Mathematics and Physics MSci

Core Modules

Year 1

• Calculus

  In this module, you will develop an understanding of the key concepts in Calculus, including differentiation and integration. You will learn how to factorise polynomials and separate rational functions into partial fractions, differentiate commonly occurring functions, and find definite and indefinite integrals of a variety of functions using substitution or integration by parts. You will also examine how to recognise the standard forms of first-order differential equations, and reduce other equations to these forms and solve them.

• Calculus II

  In this module you will develop an understanding of the calculus functions of more than one variable and how it may be used in areas such as geometry and optimisation. You learn how to manipulate partial derivatives, construct and manipulate line integrals, represent curves and surfaces in higher dimensions, calculate areas under a curve and volumes between surfaces, and evaluate double integrals, including the use of change of order of integration and change of coordinates.

• Introduction to Pure Mathematics

  In this module you will develop an understanding of the fundamental algebraic structures, including familiar integers and polynomial rings. You will learn how to apply Euclid's algorithm to find the greatest common divisor of two integers, and use mathematical induction to prove simple results. You will examine the use of arithmetic operations on complex numbers, extract roots of complex numbers, prove De Morgan's laws, and determine whether a given mapping is bijective.

• Linear Algebra I

  In this module you will develop an understanding of basic linear algebra, in particular the use of matrices and vectors. You will look at the basic theoretical and computational techniques of matrix theory, examining the power of vector methods and how they may be used to describe three-dimensional space. You will consider the notions of field, vector space and subspace, and learn how to calculate the determinant of an n x n matrix.

• Scientific Skills

  In this module you will develop an understanding of good practices in the laboratory. You will keep a notebook, recording experimental work as you do it. You will set up an experiment from a script, and carry out and record measurements. You will learn how to analyse data and plot graphs using a computer package, and present results and conclusions including error estimations from your experiments.

• Classical Mechanics

  In this module you will develop an understanding of how to apply the techniques and formulae of mathematical analysis, in particular the use of vectors and calculus, to solve problems in classical mechanics. You will look at statics, dynamics and kinematics as applied to linear and rigid bodies. You will also examine the various techniques of physical analysis to solve problems, such as force diagrams and conservation principles.

• Classical Matter

  In this module you will develop an understanding of the macroscopic properties of the various states of matter, looking at elementary ideas such as ideal gases, internal energy and heat capacity. Using classical models of thermodynamics, you will examine gases, liquids, solids, and the transitions between these states, considering phase equilibrium, the van der Waals equation and the liquefaction of gases. You will also examine other states of matter, including polymers, colloids, liquid crystals and plasmas.

• Physics of the Universe

  In this module you will develop an understanding of the building blocks of fundamental physics. You will look at Einstein’s special theory of relativity, considering time-dilation and length contraction, the basics of quantum mechanics, for example wave-particle duality, and the Schrödinger equation. You will also examine concepts in astrophysics such as the Big Bang theory and how the Universe came to be the way we observe it today.

• Academic Integrity

  This module will describe the key principles of academic integrity, focusing on university assignments. Plagiarism, collusion and commissioning will be described as activities that undermine academic integrity, and the possible consequences of engaging in such activities will be described. Activities, with feedback, will provide you with opportunities to reflect and develop your understanding of academic integrity principles.

   

Year 2

• Applications of Vector Calculus

  In this module the vector calculus methods are applied to a variety of problems in the physical sciences, with a focus on electromagnetism and optics. On completion of the module, the student should be able to calculate relevant physical quantities such as field strengths, forces, and energy distributions in static as well as dynamical electromagnetic systems and be able to treat mathematically the interactions between moving electrical charges, magnets and optical fields.

• Ordinary Differential Equation

  In this module you will develop an understanding of the concepts arising when the boundary conditions of a differential equation involves two points. You will look at eigenvalues and eigenfunctions in trigonometric differential equations, and determine the Fourier series for a periodic function. You will learn how to manipulate the Dirac delta-function and apply the Fourier transform. You will also examine how to solve differential equations where the coefficients are variable.

• Linear Algebra II

  In this module you will develop an understanding of vectors and matrices within the context of vector spaces, with a focus on deriving and using various decompositions of matrices, including eigenvalue decompositions and the so-called normal forms. You will learn how these abstract notions can be used to solve problems encountered in other fields of science and mathematics, such as optimisation theory.

• Scientific Computing Skills

  In this module you will develop an understanding of how computers are used in modern science for data analysis and visualisation. You will be introduced to the intuitive programming language, Python, and looking at the basics of numerical calculation. You will examine the usage of arrays and matrices, how to plot and visualise data, how to evaluate simple and complex expressions, how to sample using the Monte Carlo methods, and how to solve linear equations.

