# Physics, Earth Science and Maths

Natural Sciences is a multidisciplinary degree which allows you to study three subjects in the first year and continue with two subjects in the second and third year.

## Year One

You will study **40 credits of each subject** from your chosen three-subject streams.

## Physics

Students taking Physics must take 40 compulsory credits.

This module is based on the textbook "Physics for Scientists and Engineers" by Knight (all first years are provided with a copy of this book). The module aims to introduce core topics in physics which will underpin all subsequent physics modules. The module begins by discussing classical mechanics in the language of vectors and the key notion of harmonic motion which is extended to cover wave phenomena. The first semester ends with an introduction to Einstein's special theory of relativity. The second semester introduces the basic ideas of electromagnetism and electrical circuits and quantum physics.

- Vectors and Coordinate systems
- Kinematics and Motion in 1D and 2D
- Newton's Laws
- Conservation Laws
- Rotation of a Rigid Body
- Micro-macro connection
- Oscillations
- Travelling Waves
- Superposition of Waves
- Galilean Relativity
- Relativity of Time
- Spacetime
- Relativistic Energy and Momentum

**40 compulsory credits over the full year.**

## Earth Science

Students must take a total of 40 credits. 20 credits are from a compulsory module.

Bulk properties of the Earth, minerals, igneous rocks, sedimentary rocks, metamorphic rocks, geological time, tectonics, geological structures, map interpretation, geological hazards, resource geology.

**20 credits in the Spring semester.**

Select a further 20 credits from the following optional modules:

The unifying theme of this module is biogeochemical cycling - the production, distribution and cycling of materials on the Earth and their availability to, and use by, biological organisms. The module starts by covering the history of the universe, from the big bang to the evolution of the Earth's surface environment. Then you will explore the major global systems and their circulations as they are today - solids, liquids and gases. In the final section you will examine the major materials - including carbon, nitrogen, sulphur, oxygen and metals - and their budgets and cycles; and the interactions between biological and physical/chemical processes on a global scale. You will have a two-hour lecture once a week for this module.

**20 credits in the Autumn Semester.**

This module provides an understanding of the history and origins of the Earth and its life and landforms through consideration of the following topics:

- Development of life over geological time
- Environmental changes over geological time
- Field trip to the Peak District (full costs will be supplied nearer the time of the trip)

**10 credits in the Autumn Semester.**

It is built upon a structured set of paired theory lectures and practical sessions, supported by detailed theory topics delivered via Moodle, which contain linkages to associated textbook resources. It aims to ensure competency in the use of a contemporary GIS software package whilst developing transferable ICT skills.

It also encourages you to develop the analytical skills necessary for the creation of workflows that utilise the built-in analytical functionality of a GIS to solve a spatial problem.

**10 credits in the Spring Semester.**

It emphasises the role of life in creating past and present planetary environments, and conversely the role of environment and environmental change in the evolution and geography of life. The module also serves to prepare the ground for and contextualise several second and third year geography modules, especially Environmental Change and Patterns of Life.

**10 credits in the Spring Semester.**

### Maths

Students must take 40 compulsory credits.

Basic theory is extended to more advanced topics in the calculus of several variables. In addition, the basic concepts of complex numbers, vector and matrix algebra are established and extended to provide an introduction to vector spaces. Students are introduced to different types of proof, such as direct proof, proof by contradiction and proof by induction, as well as theorems and tests for determining the limits of sequences and series. An emphasis in the course is to develop general skills and confidence in applying the methods of calculus and developing techniques and ideas that are widely used and applicable in subsequent modules.

**40 compulsory credits throughout the year**

### Compulsory module

All students are required to take a compulsary module, **Academic and Transferable Skills Portfolio. **This will be taught throughout the first full year. It will support organisational and professional competancies which will be used during the course.

## Year Two

You will continue on your stream comprising of **two of your first year subjects**. You will take **60 credits of modules from each subject** and greater emphasis will be put on studying outside of formal classes.

## Physics

Students taking Physics must take a total of 60 credits.

If you've chosen Earth Science as one of your subjects you will study these compulsory modules in year three:

In the module From Newton to Einstein, you learnt about the idea of a field a quantity which is defined at every point in space. In this module, the description of fields will be extended by introducing the mathematics of vector calculus.

The module will begin with an introduction to vector calculus, illustrated in the context of the flow of ideal (non-viscous) fluids.

The mathematics will then be used to provide a framework for describing, understanding and using the laws of electromagnetism. We discuss how electric and magnetic fields are related to each other and to electrical charges and electrical currents. The macroscopic description of electric fields inside dielectric materials and magnetic fields inside magnetizable materials will be described, including the boundary conditions that apply at material interfaces.

