclear reactions and celestial objects.

This module will introduce the students to the principles and formalism of General Relativity and its applications to Black Holes and astrophysical phenomena.

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.

Quantum Mechanics topics are essential building blocks for our understanding of many physical systems. The module assumes basic knowledge in quantum mechanics from modules in earlier years, but will provide a review at the beginning. Topics include a review of quantum mechanics, operator methods and applications to the harmonic oscillator, spin & angular momentum, symmetries in quantum mechanics. The second half of the module then moves beyond isolated quantum systems by addressing the topic of Quantum Entanglement and Quantum Coherence. Here the important concepts of entanglement and decoherence will be studied by developing an understanding of open quantum systems and the density matrix formalism.

This module comprises two independent halves, on quantum simulations (Q Sim), and quantum biology (Q Bio).Quantum Simulation. The quantum simulation part of this module introduces students to the use of quantum computers in the simulation of physical systems using mapping of Hamiltonians from standard quantum mechanics to a representation suitable for application on quantum computers, along with a study of wavefunction ansatz design, algorithms, and error mitigation and correction.Quantum biology is the study of how quantum mechanical phenomena, such as quantum superposition, tunnelling, and entanglement, can be exploited by living systems to provide evolutionary and/or biological advantages. This half module will cover a range of biological molecules and processes that may exploit quantum effects, such as magnetoreception, photosynthetic light harvesting, and DNA mutations. The goal of this half module is to develop an understanding of how quantum processes could have an impact in nature and how this knowledge can be further used for applications in health and medical sciences. In addition, using research progress in the field of quantum biology and illustrative examples, this course will help to develop an experimental and theoretical understanding of how quantum processes may play a crucial role in maintaining the non-equilibrium state of the biomolecular systems. Many of the subjects acquired through this half module are likely to be of potential use in future project work,

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.

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.

The course provides an introduction to nuclear energy generation and applications of nuclear science. Nuclear reactors, their physics and operation are described. Nuclear reactor safety case work is also discussed. Future potential energy generation mechanisms such including nuclear fusion will be discussed. The module will also present a range of applications of radioactivity measurement including aspects of Environmental Science and Medical Diagnogstics and treatment therapy. The module will include some aspects of calculus and first order differential equations.

The Linear PDEs Module introduces students to linear partial differential equations, mainly in one and two space dimensions. The classical linear PDEs such as the heat, wave and Laplace equation will be analyzed in detail. Questions of existence and uniqueness of solutions will be addressed and their physical and mathematical meaning emphasized.

This module presents a complete introduction to modern astronomy and astrophysics. We start by introducing astronomy as a unique science; measurement in astronomy (position on the sky, velocity and distance); and multi-messenger probes. We then discuss acceleration due to gravity (the dominant force in the Universe) and we show that timescales are typically so long that we must use models to calculate what happened in the past and what will happen in the future. Armed with an understanding of how to make measurements in astronomy and how to model gravity, we then move from the smallest scales to the largest, studying first the interstellar medium from which stars form; stellar structure and evolution; and stellar remnants (white dwarfs, neutron stars and black holes). We compare and contrast the latest models for how planets form and we discuss the prospects for detecting life and intelligent life beyond Earth. Finally, we discuss the formation and evolution of galaxies in the Universe and the Universe as a whole. We show that the Universe appears to be mostly dark: dark matter (~22%) and dark energy (~74%). Understanding what these mysterious components are is one of the key challenges for physics in the next decade. The module includes either a computer or telescope project. Students enrolled in physics with astronomy with undertake the telescope project where (weather willing) students will gain hands on experience of taking real observational data. They will develop software tools to analyse and interpret these data and write up a final report. Students not enrolled in physics with astronomy will undertake a computational project of similar scope and length, also culminating in a final report.

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.

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 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.

In this module, students will learn key methods adopted in astrophysics to carry out advanced research: scientific computing, statistics, data analysis, machine learning. Much of the course develops highly transferrable skills that apply to science research in general. The goal is to ensure that students are well-prepared for either their research year or their future careers.