• Quantum Mechanics

  In this module you will develop an understanding of quantum mechanics and its role in and atomic, nuclear, particle and condensed matter physics. You will look at the wave nature of matter and the probabilistic nature of microscopic phenomena. You will learn how to use the key equation of quantum mechanics to describe fundamental phenomena, such as energy quantisation and quantum tunnelling. You will examine the principles of quantum mechanics, their physical consequences, and applications, considering the nature of harmonic oscillator systems and hydrogen atoms.

• Classical and Statistical Thermodynamics

  In this module you will develop an understanding of thermal physics and elementary quantum mechanics. You will look at the thermodynamic properties of an ideal gas, examining the solutions of Schrödinger’s equation for particles in a box, and phenomena such as negative temperature, superfluidity and superconductivity. You will also consider the thermodynamic equilibrium process, entropy in thermo-dynamics, and black-body radiation.

• The Solid State

  In this module you will develop an understanding of the physical properties of solids. You will look at their structure and symmetry, concepts of dislocation and plastic deformation, and the electrical characteristics of metals, alloys and semiconductors. You will examine methods of probing solids and x-ray diffraction, and the thermal properties of photons. You will also consider the quantum theory of solids, including energy bands and the Bloch theorem, as well as exploring fermiology, intrinsic and extrinsic semiconductors, and magnetism.

• Advanced Skills in Mathematics

  The aim of this module is to enable you to plan and carry out longer assignments, which require significant preparation and/or advanced data analysis. This module will lay the groundwork for the comprehensive project work undertaken in your final year project whilst introducing you to some of the research activities within the Department. You will establish skills including report writing and oral presentations in the context of employment.

• Advanced Skills in Physics

  This module builds on skills developed in the second year to prepare you for the final year project and beyond. You will plan and carry out three assignments: the first introduces advanced data analysis techniques, and the second is a group project.  Through the mini-project, you will be introduced to some of the research activities within the Department and to the methods of working within the groups. There are opportunities to practise important skills such as report writing and giving oral presentations. 

Year 3

• Optics

  In this module you develop an understanding of the properties of light, starting from Maxwell’s equations. You will look at optical phenomena such as refraction, diffraction and interference, and how they are exploited in modern applications, from virtual reality headsets to the detection of gravitational waves. You will also examine masers and lasers, and their usage in optical imaging and image processing.

• Electromagnetism

  In this module you will develop an understanding of how James Clerk Maxwell unified all known electrical and magnetic effects with just four equations, providing Einstein’s motivation for developing the special theory of relativity, explaining light as an electromagnetic phenomenon, and predicting the electromagnetic spectrum. You examine these equations and their consequences, looking at how Maxwell’s work underpins all of modern physics and technology. You will also consider how electromagnetism provides the paradigm for the study of all other forces in nature.

• Advanced Skills in Physics

  This module builds on skills developed in the second year to prepare you for the final year project and beyond. You will plan and carry out three assignments: the first introduces advanced data analysis techniques, and the second is a group project.  Through the mini-project, you will be introduced to some of the research activities within the Department and to the methods of working within the groups. There are opportunities to practise important skills such as report writing and giving oral presentations. 

Year 4

• Research Review

  In this module you will explore a topic related to current research in physics through independent study. You will investigate original scientific papers, develop a strong understanding of the chosen subject, and produce a well-structured written report with appropriate references. Through this process, you will improve your research, organisational and presentation skills, as well as develop greater confidence in working independently and engaging with advanced scientific literature.

• Major Project in Physics

  In this module you will carry out a major individual research project, forming a significant part of your final year in the MSci programme. Your project may be experimental, theoretical or computational, and will be closely linked to the department's research activity. You will gain experience in planning and managing a substantial piece of scientific work, and develop skills in research techniques, data analysis and presentation. You will also learn the value of collaboration and teamwork in scientific environments. The module culminates in a detailed project report that showcases your work and can support future applications in industry or academia.

• MSci Project in Mathematics

  You will carry out a detailed investigation on a topic of your choosing, guided by an academic supervisor. You will prepare a written report around 10,000 words in length, and give a fifteen-minute presentation outlining your findings.

Optional Modules

There are a number of optional course modules available during your degree studies. The following is a selection of optional course modules that are likely to be available. Please note that although the College will keep changes to a minimum, new modules may be offered or existing modules may be withdrawn, for example, in response to a change in staff. Applicants will be informed if any significant changes need to be made.

Year 1

• All modules are core

Year 2

• Statistical Methods

  In this module you will develop an understanding of statistical modelling, becoming familiar with the theory and the application of linear models. You will learn how to use the classic simple linear regression model and its generalisations for modelling dependence between variables. You will examine how to apply non-parametric methods, such as the Wilxocon and Kolmogorov-Smirnov goodness-of-fit tests, and learn to use the R open source software package.