The last section of the module will discuss Maxwells equations of electrodynamics and how they lead to the vector wave equation for electromagnetic waves.

**20 compulsory credits over the full year.**

In this module you will develop your experimental technique and gain experience of some key instruments and methods. The experiments will cover electrical measurements, optics and radiation. You will also learn how to use a computer to control experiments and to record data directly from measuring instruments.

**20 compulsory credits over the full year.**

This module will provide an introduction to the theory and elementary applications of quantum mechanics, a theory that is one of the key achievements of 20th-century physics.

Quantum mechanics is an elegant theoretical construct that is both beautiful and mysterious. Some of the predictions of quantum mechanics are wholly counter-intuitive and there are aspects of it that are not properly understood but it has been tested experimentally for over 50 years and, wherever predictions can be made, they agree with experiment.

**20 compulsory credits over the full year.**

If you've chosen Maths as one of your subjects you will study these compulsory modules in year two:

Macroscopic systems exhibit behaviour that is quite different from that of their microscopic constituents studied in isolation. New physics emerges from the interplay of many interacting degrees of freedom. In this module you will learn about the important physical properties of matter and the two main approaches to their description. One, thermodynamics, treats macroscopically relevant degrees of freedom (temperature, pressure and so on) and find relations between these and the fundamental laws which govern them, independent of their microscopic structure. The other approach, statistical mechanics, links the macroscopically relevant properties to the microphysics by replacing the detailed microscopic dynamics with a statistical description. The common feature of both of these methods is the introduction of two macroscopic quantities, temperature and entropy, that have no microscopic meaning.

**20 compulsory credits over the full year.**

**20 compulsory credits over the full year.**

**Plus** **select an additional 20 credits** from these optional modules to cover in year two:

You will develop your knowledge of the various physical processes occurring in stars of different types. You’ll use this knowledge to build both mathematical models and your qualitative physical understanding of stellar structure and evolution will be enhanced. You’ll have two hours per week of lectures studying this module.

**10 credits in the Autumn semester.**

You’ll be given an overview of how forces at the nanoscale are different to those observed in macroscopic systems and will consider how they can be exploited in nanometre-scale processes and devices.

You’ll focus on the physical basis and measurement of forces operating on the nanoscale, considering van der Waals, electrostatic, hydrophobic and hydrophilic interactions.

You’ll spend around three hours per week in lectures and workshops studying this module.

**10 compulsory credits in the Autumn semester.**

This module will develop your current understanding of the various physical processes that dictate the formation, evolution and structure of galaxies. You’ll explore a number of topics including The Milky Way, The Dynamics of Galaxies, Active Galaxies and Galaxy Evolution among others. You’ll spend two hours per week in lectures studying this module.

**10 compulsory credits in the Spring semester.**

This module introduces basic techniques in numerical methods and numerical analysis which can be used to generate approximate solutions to problems that may not be amenable to analysis.

Specific topics include:

- Implementing algorithms in Matlab
- Discussion of errors (including rounding errors)
- Iterative methods for nonlinear equations (simple iteration, bisection, Newton, convergence)
- Gaussian elimination, matrix factorisation, and pivoting
- Iterative methods for linear systems, matrix norms, convergence, Jacobi, Gauss-Siedel
- Interpolation (Lagrange polynomials, orthogonal polynomials, splines)
- Numerical differentiation & integration (Difference formulae, Richardson extrapolation, simple and composite quadrature rules)
- Introduction to numerical ODEs (Euler and Runge-Kutta methods, consistency, stability)

**20 credits throughout the full year.**

## Earth Science

Students take 60 credits from this list

In addition, you choose further laboratory techniques to investigate in the second semester. The ethical, safety and fieldwork limitations of geographical work are also considered.

**20 compulsory credits throughout the full year.**

This module:

- introduces the water and sediment processes that operate in rivers
- describes the characteristic forms of alluvial channels and the links between river processes and channel dynamics
- uses laboratory practicals and a field trip to deliver kinaesthetic, student-centred learning and add value to teaching and learning during lectures

Topics covered include:

- catchments and longitudinal patterns
- river planforms: braided, meandering and straight
- timescales of river change and morphological adjustments
- complex response in the fluvial system
- flow resistance, sediment transport and bank erosion
- an introduction to biogeomorphology and aquatic ecology

**20 credits in the Autumn Semester.**

This module provides a general introduction to the subject of earth observation. This involves analysing remotely sensed images, typically acquired from instruments on board satellites or aircraft, to investigate spatial phenomena on the Earth's surface.

