Mathematical and Theoretical Physics

Core modules:

Analytical Dynamics and Classical Field Theory

Newton formulated his famous laws of mechanics and gravitation in the 17th century, but only later did it become clear, through the work of mathematicians Lagrange, Hamilton and Jacobi, that underlying Newton's work are deep mathematical structures. In the first semester, we will reformulate and unify mechanics and classical field theory (e.g. Maxwell's equations), starting with the stationary-action principle. We will derive Noether's theorem, which relates the symmetries of a system to conserved quantities in physics (such as energy, momentum and charge). In the second semester, we will explore Einstein's general theory of relativity, in which gravity arises through the curvature of spacetime. We will apply the mathematical tools of Riemannian geometry to explore some radical consequences of the theory, such as black holes, the big bang, and the expanding universe.

30 credits

Dissertation

The dissertation is piece of extensive work (10-20,000 words) which provides students' with the opportunity to synthesise theoretical knowledge on a subject that is of interest to them. Students will gain experience of the phases of a relatively large piece of work: planning to a deadline; researching background information; problem specification; the carrying through of relevant analyses; and reporting, both at length through the dissertation and in summary through an oral presentation.

60 credits

Research Skills in Mathematics

The unit provides training in research-level skills used in mathematics and related technical disciplines, with a particular focus on mathematical/scientific writing and presentation skills. Students will gain experience in the use of appropriate computer packages for the presentation of mathematical and statistical material, and guidance on how to prepare a coherent, structured and accurate report. Topics covered include conducting literature searches, summarising information, accessing papers through preprint servers, and managing reference lists. Students will draw on these skills to write their own literature review in an area of interest to them, and prepare and deliver a talk on this topic.

15 credits

Maths options - between 15 and 60 credits from:

Topics in Mathematical Physics

This unit will introduce students to advanced concepts and techniques in modern mathematical physics, in preparation for research-level activities.

It is assumed that the student comes equipped with a working knowledge of analytical dynamics, and of non-relativistic quantum theory.

We will examine how key physical ideas are precisely formulated in the language of mathematics. For example, the idea that fundamental particles arise as excitations of relativistic quantum fields finds its mathematical realisation in Quantum Field Theory. In QFT, particles can be created from the vacuum, and destroyed, but certain other quantities such as charge, energy, and momentum are conserved (after averaging over quantum fluctuations).

We will examine links between conservation laws and invariants, and the underlying (discrete or continuous) symmetry groups of theories. We will also develop powerful calculation tools. For example, to find the rate of creation of new particles in a potential, one must evaluate the terms in a perturbative (Feynman-diagram) expansion.

For details of the current syllabus, please consult the module leader.

15 credits

Machine Learning

Machine learning lies at the interface between computer science and statistics. The aims of machine learning are to develop a set of tools for modelling and understanding complex data sets. It is an area developed recently in parallel between statistics and computer science. With the explosion of “Big Data”, statistical machine learning has become important in many fields, such as marketing, finance and business, as well as in science. The module focuses on the problem of training models to learn from training data to classify new examples of data.

15 credits

Advanced Topics in Waves and Fluid Dynamics A

Waves and Fluid Dynamics are cornerstones of Applied Mathematics.  Both relate to the flow of fluids, i.e., propagation of information, which include not only gas (e.g., air) and liquid (e.g., water), but also in more complex media (e.g., lubricants and blood), and other materials or even the fourth state of matter: plasma.  The scientific principles and mathematical techniques involved in studying these are of inherent interest.  Wave motions give rise to well-known class of partial differential equations, and relate to concepts such as standing, progressive, and shock (i.e., nonlinear) waves; we can study these using Fourier series, Laplace transform and the powerful method of characteristics.  Viscous fluid flow gives rise to the Navier-Stokes equations.  The first semester will cover some of these ideas, while the second will move onto more advanced topics, such as three-dimensional flows, boundary layers, vortex dynamics, or magnetohydrodynamics.

30 credits

Financial Mathematics

The discovery of the Capital Asset Pricing Model by William Sharpe in the 1960's and the Black-Scholes option pricing formula a decade later mark the beginning of a very fruitful interaction between mathematics and finance. The latter obtained new powerful analytical tools while the former saw its knowledge applied in new and surprising ways. (A key result used in the derivation of the Black-Scholes formula, Ito's Lemma, was first applied to guide missiles to their targets; hence the title 'rocket science' applied to financial mathematics). This course describes the mathematical ideas behind these developments together with their application in modern finance, and includes a project.

15 credits

Mathematical Modelling of Natural Systems

Mathematical modelling enables insight into a wide range of scientific problems. This module will provide a practical introduction to techniques for modelling natural systems. Students will learn how to construct, analyse and interpret mathematical models, using a combination of differential equations, scientific computing and mathematical reasoning. Students will learn the art of mathematical modelling: translating a scientific problem into a mathematical model, identifying and using appropriate mathematical tools to analyse the model, and finally relating the significance of the mathematical results back to the original problem. Study systems will be drawn from throughout the environmental and life sciences.

15 credits

Probability with Measure Theory

Probability is a relatively new part of mathematics, first studied rigorously in the early part of 20th century. This module introduces the modern basis for probability theory, coming from the idea of 'measuring' an object by attaching a non-negative number to it. This might refer to its length or volume, but also to the probability of an event happening. We therefore find a close connection between integration and probability theory, drawing upon real analysis. This rigorous theory allows us to study random objects with complex or surprising properties, which can expand our innate intuition for how probability behaves. The precise material covered in this module may vary according to the lecturer's interests.