• Probability Theory

  In this module you will develop an understanding of the basic principles of the mathematical theory of probability. You will use the fundamental laws of probability to solve a range of problems, and prove simple theorems involving discrete and continuous random variables. You will learn how to forumulate an explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

• Graphs and Optimisation

  In this module you will develop an understanding of the basic concepts of graph theory and linear programming. You will consider how railroad networks, electrical networks, social networks, and the web can be modelled by graphs, and look at basic examples of graph classes such as paths, cycles and trees. You will examine the flows in networks and how these are related to linear programming, solving problems using the simplex algorithm and the strong duality theorem.

• Rings and Factorisation

  In this module you will develop an understanding of ring theory and how this area of algebra can be used to address the problem of factorising integers into primes. You will look at how these ideas can be extended to develop notions of 'prime factorisation' for other mathematical objects, such as polynomials. You will investigate the structure of explicit rings and learn how to recognise and construct ring homomorphisms and quotients. You will examine the Gaussian integers as an example of a Euclidean ring, Kronecker's theorem on field extensions, and the Chinese Remainder Theorem.

• Groups and Group Actions

  In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. You will see how groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory. You also will also consider how various counting problems concerning discrete patterns can be solved by means of group actions.

• Further Linear Algebra and Modules

  In this module you will develop an understanding of the language and concepts of linear algebra that are used within Mathematics. You will look at topics in linear algebra and the theory of modules, which can be seen as generalisations of vector spaces. You will learn how to use alternative matrix representations, such as the Jordan canonical or the rational canonical form, and see why they are important in mathematics.

• Real Analysis

  In this module you will develop an understanding of the convergence of series. You will look at the Weierstrass definition of a limit and use standard tests to investigate the convergence of commonly occuring series. You will consider the power series of standard functions, and analyse the Intermediate Value and Mean Value Theorems. You will also examine the properties of the Riemann integral.

Year 3

• Mathematics Project

  You will carry out a detailed investigation on a topic of your choosing, guided by an academic supervisor. You will prepare a written report around 7,000 words in length, and give a ten-minute presentation outlining your findings.

• Mathematics in the Classroom

  In this module you will develop an understanding of a range of methods for teaching children up to A-level standard. You will act act as a role model for pupils, devising appropriate ways to convey the principles and concepts of mathematics. You will spend one session a week in a local school, taking responsibility for preparing lesson plans, putting together relevant learning aids, and delivering some of the classes. You will work with a specific teacher, who will act as a trainer and mentor, gaining valuable transferable skills.

• Number Theory

  In this module you will develop an understanding of how prime numbers are the building blocks of the integers 0, ±1, ±2, … You will look at how simple equations using integers can be solved, and examine whether a number like 2017 should be written as a sum of two integer squares. You will also see how Number Theory can be used in other areas such as Cryptography, Computer Science and Field Theory.

• Computational Number Theory

  In this module you will develop an understanding of a range methods used for testing and proving primality, and for the factorisation of composite integers. You will look at the theory of binary quadratic forms, elliptic curves, and quadratic number fields, considering the principles behind state-of-the art factorisation methods. You will also look at how to analyse the complexity of fundamental number-theoretic algorithms.

• Complexity Theory

  In this module you will develop an understanding the different classes of computational complexity. You will look at computational hardness, learning how to deduce cryptographic properties of related algorithms and protocols. You will examine the concept of a Turing machine, and consider the millennium problems, including P vs NP, with a $1,000,000 prize on offer from the Clay Mathematics Institute if a correct solution can be found.

• Principles of Algorithm Design

  In this module you will develop an understanding of efficient algorithm design and its importance for handling large inputs. You will look at how computers have changed the world in the last few decades, and examine the mathematical concepts that have driven these changes. You will consider the theory of algorithm design, including dynamic programming, handling recurrences, worst-case analysis, and basic data structures such as arrays, stacks, balanced search trees, and hashing.

• Quantum Theory 1

  In this module you will develop an understanding of quantum theory, and the development of the field to explain the behaviour of particles at the atomic level. You will look at the mathematical foundations of the theory, including the Schrodinger equation. You will examine how the theory is applied to one and three dimensional systems, including the hydrogen atom, and see how a probabilistic theory is required to interpret what is measured.

• Quantum Theory 2

  In this module you will develop an understanding of how the Rayleigh-Ritz variational principle and perturbation theory can be used to obtain approximate solutions of the Schrödinger equation. You will look at the mathematical basis of the Period Table of Elements, considering spin and the Pauli exclusion principle. You will also examine the quantum theory of the interaction of electromagnetic radiation with matter.

• Dynamics of Real Fluids

  In this module you will develop an understanding of how the theory of ideal fluids can be used to explain everyday phenomena in the world around us, such as how sound travels, how waves travel over the surface of a lake, and why golden syrup (or volcanic lava) flows differently from water. You will look at the essential features of compressible flow and consider basic vector analysis techniques.