Example topics include the use of global image data sets to investigate climate change, analysis of satellite sensor imagery to identify wildlife habitats and conservation concerns, and urban land use mapping from detailed aerial photography. Theoretical lectures cover the concepts underpinning remote sensing, including the physical principles determining image creation, fundamental image characteristics, methods of image analysis and uses or applications of earth observation.

There is also a strong practical component to the module, with regular practical exercises on various forms of digital image analysis.

**20 credits in the Autumn Semester.**

Overview: Soils are the most complex biomaterial on earth. An understanding of the basic concepts concerning the form and function of soils is important for future management strategies such as mitigating the effects of climate change and providing safe and sustainable food. This module focuses on the important soil properties from physical, chemical and biological perspectives including soil organic matter (microbiology and chemistry); soil chemical reactions (acidity, redox); soil fauna and flora; soil-water relations (irrigation and drainage).

**10 credits**

This module will develop understanding of the important chemical and physical processes that operate in the terrestrial environment, principally within soils and fresh water systems. It includes the study of the hydrological cycle, surface and sub-surface water chemistry including rainfall, rivers and lakes, processes that govern the movement of solutes and colloidal materials, adsorption, redox, solubility, diffusion and kinetics.

**10 credits.**

Overview: This module provides a consideration of:

- Spatial Decision Making & the role that GIS has in this
- Spatial Data Types and Sources
- Vector and Raster Processing Algorithms
- Professional Training in ArcGIS
- Project planning, implementation and reporting

**20 credits .**

## Maths

Students taking Maths must take 60 credits from their chosen specialism:

#### Applied, Computation and Statistics specialism

20 compulsory credits:

This course aims to give students a sound grounding in the application of both differential and integral calculus to vectors, and to apply vector calculus methods and separation of variables to the solution of partial differential equations. The module is an important pre-requisite for a wide range of other courses in Applied Mathematics.

**10 credits in the Autumn Semester**

This course is an introduction to Fourier series and integral transforms and to methods of solving some standard ordinary and partial differential equations which occur in applied mathematics and mathematical physics.

The course describes the solution of ordinary differential equations using series and introduces Fourier series and Fourier and Laplace transforms, with applications to differential equations and signal analysis. Standard examples of partial differential equations are introduced and solution using separation of variables is discussed.

**10 credits in the Spring Semester**

And 40 optional credits from the following modules:

The module covers introductory topics in statistics and probability that could be applied to data analysis in a broad range of subjects. Topics include probability distributions, parameter estimation, confidence intervals,hypothesis testing and an introduction to statistical modelling. Consideration is given to issues in applied statistics such as sample size calculations, the multiple comparison problem,data collection, design of experiments, critiquing and interpreting statistical reports and papers.

**20 credits throughout the full year.**

This course aims to provide students with tools which enable them to develop and analyse linear and nonlinear mathematical models based on ordinary and partial differential equations. Furthermore, it aims to introduce students to the fundamental mathematical concepts required to model the flow of liquids and gases and to apply the resulting theory to model physical situations.

**20 credits throughout the full year.**

This module introduces basic techniques in numerical methods and numerical analysis which can be used to generate approximate solutions to problems that may not be amenable to analysis.

Specific topics include:

- Implementing algorithms in Matlab
- Discussion of errors (including rounding errors)
- Iterative methods for nonlinear equations (simple iteration, bisection, Newton, convergence)
- Gaussian elimination, matrix factorisation, and pivoting
- Iterative methods for linear systems, matrix norms, convergence, Jacobi, Gauss-Siedel
- Interpolation (Lagrange polynomials, orthogonal polynomials, splines)
- Numerical differentiation & integration (Difference formulae, Richardson extrapolation, simple and composite quadrature rules)
- Introduction to numerical ODEs (Euler and Runge-Kutta methods, consistency, stability)

**20 credits throughout the full year.**

#### Mathematical Physics specialism

60 compulsory credits:

This course develops Newtonian mechanics into the more powerful formulations due to Lagrange and Hamilton and introduces the basic structure of quantum mechanics. The course provides the foundation for a wide range of more advanced courses in mathematical physics.

**20 credits through the full year.**

This course aims to provide students with tools which enable them to develop and analyse linear and nonlinear mathematical models based on ordinary and partial differential equations. Furthermore, it aims to introduce students to the fundamental mathematical concepts required to model the flow of liquids and gases and to apply the resulting theory to model physical situations.