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 six advanced topics in physics. Students will be assessed on only four of these topics, with individuals self-selecting the contact sessions, coursework options and exam questions that reflect their preference. An indicative list of these topics include: biological physics, special relativity, particle physics, cosmology, nuclear astrophysics, and quantum computing.

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.

Quantum computers are built around a central processing unit which is a physical device operating according to the rules of quantum mechanics. This module teaches the principles of how quantum processors are built and how they function. The module follows the superconducting architecture, which currently has a significant role in the industry and has been widely studied and implemented. The module will start with reviewing the basic physics required to understand superconducting qubits and the theory of superconducting circuits. It will then continue to the operation of such qubits to perform gates, storage, and readout. After studying the basic building blocks and operations, the module will discuss early demonstrations of important algorithms on superconducting processors. These will lead us naturally to the important topics of improving performance through protection of coherence, advanced control, and validation. The last part of the module will focus on state-of-the-art superconducting processors, focusing on the various challenges in scaling up and the strategies that the industry is pursuing to overcome them.

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.

The module will introduce students to research level equipment and techniques that are used within the research groups of Physics via extended projects. The four-week projects will cover astrophysics, experimental and theoretical nuclear physics, experimental and theoretical soft matter physics, and quantum technologies. Students will gain experience using state-of-the-art equipment and software, analysing and working with large data sets and in problem solving. The module builds upon experience gained during first- and second-year laboratory and computing classes with project-based work that is typically more open ended and less structured. Students are expected to take more responsibility for the planning and direction of work than in previous years. The goal is to help prepare students for independent research within a team and for future project work (e.g. Final Year Projects and MPhys research years). Numbers will be limited on certain projects.

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.

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.

The course will follow the historical development of the main medical imaging techniques.The first part will consider, from a theoretical perspective, the fundamentals of X-ray image formation both in the planar modality and in the Computed Tomography modality. Elements of image processing and image reconstruction will be addressed.The second will look at the physical principles and methods of Nuclear Medicine.The third will look at the principles underlying the application of diagnostic ultrasound in medicine.The fourth will consider Magnetic Resonance Imaging (MRI), one of the most important techniques of medical imaging used in hospitals today.In parallel to the related theoretical classes, students will do X-ray imaging modelling sessions and ultrasound practicals.

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 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.

This module covers circuit-based quantum computers and some of the algorithms that can be implemented using them. The idea of a classical computational algorithm and its complexity is introduced, and used as a language to discuss quantum computational algorithms. Components of quantum circuits are discussed individually, and then built up to show how algorithms such as the quantum Fourier transform and Grover's search algorithm can be implemented, and how such algorithms can be used in applications such as factorizing integers using Shor's algorithm. A suitable computational framework is introduced and used as a tool to implement the algorithms on simulated quantum computers.

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.

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.

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.

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

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 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.

This module aims to provide an advanced level understanding of explosive nuclear astrophysics and the physics of stars. In particular, the course will provide an analytical underpinning of resonant reaction rates, together with the experimental techniques involved in their determination, as well as a theoretical treatment of nuclear reactions and celestial objects.

This module will introduce the students to the principles and formalism of General Relativity and its applications to Black Holes and astrophysical phenomena.

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.

Quantum Mechanics topics are essential building blocks for our understanding of many physical systems. The module assumes basic knowledge in quantum mechanics from modules in earlier years, but will provide a review at the beginning. Topics include a review of quantum mechanics, operator methods and applications to the harmonic oscillator, spin & angular momentum, symmetries in quantum mechanics. The second half of the module then moves beyond isolated quantum systems by addressing the topic of Quantum Entanglement and Quantum Coherence. Here the important concepts of entanglement and decoherence will be studied by developing an understanding of open quantum systems and the density matrix formalism.