• Non-Linear Dynamic Systems

  In this module you will develop an understanding of non-linear dynamical systems. You will investigate whether the behaviour of a non-linear system can be predicted from the corresponding linear system and see how dynamical systems can be used to analyse mechanisms such as the spread of disease, the stability of the universe, and the evolution of economic systems. You will gain an insight into the 'secrets' of the non-linear world and the appearance of chaos.

• Statistical Inference

  In this module you will develop an understanding of the main priciples and methods of statstics, in particular the theory of parametric estimation and hypotheses testing.You will learn how to formulate statistical problems in mathematical terms, looking at concepts such as Bayes estimators, the Neyman-Pearson framework, likelihood ratio tests, and decision theory.

• Time Series Analysis

  In this module you will develop an understanding of some of the descriptive methods and theoretical techniques that are used to analyse time series. You will look at the standard theory around several prototype classes of time series models and learn how to apply appropriate methods of times series analysis and forecasting to a given set of data using Minitab, a statistical computing package. You will examine inferential and associated algorithmic aspects of time-series modelling and simulate time series based on several prototype classes.

• Markov Chains and Applications

  In this module you will develop an understanding of the probabilistic methods used to model systems with uncertain behaviour. You will look at the structure and concepts of discrete and continuous time Markov chains with countable stable space, and consider the methods of conditional expectation. You will learn how to generate functions, and construct a probability model for a variety of problems.

• Channels

  In this module you will develop an understanding of the mathematics of communication, focusing on digital communication as used across the internet and by mobile telephones. You looking at compression, considering how small a file, such as a photo or video, can be made, and therefore how the use of data can be minimised. You will examine error correction, seeing how communications may be correctly received even if something goes wrong during the transmission, such as intermittent wifi signal. You will also analyse the noiseless coding theorem, defining and using the concept of channel capacity.

• Quantum Information Theory

  In this module you will develop an understanding of how the behaviour of quantum systems can be harnessed to perform information processing tasks that are otherwise difficult, or impossible, to carry out. You will look at basic phenomena such as quantum entanglement and the no-cloning principle, seeing how these can be used to perform, for example, quantum key distribution. You will also examine a number of basic quantum computing algorithms, observing how they outperform their classical counterparts when run on a quantum computer.

• Mathematics of Financial Markets

  In this module you will develop an understanding of how financial markets operate, with a focus on the ideas of risk and return and how they can be measured. You will look at the random behaviour of the stock market, Markowitz portfolio optimisation theory, the Capital Asset Pricing Model, the Binomial model, and the Black-Scholes formula for the pricing of options.

• Financial Mathematics II

  This module continues the study of financial mathematics begun in Financial Mathematics I. In this module, students will develop a further understanding of the role of mathematics in securities markets. The topics in the module may include models of interest rates, elements of portfolio theory and models for financial returns.

• Combinatorics

  In this module you will develop an understanding of some of the techniques and concepts of combinatorics, including methods of counting, generating functions, probabilistic methods, permutations, and Ramsey theory. You will see how algebra and probability can be used to count abstract mathematical objects, and how to calculate sets by inclusion and exclusion. You will examine the applications of number theory and consider the use of simple probabilistic tools for solving combinatorial problems.

• Error Correcting Codes

  In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

• Cipher Systems

  In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

• Public Key Cryptography

  In this module you will develop an understanding of public key cryptography and the mathematical ideas that underpin it, including discrete logarithms, lattices and elliptic curves. You will look at several important public key cryptosystems, including RSA, Rabin, ElGamal encryption and Schnorr signatures. You will consider notions of security and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen ciphertext attack.

• Applications of Field Theory

  In this module you will develop an understanding of Field Theory. You will learn how to express equations such as X²⁰¹⁷=1 in a formal algebraic setting, how to classify finite fields, and how to determine the number of irreducible polynomials over a finitie field. You will also consider some the applications of fields, including ruler and compass constructions and why it is impossible to generically trisect an angle using them.

• Nonlinear Dynamical Systems

  This module introduces you to the fundamentals of the analysis of non-linear dynamical systems and, in particular, investigates whether the behaviour of a non-linear system can be predicted from the corresponding linear system. You will cover systems of first-order linear differential equations, similarity types for matrices and their connection with linear systems, the classification of two-dimensional linear phase portraits with extension to three dimensions, non-linear differential equations through Liapunov’s stability analysis, periodic solutions and limit cycles, and the Poincaré–Bendixson theorem, as well as applications to problems from physics, biology and economics. You will also cover non-linear difference equations, Poincaré surface of section, stability of critical points, and routes to chaos.

Year 4

• Computational Number Theory

  In this module you will develop an understanding of a range of methods used for testing and proving primality, and for the factorisation of composite integers. You will look at the theory of binary quadratic forms, elliptic curves, and quadratic number fields, considering the principles behind state-of-the-art factorisation methods. You will also look at how to analyse the complexity of fundamental number-theoretic algorithms.