**20 credits throughout the full year.**

This course aims to give students a sound grounding in the application of both differential and integral calculus to vectors, and to apply vector calculus methods and separation of variables to the solution of partial differential equations. The module is an important pre-requisite for a wide range of other courses in Applied Mathematics.

**10 credits in the Autumn Semester.**

This course is an introduction to Fourier series and integral transforms and to methods of solving some standard ordinary and partial differential equations which occur in applied mathematics and mathematical physics.

The course describes the solution of ordinary differential equations using series and introduces Fourier series and Fourier and Laplace transforms, with applications to differential equations and signal analysis. Standard examples of partial differential equations are introduced and solution using separation of variables is discussed.

**10 credits in the Spring Semester.**

## Year Three

You will continue with the **same two subjects studied in the second year**, taking** 50 credits in each**.

### Compulsory year three module

Alongside subject-specific study, you will undertake a **20-credit compulsory synoptic module** which aims to tie together the subjects you are studying through an interdisciplinary group project.

The Natural Sciences programme is by nature interdisciplinary but is mostly taught via specialized modules delivered by individual Schools with little exploration of the interfaces between the sciences. The synoptic module (C13602) gives students the opportunity to combine knowledge and skills acquired whilst on their pathway to carry out a (number of) interdisciplinary piece(s) of work.

**20 credits throughout the full year.**

## Physics

Students taking Physics must take 50 compulsory credits from your chosen specialism.

If you've chosen Earth Science as one of your subjects you will study these compulsory modules in year three:Macroscopic systems exhibit behaviour that is quite different from that of their microscopic constituents studied in isolation. New physics emerges from the interplay of many interacting degrees of freedom. In this module you will learn about the important physical properties of matter and the two main approaches to their description. One, thermodynamics, treats macroscopically relevant degrees of freedom (temperature, pressure and so on) and find relations between these and the fundamental laws which govern them, independent of their microscopic structure. The other approach, statistical mechanics, links the macroscopically relevant properties to the microphysics by replacing the detailed microscopic dynamics with a statistical description. The common feature of both of these methods is the introduction of two macroscopic quantities, temperature and entropy, that have no microscopic meaning.

**20 compulsory credits over the full year.**

This module will introduce students to the physics of atoms, nuclei and the fundamental constituents of matter and their interactions. The module will also develop the quantum mechanical description of these.

Topics to be covered are:

- Approximation techniques first order perturbation theory, degeneracies, second order perturbation theory, transition rates, time-dependent perturbation theory, Fermi's golden rule
- Particle Physics protons and neutrons, antiparticles, particle accelerators and scattering experiments, conservation laws, neutrinos, leptons, baryons and hadrons, the quark model and the strong interaction, weak interactions, standard model
- Introduction to atomic physics review of simple model of hydrogen atom, Fermi statistics and Pauli principle, aufbau principle, hydrogenic atoms, exchange, fine structure and hyperfine interactions, dipole interaction, selection rules and transition rates
- Lasers optical polarization and photons, optical cavities, population inversions, Bose statistics and stimulated emission, Einstein A and B coefficients
- Nuclear Physics Radioactivity, decay processes, alpha, beta and gamma emission, detectors, stability curves and binding energies, nuclear fission, fusion, liquid drop and shell models.

**20 credits over the full year.**

You will carry out a project drawn from one of several areas of physics. The project may be experimental or theoretical in nature. Many of the projects reflect the research interests of members of academic staff. You’ll work in pairs and will be expected to produce a plan of work and to identify realistic goals for your project. Each pair has a project supervisor responsible for setting the project.

**10 credits in the Autumn semester.**

If you've chosen Maths as one of your subjects you will study these compulsory modules in year three:

This module will provide a general introduction to solid state physics. Topics covered include:

- Bonding nature of chemical bonds, thermodynamics of solid formation
- Crystal structures description of crystal structures, k-space, reciprocal lattice, Bragg diffraction, Brillouin zones
- Nearly-free electron model - Bloch's theorem, band gaps from electron Bragg scattering, effective masses
- Band theory Fermi surfaces, qualitative picture of transport, metals, insulators and semiconductors
- Semiconductors - doping, inhomogeneous semiconductors, basic description of pn junction
- Phonons normal modes of ionic lattice, quantization, Debye theory of heat capacities, acoustic and optical phonons
- Optical properties of solids absorption and reflection of light by metals, Brewster angle, dielectric constants, plasma oscillations
- Magnetism- Landau diamagnetism, paramagnetism, exchange interactions, Ferromagnetism, antiferromagnetism, neutron scattering, dipolar interactions and domain formation, magnetic technology

**20 compulsory credits over the full year.**

This module will introduce students to the physics of atoms, nuclei and the fundamental constituents of matter and their interactions. The module will also develop the quantum mechanical description of these.