This module comprises two independent halves, on quantum simulations (Q Sim), and quantum biology (Q Bio).Quantum Simulation. The quantum simulation part of this module introduces students to the use of quantum computers in the simulation of physical systems using mapping of Hamiltonians from standard quantum mechanics to a representation suitable for application on quantum computers, along with a study of wavefunction ansatz design, algorithms, and error mitigation and correction.Quantum biology is the study of how quantum mechanical phenomena, such as quantum superposition, tunnelling, and entanglement, can be exploited by living systems to provide evolutionary and/or biological advantages. This half module will cover a range of biological molecules and processes that may exploit quantum effects, such as magnetoreception, photosynthetic light harvesting, and DNA mutations. The goal of this half module is to develop an understanding of how quantum processes could have an impact in nature and how this knowledge can be further used for applications in health and medical sciences. In addition, using research progress in the field of quantum biology and illustrative examples, this course will help to develop an experimental and theoretical understanding of how quantum processes may play a crucial role in maintaining the non-equilibrium state of the biomolecular systems. Many of the subjects acquired through this half module are likely to be of potential use in future project work,

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.

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.

The course provides an introduction to nuclear energy generation and applications of nuclear science. Nuclear reactors, their physics and operation are described. Nuclear reactor safety case work is also discussed. Future potential energy generation mechanisms such including nuclear fusion will be discussed. The module will also present a range of applications of radioactivity measurement including aspects of Environmental Science and Medical Diagnogstics and treatment therapy. The module will include some aspects of calculus and first order differential equations.

The Linear PDEs Module introduces students to linear partial differential equations, mainly in one and two space dimensions. The classical linear PDEs such as the heat, wave and Laplace equation will be analyzed in detail. Questions of existence and uniqueness of solutions will be addressed and their physical and mathematical meaning emphasized.

This module presents a complete introduction to modern astronomy and astrophysics. We start by introducing astronomy as a unique science; measurement in astronomy (position on the sky, velocity and distance); and multi-messenger probes. We then discuss acceleration due to gravity (the dominant force in the Universe) and we show that timescales are typically so long that we must use models to calculate what happened in the past and what will happen in the future. Armed with an understanding of how to make measurements in astronomy and how to model gravity, we then move from the smallest scales to the largest, studying first the interstellar medium from which stars form; stellar structure and evolution; and stellar remnants (white dwarfs, neutron stars and black holes). We compare and contrast the latest models for how planets form and we discuss the prospects for detecting life and intelligent life beyond Earth. Finally, we discuss the formation and evolution of galaxies in the Universe and the Universe as a whole. We show that the Universe appears to be mostly dark: dark matter (~22%) and dark energy (~74%). Understanding what these mysterious components are is one of the key challenges for physics in the next decade. The module includes either a computer or telescope project. Students enrolled in physics with astronomy with undertake the telescope project where (weather willing) students will gain hands on experience of taking real observational data. They will develop software tools to analyse and interpret these data and write up a final report. Students not enrolled in physics with astronomy will undertake a computational project of similar scope and length, also culminating in a final report.

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.

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 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.

In this module, students will learn key methods adopted in astrophysics to carry out advanced research: scientific computing, statistics, data analysis, machine learning. Much of the course develops highly transferrable skills that apply to science research in general. The goal is to ensure that students are well-prepared for either their research year or their future careers.

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 six advanced topics in physics. Students will be assessed on only four of these topics, with individuals self-selecting the contact sessions, coursework options and exam questions that reflect their preference. An indicative list of these topics include: biological physics, special relativity, particle physics, cosmology, nuclear astrophysics, and quantum computing.

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.

Quantum computers are built around a central processing unit which is a physical device operating according to the rules of quantum mechanics. This module teaches the principles of how quantum processors are built and how they function. The module follows the superconducting architecture, which currently has a significant role in the industry and has been widely studied and implemented. The module will start with reviewing the basic physics required to understand superconducting qubits and the theory of superconducting circuits. It will then continue to the operation of such qubits to perform gates, storage, and readout. After studying the basic building blocks and operations, the module will discuss early demonstrations of important algorithms on superconducting processors. These will lead us naturally to the important topics of improving performance through protection of coherence, advanced control, and validation. The last part of the module will focus on state-of-the-art superconducting processors, focusing on the various challenges in scaling up and the strategies that the industry is pursuing to overcome them.

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.