• Complexity Theory

  In this module you will develop an understanding the different classes of computational complexity. You will look at computational hardness, learning how to deduce cryptographic properties of related algorithms and protocols. You will examine the concept of a Turing machine, and consider the millennium problems, including P vs NP, with a $1,000,000 prize on offer from the Clay Mathematics Institute if a correct solution can be found.

• Principles of Algorithm Design

  In this module you will develop an understanding of efficient algorithm design and its importance for handling large inputs. You will look at how computers have changed the world in the last few decades, and examine the mathematical concepts that have driven these changes. You will consider the theory of algorithm design, including dynamic programming, handling recurrences, worst-case analysis, and basic data structures such as arrays, stacks, balanced search trees, and hashing.

• Quantum Theory 2

  In this module you will develop an understanding of how the Rayleigh-Ritz variational principle and perturbation theory can be used to obtain approximate solutions of the Schrödinger equation. You will look at the mathematical basis of the Period Table of Elements, considering spin and the Pauli exclusion principle. You will also examine the quantum theory of the interaction of electromagnetic radiation with matter.

• Inference

  In this module you will develop an understanding of the main priciples and methods of statstics, in particular the theory of parametric estimation and hypotheses testing.You will learn how to formulate statistical problems in mathematical terms, looking at concepts such as Bayes estimators, the Neyman-Pearson framework, likelihood ratio tests, and decision theory.

• Applied Probability

  In this module you will develop an understanding of the probabilistic methods used to model systems with uncertain behaviour. You will look at the structure and concepts of discrete and continuous time Markov chains with countable stable space, and consider the methods of conditional expectation. You will learn how to generate functions, and construct a probability model for a variety of problems.

• Channels

  In this module you will develop an understanding of the mathematics of communication, focusing on digital communication as used across the internet and by mobile telephones. You looking at compression, considering how small a file, such as a photo or video, can be made, and therefore how the use of data can be minimised. You will examine error correction, seeing how communications may be correctly received even if something goes wrong during the transmission, such as intermittent wifi signal. You will also analyse the noiseless coding theorem, defining and using the concept of channel capacity.

• Quantum Information and Coding

  In this module you will develop an understanding of how the behaviour of quantum systems can be harnessed to perform information processing tasks that are otherwise difficult, or impossible, to carry out. You will look at basic phenomena such as quantum entanglement and the no-cloning principle, seeing how these can be used to perform, for example, quantum key distribution. You will also examine a number of basic quantum computing algorithms, observing how they outperform their classical counterparts when run on a quantum computer.

• Financial Mathematics II

  This module continues the study of financial mathematics begun in Financial Mathematics I. In this module, students will develop a further understanding of the role of mathematics in securities markets. The topics in the module may include models of interest rates, elements of portfolio theory and models for financial returns.

• Combinatorics

  In this module you will develop an understanding of some of the techniques and concepts of combinatorics, including methods of counting, generating functions, probabilistic methods, permutations, and Ramsey theory. You will see how algebra and probability can be used to count abstract mathematical objects, and how to calculate sets by inclusion and exclusion. You will examine the applications of number theory and consider the use of simple probabilistic tools for solving combinatorial problems.

• Error Correcting Codes

  In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

• Cipher Systems

  In this module you will develop an understanding of secure communication and how cryptography is used to achieve this. You will look at some of the historical cipher systems, considering what security means and the kinds of attacks an adversary might launch. You will examine the structure of stream ciphers and block ciphers, and the concept of public key cryptography, including details of the RSA and ElGamal cryptosystems. You will see how these techniques are used to achieve privacy and authentication, and assess the problems of key management and distribution.

• Public Key Cryptography

  In this module you will develop an understanding of public key cryptography and the mathematical ideas that underpin it, including discrete logarithms, lattices and elliptic curves. You will look at several important public key cyptosystems, including RSA, Rabin, ElGamal encryption and Schnorr signatures. You will consider notions of security and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen ciphertext attack.

• Applications of Field Theory

  In this module you will develop an understanding of Field Theory. You will learn how to express equations such as X²⁰¹⁷=1 in a formal algebraic setting, how to classify finite fields, and how to determine the number of irreducible polynomials over a finitie field. You will also consider some the applications of fields, including ruler and compass constructions and why it is impossible to generically trisect an angle using them.

• Topology

  In this module you will develop an understanding of geometric objects and their properties. You will look at objects that are preserved under continuous deformation, such as through stretching or twisting, and will examine knots and surfaces. You will see how colouring a knot can be used to determine whether or not it can be transformed into the unknot without any threading. You will also consider why topologists do not distinguish between a cup and a donut.

• Lie Groups and Lie Algebras

  In this module you will study the structure and representation theory of Lie groups and Lie algebras. Topics include matrix Lie groups, the exponential map, the Baker–Campbell–Hausdorff formula, representations and tensor products, root systems, the Cartan–Weyl basis, and the classification of simple Lie algebras.