Topics to be covered are:

- Approximation techniques first order perturbation theory, degeneracies, second order perturbation theory, transition rates, time-dependent perturbation theory, Fermi's golden rule
- Particle Physics protons and neutrons, antiparticles, particle accelerators and scattering experiments, conservation laws, neutrinos, leptons, baryons and hadrons, the quark model and the strong interaction, weak interactions, standard model
- Introduction to atomic physics review of simple model of hydrogen atom, Fermi statistics and Pauli principle, aufbau principle, hydrogenic atoms, exchange, fine structure and hyperfine interactions, dipole interaction, selection rules and transition rates
- Lasers optical polarization and photons, optical cavities, population inversions, Bose statistics and stimulated emission, Einstein A and B coefficients
- Nuclear Physics Radioactivity, decay processes, alpha, beta and gamma emission, detectors, stability curves and binding energies, nuclear fission, fusion, liquid drop and shell models.

**20 credits over the full year.**

You will carry out a project drawn from one of several areas of physics. The project may be experimental or theoretical in nature. Many of the projects reflect the research interests of members of academic staff. You’ll work in pairs and will be expected to produce a plan of work and to identify realistic goals for your project. Each pair has a project supervisor responsible for setting the project.

**10 credits in the Autumn semester.**

## Earth Science

Y3 students take 40-60 credits from the following:

A geohazard is a natural process or phenomenon that has the potential to adversely affect humanity by endangering life or property. A geo resource is a substance or commodity that can be extracted from the subsurface for use by humanity. This module will spend one semester focussing on these two important issues for Environment and Society.

**20 credits throughout the full year.**

This module considers human attempts to manage and restore freshwater environments, specifically rivers, lakes and wetlands. It considers changes in the fluvial system that occur in response to river management and engineering and examines approaches to restoring the natural functions of rivers that have been heavily degraded by human impacts.

The module examines some of the main stressors on lakes and wetlands lake management, and approaches for their management using an ecosystem-scale approach. The principles by which restoration practice is guided will be considered, and criteria for selection between alternative strategies will be introduced. The module will consider water quality and legislative requirements for freshwater bodies.

The module includes a field trip where you will visit a local nature reserve and develop a management plan with input from management practitioners and land-owners. You will also be able to engage with river management practitioners in a series of guest lectures.

**20 credits in the Autumn Semester.**

Palaeobiology explores the relationship between life and the Earth's physical and chemical environment over geological/ evolutionary time. The module will focus on the geological consequences of evolution and how life has influenced physical and chemical environment. Topics covered will include: Origins and evolution of life; Evolution of the atmosphere and biosphere; the geobiology of critical intervals in both palaeobiology and evolutionary ecology. Students will gain an in depth knowledge of the mechanisms that control changes in the physiochemical environmental and their impact upon evolution. In order to gain a broad understanding the module will explore past changes as seem in the fossil record, together with present day processes that underpin these responses. The lectures and course work will give students knowledge of the tools that are used to reconstruct past environmental conditions and the effect of future changes in the abiotic stimuli that drive environmental change.

**10 credits in the Autumn semester.**

## Maths

Students taking Maths must take a total of 50 credits from their chosen specialism:

#### Applied, Computation & Statistics specialism

Select 50 credits from the below modules:

In this module a variety of techniques and areas of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming. You’ll develop techniques for application which can be used outside the mathematical arena.

**20 credits in the Autumn Semester.**

Mathematics can be usefully applied to a wide range of applications in medicine and biology. Without assuming any prior biological knowledge, this course describes how mathematics helps us understand topics such as population dynamics, biological oscillations, pattern formation and nonlinear growth phenomena. There is considerable emphasis on model building and development.

**20 credits in the Autumn Semester.**

This course provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. It also provides an introduction to to cryptography, including classical mono and polyalphabetic ciphers as well as modern public key cryptography and digital signatures, their uses and applications.

**10 credits in the Autumn Semester.**

Game theory contains many branches of mathematics (and computing); the emphasis here is primarily algorithmic. The module starts with an investigation into normal-form games, including strategic dominance, Nash equilibria, and the Prisoner’s Dilemma. We look at tree-searching, including alpha-beta pruning, the ‘killer’ heuristic and its relatives. It then turns to mathematical theory of games; exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.

**10 credits in the Spring Semester.**

This course aims to extend previous knowledge of fluid flow by introducing the concept of viscosity and studying the fundamental governing equations for the motion of liquids and gases. Methods for solution of these equations are introduced, including exact solutions and approximate solutions valid for thin layers. A further aim is to apply the theory to model fluid dynamical problems of physical relevance.