15 credits

Physics options – between 15 and 60 credits from:

Advanced Particle Physics

The module provides students with a comprehensive understanding of modern particle physics. Focusing on the standard model, it provides a theoretical underpinning of this model and discusses its predictions. Recent developments including the discovery of the Higgs Boson and neutrino oscillation studies are covered. A description of the experiments used to probe the standard model is provided. Finally the module looks at possible physics beyond the standard model.

15 credits

Advanced Electrodynamics

This module gives a detailed mathematical foundation for modern electrodynamics, starting from Maxwell's equations, charge conservation and the wave equation, to gauge invariance, waveguides, cavities and antennas, and an introduction to quantum electrodynamics. After a brief recap of vector calculus, we explore the role of the scalar and vector potential, the multi-pole expansion of the field, the Poisson and Laplace equations, energy and momentum conservation of the fields, and waveguides and cavities. After a relativistic treatment of the fields we consider the quantisation of the electromagnetic field modes, the Hamiltonian for the dipole coupling between a field and a radiation emitter, and finally we explore the Aharonov-Bohm effect.

15 credits

Advanced Quantum Mechanics

Quantum mechanics at an intermediate to advanced level, including the mathematical vector space formalism, approximate methods, angular momentum, and some contemporary topics such as entanglement, density matrices and open quantum systems. We will study topics in quantum mechanics at an intermediate to advanced level, bridging the gap between the physics core and graduate level material.   The syllabus includes a formal mathematical description in the language of vector spaces; the description of the quantum state in Schrodinger and Heisenberg pictures, and using density operators to represent mixed states; approximate methods: perturbation theory,  variational method and time-dependent perturbation theory;  the theory of angular momentum and spin; the treatment of identical particles; entanglement; open quantum systems and decoherence. The problem solving will provide a lot of practice at using vector and matrix methods and operator algebra techniques. The teaching will take the form of traditional lectures plus weekly problem classes where you will be provided with support and feedback on your attempts.

15 credits

Advanced Soft Matter and Biological Physics

Fascinating behaviour of soft matter and biological systems often occurs at thermal energy scales and can be described by statistical mechanical models. In addition, living biological matter is driven out of equilibrium due to internal biochemical sources of energy. Mathematical models and modern advanced experimental techniques are revealing the physical principles underpinning the biological world and the technological possibilities of complex soft materials.Much recent progress in soft matter and biology has been made thanks to the advent of advanced experimental  techniques which we will show are based on elegant physical principles. We will also study the physical principles underpinning the behaviour of complex soft matter and biological materials. We will describe phase transitions in multiple soft matter systems by calculating free energies. We will use random walk models to describe the shape of polymer molecules and the Brownian motion of colloids. We will also study the dynamics of polymers and the kinetics of polymerisation. We will then consider how polymerisation of protein filaments and action of molecular motors can generate forces in biological cells. This will involve us introducing concepts of systems that are in equilibrium versus out of equilibrium. Using a mathematical  framework we can describe behaviour at different length scales for example from the cytoskeleton to tissues, bacteria colonies and flocking. We will also investigate how the energy required for life is captured in photosynthesis.

15 credits

Introduction to Cosmology

The aim of this course is to provide students with an understanding of the Universe as its own entity. Students will learn how the contents of the Universe affect its dynamic evolution, and how we can use observations of Type 1a Supernovae and the Cosmic Microwave Background to constrain the properties of the Universe. Students will also learn about key epochs during the history of the Universe, from inflation through to nucleosynthesis, recombination, and reionisation, before learning how the first stars and galaxies started to form. Throughout a series of lectures, students will first learn that spacetime forms the fabric of the Universe, and how the contents of the Universe in the form of dark energy, dark and baryonic matter, and radiation dictate the dynamic evolution of the Universe. Students will next learn about modern precision cosmology, whereby cosmologists use observations of Type 1a Supernovae and the Cosmic Microwave Background to measure various cosmological parameters. This aspect of the course will form the basis of a computer programming-based assessment. Toward the end of the lecture course, students will learn about the epochs of inflation, nucleosynthesis, recombination and reionisation, before learning how today's stars and galaxies began to form. Finally, students will learn about current cosmological research via a literature review.

15 credits

Quantum Optics and Quantum Computing

Quantum computing is introduced through the fundamental concepts of quantum gates and circuits before moving to cover more advanced topics such as quantum programming, quantum algorithms and quantum error correction. These concepts are then applied by studying how programming quantum circuits can be done using cloud computers (e.g. using openQASM format) and the implementation of quantum algorithms (including examples) and quantum error correction using stabiliser formalism and graph states and quantum error correction codes.

The second part of the module covers quantum optics and quantum optical applications at the forefront of current research in the field. This includes topics such as weak and strong coupling of dipole sources in a cavity, single photon sources, protocols of quantum optical communications and linear optics computation. The module then progresses to quantum optical applications. Cavity electrodynamics is studied in the regimes of strong and weak coupling of matter excitations to the electromagnetic field in optical microstructures. This will lead to the physics of highly efficient single photon devices necessary for linear optics quantum computation. The effects of entanglement and quantum teleportation will be also considered.

15 credits