• Statistical Mechanics

  In this module you will deepen your understanding of statistical mechanics and its application to both equilibrium and non-equilibrium systems. You will review core concepts such as the grand canonical ensemble and distribution functions for bosons and fermions. The module explores weakly interacting systems through the virial expansion and models like the imperfect and van der Waals gases, and introduces mean field theory for magnetic systems. You will examine strongly interacting systems and the nature of phase transitions, studying models such as the Ising model and Landau theory, and learn to distinguish between first- and second-order transitions. Topics such as broken symmetry, Goldstone bosons and critical fluctuations will be covered. The module concludes with an introduction to dissipative systems, including the fluctuation–dissipation theorem, Langevin dynamics and correlation functions, providing insight into how systems evolve towards equilibrium.

• Advanced Quantum Theory

  In this module you will develop a deeper understanding of the mathematical formalism of quantum mechanics, including the use of density operators, evolution operators and different representations of quantum dynamics. You will explore the role of symmetry and conservation laws in quantum systems, focusing on translations, parity and rotations. The module introduces semiclassical methods such as the Wentzel–Kramers–Brillouin approximation for solving the Schrödinger equation in one dimension. You will study the physics of identical particles and learn the basics of second quantisation, a key tool in many-particle quantum theory. The concept of quantum entanglement is introduced, along with its fundamental implications. The module also covers selected advanced topics, such as Feynman path integrals and quantum scattering, providing insight into the foundations and applications of modern quantum theory.

• Photonics and Metamaterials

  In this module you will explore key theoretical and practical aspects of modern photonic technologies, with a focus on emerging trends in nanophotonics and their real-world applications. You will study the physics underlying technologies such as lasers, plasmonics, metamaterials, biosensing and bio-imaging. The module introduces both fundamental and advanced concepts in photonics, helping you understand how light interacts with matter at micro- and nano-scales. You will apply these concepts to solve model problems, develop skills in analytical and numerical methods, and work effectively in group settings. By the end of the module, you will be able to relate photonic principles to cutting-edge technologies used in research, industry and healthcare.

• Particle Accelerator Physics

  This module introduces you to particle accelerator physics, starting with the history of accelerators and their basic principles, including centre-of-mass energy, luminosity and the accelerating gradient. You will then cover the characteristics of modern accelerators and colliders, such as LEP, the LHC, and b-factories. Focus will then shift to the propagation of particle beams and bunches, how they behave and how they are controlled, covered through topics on transverse motion, principles of beam cooling, strong focusing, simple lattices, circulating beams, synchrotron radiation, and longitudinal dynamics. You will learn about accelerating structures such as radio-frequency cavities and how superconductivity is applied in particle acceleration. You will cover contemporary applications of accelerators as light sources and their medical uses before moving on to discussions of the future of accelerators and colliders, including the ILC, neutrino factories, muon colliders, and laser plasma acceleration.

• Theoretical Treatments of Nano-systems

  In this module you will study the quantum mechanical modelling of materials at the nanoscale. You will explore key theoretical methods such as Density Functional Theory, Hartree-Fock, molecular dynamics and Monte Carlo techniques. The course focuses on how these tools are used to understand and predict properties of complex systems.

• Nano-Electronics and Quantum Technology

  In this module you will explore the rapidly advancing field of nano-electronic devices, with a focus on their applications in quantum computation and metrology. You will study low-dimensional electronic transport in systems such as two-dimensional electron gases, quantum wires and quantum dots, and examine phenomena including ballistic conduction, Landau levels and the Quantum Hall effect. The module covers tunnel junction devices such as single-electron transistors and Josephson junctions, and their roles in defining quantum standards of resistance, voltage and current. You will also study quantum devices based on two-level systems, including transmon qubits and superconducting circuits, and gain insight into quantum coherence and control. In addition, the module introduces nano-fabrication techniques such as lithography, deposition and etching, alongside nano-analysis methods using scanning probe and electron microscopy, and spectroscopic tools.

• Standard Model Physics and Beyond

  In this module you will study advanced particle physics, focusing on field theory, symmetries and the Standard Model. Topics include gauge theories, spontaneous symmetry breaking, the Higgs mechanism and decay processes. You will develop analytical skills, apply theoretical tools, and communicate complex ideas effectively in both specialist and general contexts.

• Statistical Data Analysis

  In this module you will develop a solid foundation in probability and statistical methods used in the analysis of experimental data in physics. You will learn key concepts such as random variables, probability density functions, expectation values and error propagation. The module introduces the Monte Carlo method, including random number generation, transformation techniques and Markov Chain Monte Carlo. You will study statistical testing using both frequentist and Bayesian approaches, exploring significance tests, p-values, and the treatment of nuisance parameters. You will also learn methods for parameter estimation, such as maximum likelihood, least squares and Bayesian inference. Through these tools, you will gain the skills needed to analyse and interpret data in modern physics research.