**20 credits in the Spring Semester.**

Differential equations play a crucial modelling role in many applications, such as fluid dynamics, electromagnetism, biomedicine, astrophysics and financial modelling. Typically, the equations under consideration are so complicated that their solution may not be determined by purely analytical techniques; instead one has to resort to computing numerical approximations to the unknown analytical solution. In this module we study numerical techniques for approximating data, ordinary and partial differential equations, and solving, or finding eigenvalues and eigenvectors of, the large linear systems of equations that result from these approximations. The module covers:

- Initial value problems (ODEs): multistage and multistep methods; convergence and stability; higher order ODEs; systems of first order ODEs; implicit methods
- Partial differential equations: finite differences for elliptic, parabolic and hyperbolic PDEs; truncation error and stability analysis; finite volume methods
- Approximation theory: least squares approximation; trigonometric polynomial approximation
- Eigenvalues and eigenvectors: power method; inverse iteration; Householder transformations; QR algorithm; singular value decomposition
- Large linear systems: Krylov subspace methods; conjugate gradient method; preconditioning

**20 credits in the Spring Semester.**

#### Mathematical Physics specialism

Select 50 credits from the below modules

**20 credits in the Autumn Semester**

This course introduces various analytical methods for the solution of ordinary and partial differential equations, focussing on asymptotic techniques and dynamical systems theory. Students taking this course will build on their understanding of differential equations covered in Modelling with Differential Equations.

**20 credits in the Autumn Semester.**

This course aims to extend previous knowledge of fluid flow by introducing the concept of viscosity and studying the fundamental governing equations for the motion of liquids and gases. Methods for solution of these equations are introduced, including exact solutions and approximate solutions valid for thin layers. A further aim is to apply the theory to model fluid dynamical problems of physical relevance.

**20 credits in the Spring Semester.**

In this module you’ll have an introduction to Einstein’s theory of general and special relativity. The relativistic laws of mechanics will be described within a unified framework of space and time. You’ll learn how to compare other theories against this work and you’ll be able to explain new phenomena which occur in relativity.

**20 credits in the Spring Semester.**

This course provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. It also provides an introduction to to cryptography, including classical mono and polyalphabetic ciphers as well as modern public key cryptography and digital signatures, their uses and applications.

**10 credits in the Autumn Semester.**

Game theory contains many branches of mathematics (and computing); the emphasis here is primarily algorithmic. The module starts with an investigation into normal-form games, including strategic dominance, Nash equilibria, and the Prisoner’s Dilemma. We look at tree-searching, including alpha-beta pruning, the ‘killer’ heuristic and its relatives. It then turns to mathematical theory of games; exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.

**10 credits in the Spring Semester.**

## Year Four (MSci students only)

You will choose one of your third-year subjects to focus on in the fourth year, spending half your time working on an independent research project aiming to develop the skills needed to pursue a career in research.

All students take 120 credits of modules in the fourth year and each subject has a minimum number of credits listed. Students can take 120 credits from a single subject (where available) or they can use modules from their second subject to make up the difference between the minimum and the required number of credits.

## Physics

**You must take a total of 120 credits from physics throughout the year.**

60 compulsory credits:

In this year-long module you’ll aim to solve a theoretical or practical problem. You’ll spend semester one researching your chosen project and carry out your original research in semester two. You’ll have the opportunity to work with external parties such as an industrial laboratory, school or hospital if appropriate to your topic.

**60 credits over the full year.**

Plus 20 compulsory credits if you haven't chosen Maths from year two or three.

This module will provide a general introduction to solid state physics. Topics to be covered will include:

- Fermi Dirac and Bose-Einstein Statistics, Fermi Wave-vector, temperature
- Introduction to Fourier Transforms and Associated Techniques
- bonding nature of chemical bonds, thermodynamics of solid formation
- crystal structures description of crystal structures, k-space, reciprocal lattice, Bragg diffraction, Brillouin zones
- Nearly-free electron model - Bloch's theorem, band gaps from electron Bragg scattering, effective masses
- Band theory Fermi surfaces, qualitative picture of transport, metals, insulators and semiconductors
- Semiconductors - doping, inhomogeneous semiconductors, basic description of pn junction
- Phonons normal modes of ionic lattice, quantization, Debye theory of heat capacities, acoustic and optical phonons
- Optical properties of solids absorption and reflection of light by metals, Brewster angle, dielectric constants, plasma oscillations
- Magnetism, Landau diamagnetism, paramagnetism, exchange interactions, Ferromagnetism, antiferromagnetism, neutron scattering, dipolar interactions and domain formation, magnetic technology

**10 credits in the Autumn semester.**

And an additional 40 to 60 credits to be selected from the below optional modules:

In this module you’ll explore the theoretical aspect of atmospheric physics. Topics will include planetary atmosphere, troposphere, solar radiation and the Energy budget, radiation transfer and Photochemistry among others. You’ll have two hours of lectures per week studying this module.