The module will introduce students to research level equipment and techniques that are used within the research groups of Physics via extended projects. The four-week projects will cover astrophysics, experimental and theoretical nuclear physics, experimental and theoretical soft matter physics, and quantum technologies. Students will gain experience using state-of-the-art equipment and software, analysing and working with large data sets and in problem solving. The module builds upon experience gained during first- and second-year laboratory and computing classes with project-based work that is typically more open ended and less structured. Students are expected to take more responsibility for the planning and direction of work than in previous years. The goal is to help prepare students for independent research within a team and for future project work (e.g. Final Year Projects and MPhys research years). Numbers will be limited on certain projects.

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.

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.

The course will follow the historical development of the main medical imaging techniques.The first part will consider, from a theoretical perspective, the fundamentals of X-ray image formation both in the planar modality and in the Computed Tomography modality. Elements of image processing and image reconstruction will be addressed.The second will look at the physical principles and methods of Nuclear Medicine.The third will look at the principles underlying the application of diagnostic ultrasound in medicine.The fourth will consider Magnetic Resonance Imaging (MRI), one of the most important techniques of medical imaging used in hospitals today.In parallel to the related theoretical classes, students will do X-ray imaging modelling sessions and ultrasound practicals.

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 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.

This module covers circuit-based quantum computers and some of the algorithms that can be implemented using them. The idea of a classical computational algorithm and its complexity is introduced, and used as a language to discuss quantum computational algorithms. Components of quantum circuits are discussed individually, and then built up to show how algorithms such as the quantum Fourier transform and Grover's search algorithm can be implemented, and how such algorithms can be used in applications such as factorizing integers using Shor's algorithm. A suitable computational framework is introduced and used as a tool to implement the algorithms on simulated quantum computers.

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.

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.

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.

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

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 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.

This module aims to provide an advanced level understanding of explosive nuclear astrophysics and the physics of stars. In particular, the course will provide an analytical underpinning of resonant reaction rates, together with the experimental techniques involved in their determination, as well as a theoretical treatment of nuclear reactions and celestial objects.

This module will introduce the students to the principles and formalism of General Relativity and its applications to Black Holes and astrophysical phenomena.

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.

Quantum Mechanics topics are essential building blocks for our understanding of many physical systems. The module assumes basic knowledge in quantum mechanics from modules in earlier years, but will provide a review at the beginning. Topics include a review of quantum mechanics, operator methods and applications to the harmonic oscillator, spin & angular momentum, symmetries in quantum mechanics. The second half of the module then moves beyond isolated quantum systems by addressing the topic of Quantum Entanglement and Quantum Coherence. Here the important concepts of entanglement and decoherence will be studied by developing an understanding of open quantum systems and the density matrix formalism.

This module comprises two independent halves, on quantum simulations (Q Sim), and quantum biology (Q Bio).Quantum Simulation. The quantum simulation part of this module introduces students to the use of quantum computers in the simulation of physical systems using mapping of Hamiltonians from standard quantum mechanics to a representation suitable for application on quantum computers, along with a study of wavefunction ansatz design, algorithms, and error mitigation and correction.Quantum biology is the study of how quantum mechanical phenomena, such as quantum superposition, tunnelling, and entanglement, can be exploited by living systems to provide evolutionary and/or biological advantages. This half module will cover a range of biological molecules and processes that may exploit quantum effects, such as magnetoreception, photosynthetic light harvesting, and DNA mutations. The goal of this half module is to develop an understanding of how quantum processes could have an impact in nature and how this knowledge can be further used for applications in health and medical sciences. In addition, using research progress in the field of quantum biology and illustrative examples, this course will help to develop an experimental and theoretical understanding of how quantum processes may play a crucial role in maintaining the non-equilibrium state of the biomolecular systems. Many of the subjects acquired through this half module are likely to be of potential use in future project work,

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.

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.

The course provides an introduction to nuclear energy generation and applications of nuclear science. Nuclear reactors, their physics and operation are described. Nuclear reactor safety case work is also discussed. Future potential energy generation mechanisms such including nuclear fusion will be discussed. The module will also present a range of applications of radioactivity measurement including aspects of Environmental Science and Medical Diagnogstics and treatment therapy. The module will include some aspects of calculus and first order differential equations.