• String Theory and Branes

  In this module you will study the classical and quantum dynamics of particles and strings. Topics include the quantisation of open and closed bosonic strings, brane dynamics and the supersymmetric particle. Advanced topics may cover D-branes, covariant string theory, the Brink–Schwarz particle and electromagnetic duality symmetries.

• Supersymmetry and Conformal Field Theory

  In this module you will explore advanced topics in supersymmetry and conformal field theory. You will study extensions of the Poincaré algebra, spinors, Clifford algebras, and representations of supersymmetry and conformal algebras. The module includes N=1 supersymmetric models, scale dependence, and optional advanced topics such as supergravity and supersymmetric localisation.

• Astroparticle Cosmology

  The module will focus on cosmology and gravitational and high-energy physics in the context of the early Universe. The standard cosmological model will be developed, the need for inflation and the various inflationary models will be studied in depth, and baryogenesis, leptogenesis and nucleosynthesis will be discussed. Spontaneously broken symmetries and the formation of topological defects will be covered, with the physics of topological defects—particularly cosmic strings—explored in detail. The hot topics of dark matter and dark energy will also be examined, along with the origin of large-scale structure and cosmic microwave background temperature anisotropies.

• Theory of Complex Networks

  In this module you will study the structure and dynamics of complex networks. Topics include microscopic properties such as degree, clustering and centrality; macroscopic features like degree distributions and modularity; and processes on networks including diffusion, random walks and PageRank. You will also explore random graph models and percolation transitions.

• Equilibrium Analysis of Complex Systems

  In this module you will study advanced methods in statistical mechanics, including canonical ensembles, transfer matrices and asymptotic techniques. You will explore approximation methods and tools for disordered systems, such as the replica and cavity methods. Applications include physical, biological, informational and economic systems, optimisation problems, and network-based risk models.

• Dynamical Analysis of Complex Systems
• Mathematical Biology

  In this module you will explore mathematical models in biology, focusing on population dynamics, infectious disease spread and molecular evolution. You will study continuous and discrete models, ODE-based epidemic models, reaction kinetics and probabilistic approaches. The course also introduces nonlinear dynamics and chaos in biological systems.

• Elements of Statistical Learning
• Advanced Classical Physics

  This module introduces you to advanced topics and methods in classical physics (including mechanics, electromagnetism and relativity). You will cover the Calculus of Variations including the Euler-Lagrange equation, constraints and the method of Lagrange multipliers; non-relativistic classical physics through the principle of stationary action and Lagrangian mechanics, integration of the equation of motion in one dimension, motion in a non-inertial reference frame, Hamiltonian mechanics, symmetries, constants of motion and conservation laws, canonical transformations and Liouville's theorem; the module will end with a dive into relativitstic physics covering the principle of relativity and relativistic formalism, Lagrangian of a charged particle in an electromagnetic field, equations of motion of a charged paritcle in an EM field and gauge invariance, electromagnetic field tensor, relativistic invariants of the field, covariant formulation of the Lorentz force and Maxwell's equations.

• Further Mathematical Methods

  This module introduces you to the mathematical areas of complex analysis, matrix theory and linear transformations, and group theory, and enables you to apply related advanced mathematical formalisms and methods to formulate and solve physics problems. In the complex analysis section, you will cover holomorphic functions, power series, logarithms, isolated singular points, complex integration, Cauchy’s theorem and integral formula, Laurent series, Cauchy’s residue theorem, and applications of contour integration. For the matrix theory and linear transformations topic, you will cover the fundamentals of matrix theory, unitary diagonalisation, vector spaces and linear transformations, matrix representations, equivalence and similarity, inner product, norm, orthonormal bases, dual space and bra–ket notation. For the final chapter, you will cover the fundamentals of group theory, crystal lattice symmetries, Lie groups and Lie algebra, reducible and irreducible group representations.

• Quantum Field Theory

  In this module you will develop an understanding of quantum field theory through the study of relativistic wave equations and field quantisation. You will begin with an introduction to the Lorentz group and four-vector notation, followed by a review of classical field theory using relativistic formalism and Noether’s theorem. You will explore scalar fields through the Klein–Gordon equation and then move on to canonical quantisation of free scalar fields, introducing commutators, time-ordered products and the Feynman propagator. The module also covers classical spinor fields and the Dirac equation, including its symmetries, solutions and interaction with electromagnetic fields. You will study interacting quantum field theories using the interaction picture and S-matrix, and learn how to apply Feynman rules to calculate tree-level diagrams for particle decays and scattering processes.

• Modelling Quantum Many-Body Systems

  The purpose of this module is to provide an introduction to the theory and applications of quantum many-body systems. The module will focus primarily on systems at or close to equilibrium, with a view towards non-equilibrium quantum systems.