**10 credits in the Autumn semester.**

Cosmology is the scientific study of the universe as a whole. The module provides an introduction to modern cosmology, including some of the more recent observational and theoretical developments. No prior knowledge of General Relativity is required. Topics covered include: observed features of the universe, the Cosmological Principle, Newtoniaan and Relativistic cosmology, the Friedmann Models, cosmic expansion, the cosmological constant, evidence for the big bang model, the thermal history of the big bang, the early universe and inflation, the classical cosmological tests, structure formation (brief treatment only).

**10 credits in the Autumn semester.**

To make models of extreme astrophysical sources and environments basedon physical theory.

To interpret observational data in the light of relevant physical theory.

**10 credits in the Spring semester.**

This module introduces you to the physical properties of semiconductors and low-dimensional systems, such as quantum wells, wires and dots. The aim is to explain the physics that underlies optical and transport properties of these structures and and their applications in advanced technologies.

This course is structured in two main parts. The first part focuses on the foundation of quantum mechanics and solid state physics needed to describe a low dimensional system. The module then moves on describing the physical principles of semiconductor junction and devices.

**10 credits in the Spring semester.**

To introduce the key theoretical ideas of elementary particle physics, such as symmetry and conservation laws, and to build the foundations for a mathematical description of particle properties and interactions.

**10 credits in the Spring semester.**

The first half of this module will describe radiation sources and detectors, with particular reference to those used in the medical imaging applications described in the second half. It will include the physics of accelerators such as linacs, cyclotrons and synchrotrons, of detectors such as ionization chambers, scintillators and solid state detectors and of X-ray imaging, nuclear imaging and positron emission tomography (PET).

**10 credits in the Autumn semester.**

- Introduction to Soft Matter
- Forces, energies and timescales in soft matter
- Liquids and glasses
- Phase transitions in soft matter (solid-liquid and liquid-liquid demixing)
- Polymeric materials
- Gelation
- Crystallisation in soft systems
- Liquid crystals
- Molecular order in soft systems
- Soft Nanotechnology

**10 credits in the Autumn semester.**

## Earth Science

**You must a take a minimum of 80 and a maximum of 120 credits** **from earth science throughout the year.**

60 compulsory credits:

The aim of the module is to provide training for the description, planning and conduct of a programme of research in order to solve or report on a specific scientific problem. The MSci project is taken in both the autumn and spring semesters and comprises 60 credits. In the autumn the student will work with the supervisor to devise a projectby identifying an appropriate topic before focusing on a specific scientific problem. This will involve regular planning meetings and individual research by the student. In the spring semester the students will undertake the main body of work for the project which may be experimental, computer, literature or theoretically based (or various combinations of these). The student will continue to have, as a minimum, monthly supervisor meetings and document all progression in their project notebooks. The module is assessed by a project write up in the style of a scientific paper, the project notebook and a poster presentation with an oral component to the staff and the student cohort.

**60 compulsory credits throughout the full year.**

**A minimum of 20 and up to a maximum of 60 credits** can be selected from the following:

This module provides an overview of water quality assessment techniques, including chemical, biological, sensor and long-term reconstruction methods. The majority of the module will be taught as part of a residential three-day field course, supplemented with laboratory practical work. You will get hands-on experience in techniques and their application.

The module will be underpinned training in the by a theoretical understanding of the water quality and its drivers at a variety of spatial and temporal scales. Training will include how to devise appropriate water quality management plans and use of benchmarking tools from environmental monitoring agencies.

**10 credits in the Autumn Semester.**

This module will focus on the following themes:

- Key river and catchment processes
- Impacts of anthropogenic (ie climate, land-use) change on rivers and catchments
- Current and historic river/catchment management practises
- Tools and techniques for monitoring and mapping rivers and catchments
- Modelling rivers and catchments to test management scenarios

**10 credits in the Spring Semester.**

Principles of experimentation in crop science, basic statistical principles, experimental design, hypothesis testing, sources of error, analysis of variance, regression techniques, presentation of data, use of Genstat for data analysis. There are two routes through the module; one focusing on crop improvement and one focusing on more general issues.

**10 compulsory credits throughout the year.**

This module considers:

- The importance of engaging publics with cutting edge research
- Methods of engagement that are suitable for varying audiences
- How to write for varied audiences
- How to engage with policymakers and industry
- Public speaking skills
- The planning, development and delivery of an engagement event for the public/policymakers

**10 compulsory credits in the Spring semester.**

## Maths

**You must take a minimum of 80 and a maximum of 120 credits from maths throughout the year.**

40 compulsory credits:

This module consists of a self-directed investigation of a project selected from a list of projects or, subject to prior approval of the School, from elsewhere.

The project will be supervised by a member of staff and will be based on a substantial mathematical problem, an application of mathematics or investigation of an area of mathematics not previously studied by the student. The course includes training in the use of IT resources, the word-processing of mathematics and report writing.

**40 compulsory credits throughout the year**

And select a minimum of 40 credits and a maximum of 80 credits from the optional modules:

The development of techniques for the study of nonlinear differential equations is a major worldwide research activity to which members of the School have made important contributions. This course will cover a number of state-of-the-art methods, namely:

- use of green function methods in the solution of linear partial differential equations
- characteristic methods, classification and regularization of nonlinear partial differentiation equations
- bifurcation theory

These will be illustrated by applications in the biological and physical sciences.

**20 credits in the Autumn Semester**

The course introduces notions of topology and differential geometry which are required for modern research in relativity and other topics involving geometry. The course will be illustrated with a body of concrete geometrical examples drawn from general relativity. The modern study of general relativity requires familiarity with a number of tools of differential geometry, including manifolds, symmetries, Lie Groups, differentiation and integration on manifolds. These are introduced using examples of curved space-times whose context is familiar from the study of general relativity, the presentation of geometric concepts will be significantly more abstract and powerful than in Relativity MATH3018

**20 credits in the Autumn Semester**

The first part of the module introduces no-arbitrage pricing principle and financial instruments such as forward and futures contracts, bonds and swaps, and options. The second part of the module considers the pricing and hedging of options and discrete-time discrete-space stochastic processes. The final part of the module focuses on the Black-Scholes formula for pricing European options and also introduces the Wiener process. Ito integrals and stochastic differential equations.

**20 credits in the Autumn Semester**

The purpose of this course is to introduce concepts of scientific programming using the object oriented language C++ for applications arising in the mathematical modelling of physical processes. Students taking this module will develop knowledge and understanding of a variety or relevant numerical techniques and how to efficiently implement them in C++.

**20 credits in the Autumn Semester**

In this course the fundamental principles and techniques underlying modern statistical and data analysis will be introduced. The course will cover a 'common core' consisting of statistical concepts and methods, linear models, probability techniques and Markov chains.

You will gain experience of using a statistical package and interpreting its output. The common core material will be covered primarily at the beginning of the semester.

**20 credits in the Autumn Semester**

General relativity predicts the existence of black holes which are regions of space-time into which objects can be sent but from which no classical objects can escape. This course uses techniques learnt in MATH4015 to systematically study black holes and their properties, including horizons and singularities. Astrophysical processes involving black holes are discussed, and there is a brief introduction to black hole radiation discovered by Hawking.

This course aims to introduce the physics of black holes and its mathematical description, giving insight into problems of research interest. It provides an opportunity to apply techniques and ideas learned in previous modules to important astrophysical problems. Students will acquire knowledge and skills to a level sufficient to begin research in general relativity.

**20 credits in the Spring Semester**

This module illustrates the applications of advanced techniques of mathematical modelling using ordinary and partial differential equations. A variety of medical and biological topics are treated bringing students close to active fields of mathematical research.

**20 credits in the Spring Semester**

This module will provide a general introduction to the analysis of data that arise sequentially in time. You will discuss several commonly-occurring models, including methods for model identification for real-time series data. You will develop techniques for estimating the parameters of a model, assessing its fit and forecasting future values. You will gain experience of using a statistical package and interpreting its output.

**20 credits in the Spring Semester**

This course introduces computational methods for solving problems in applied mathematics. Students taking this course will develop knowledge and understanding to design, justify and implement relevant computational techniques and methodologies.

**20 credits in the Spring Semester**

**Disclaimer**This online prospectus has been drafted in advance of the academic year to which it applies. Every effort has been made to ensure that the information is accurate at the time of publishing, but changes (for example to course content) are likely to occur given the interval between publishing and commencement of the course. It is therefore very important to check this website for any updates before you apply for the course where there has been an interval between you reading this website and applying.