• Advanced Condensed Matter

  In this module you will explore advanced topics in condensed matter physics, focusing on the quantum phenomena that arise in complex materials. You will learn how to describe electronic properties using theoretical frameworks such as second quantisation, tight-binding models and effective low-energy Hamiltonians. The course covers key quantum effects including the Kondo effect, magnetic frustration and Wigner crystallisation, and explains how they are probed through experimental techniques like ARPES, STM and X-ray spectroscopy. You will gain experience with mathematical tools and numerical techniques, including Green’s functions, linear response theory and mean-field methods. The module introduces foundational models such as the Hubbard and t-J models, and explains their role in understanding correlated electron systems and unconventional superconductivity. You will also be introduced to density functional theory (DFT) and its extensions, with practical insight into tools like Quantum ESPRESSO and Wannier90 for constructing simplified models from first-principles calculations. Throughout the module, you will connect theoretical ideas to experimental data and develop the skills to analyse and model modern quantum materials.

• Gravitational Wave Physics

  In this module you will be introduced to the theory of gravitational waves, from their origins in Einstein’s General Relativity to their role in modern astrophysics. You will explore how gravitational waves are generated, detected and analysed, preparing you for research in this rapidly growing area of physics.

• Dark Matter & Dark Energy

  Dark matter and dark energy make up 95% of the energy content of the Universe. This course takes a research- and literature-focused view, covering astrophysics and theoretical physics, with a coding element. It begins with astrophysical and cosmological evidence for dark matter and dark energy, from rotation curves of galaxies, gravitational lensing and the large-scale structure of the Universe. It then focuses on theories of dark matter, including WIMPs, axions and primordial black holes. Each of these types of dark matter can be searched for, and the main channels of direct and indirect detection are introduced.

• Research Topics in Astrophysics
• Advanced Particle Physics

  In this module you will develop a detailed understanding of the mathematical framework of the Standard Model of Particle Physics. You will study the relativistic wave equations that describe fundamental particles, including the Klein–Gordon and Dirac equations, and explore their symmetries and physical interpretations. You will learn how to apply Feynman diagrams and perturbation theory to calculate cross-sections and decay rates for high-energy processes. The module covers the development of the Standard Model through quantum electrodynamics, weak interactions, and electroweak unification, including the Higgs mechanism. You will also study quantum chromodynamics, colour charge, parton models and the physics of hadron colliders. The theoretical tools developed in the course are used to compute predictions for key particle reactions, and these are compared with experimental results.

• Gravitational Waves Astronomy

  This module introduces students to the data analysis of gravitational-wave astronomy. The module begins with short general background of the field: the theory of gravitational waves, interferometric detectors, and gravitational-wave sources. Then, in turn it covers the topics of gravitational-wave data, source modelling, searching for signals, and estimating the parameters of the signals. For each topic, the basic data analysis theory is covered with examples. Following this, the module then turns to the astrophysical implications of the discoveries to date: cosmological measurements of the expansion rate of the Universe and searches for evidence of beyond-General Relativity physics.

• Advanced Astrophysics

  In this module you will build on previous knowledge to develop a deeper understanding of advanced astrophysics, preparing you for independent research. You will study radiative processes in astrophysical environments, including radiative transfer, interaction of radiation with matter, optical depth and thermal emission. Key concepts such as radiative flux, specific intensity and Eddington luminosity will be explored, along with mechanisms like Thomson scattering and Bremsstrahlung. The module also covers astrophysical fluid dynamics, examining the behaviour of collisional and collisionless fluids, conservation laws, pressure forces and gravitational effects through the virial theorem and Poisson’s equation. You will analyse fluid instabilities and apply mathematical tools to interpret and model radiative and dynamical processes in a variety of astrophysical settings

• Order and Excitations in Quantum Materials

  In this module you will explore the atomic-scale structure and excitations of condensed matter, gaining insight into the physical principles that govern a wide range of materials. You will study the structure of crystalline materials using reciprocal space and scattering theory, including X-ray and neutron techniques, and learn how to interpret experimental data through concepts such as the Laue condition and Ewald construction. The module covers lattice vibrations, phonons and anharmonic interactions, with applications to both one-dimensional and three-dimensional systems. You will also examine magnetism in solids, including magnetic moments, ordering, and excitations including spin-density waves and crystal field effects. The module includes a study of phase transitions and critical phenomena using models such as Landau theory and the Ising model. You will explore local structural order in liquids and amorphous solids, and be introduced to key spectroscopic tools and sources such as synchrotrons and neutron facilities, supported by a visit to a national research centre.

• Superconductors and Superfluids

  Superconductors demonstrate remarkable macroscopic quantum behaviour and underpin quantum technologies. This module introduces the superconductivity in the context of the more general phenomenon of superfluidity, presents the unique electrical, magnetic and thermodynamic properties of the superconductors in conjunction with phenomenological theory and technological applications, and discusses the microscopic physics that leads to the superconductivity and various classes of superconducting materials.