Appendices

B - Hilary Courses

Advanced Fluid Dynamics

Department: Physics

Lecturers: Prof Paul Dellar and Prof Michael Barnes

Course Weight: 1 unit/ 16 lectures

Assessment Method: written exam in week 0 TT or homework completion

General Prerequisites: Basic familiarity with fluid equations and stress tensors as provided, e.g., by Kinetic Theory

Course Synopsis:  (Part 1) Low Reynolds number hydrodynamics. The Stokes flow regime, general mathematical results, flow past a sphere. Stresses due to suspended rigid particles. Calculation of the Einstein viscosity for a dilute suspension. Stresses due to Hookean bead-spring dumb-bells. Derivation of the upper convected Maxwell and Oldroyd-B models for viscoelastic fluids. Properties of such fluids. Suspensions of orientable particles, Jeffery's equation, very brief introduction to active suspensions and liquid crystals.
(Part 2) Validity of the MHD approximation. Conservation equations. Magnetic force.  Evolution of the magnetic field. MHD waves. Static MHD equilibria. Relaxation.  MHD stability (normal modes, energy principle, application to a z-pinch). Non-ideal MHD. 

Advanced Quantum Field Theory

Department: Physics

Lecturer: Prof John Wheater (Lectures 1-12), Prof John March-Russell (lectures 13-24)

Course Weight: 1.5 units/24 lectures

Assessment Method: written exam in TT week 0

Prerequisites: Quantum Field Theory (MT), Groups and Representations (MT)

Course Synopsis: 

1. Feynman Path integral and generating functionals
2. Quantising the Abelian gauge field,  Fadeev-Popov mechanism and ghost fields
3. Scalar and fermionic QED:  tree graph processes
4. QED at one loop: dimensional regularization, BRST, Ward identities, renormalization
5. Introduction to non-abelian gauge theory, gauge fixing, Feynman rules and scattering processes in QCD
6. Renormalization group and effective field theory
7. Spontaneous symmetry breaking. The Higgs mechanism.
8. Introduction to non-perturbative QFT: Basics of confinement and chiral symmetry breaking. Solitons

Sequels: The Standard Model and Beyond 1 & 2 (TT), Conformal Field Theory (TT), Quantum Field Theory in Curved Space-Time (TT)

Algorithms and Computations in Theoretical Physics

Department: Physics

Lecturer: Prof Werner Krauth 

Course Weight: 1 unit/16 lectures

Assessment Method: homework completion only

Course Synopsis: This course introduces to algorithms and scientific computing from the viewpoint of statistical physics. It is also a practical, example-based, primer on subjects such as Markov chains, molecular dynamics, phase transitions, path integrals, superfluids and Bose-Einstein condensation, among others. The course stresses rigorous foundations and modern developments in mathematics (mixing, non-reversibility, perfect sampling,... ) and physics (entropic phase transition, KosterlitzThouless physics, cold atoms,... ), yet is entirely based on short Python programs. It will prepare advanced undergraduates in theoretical physics/math for the requirements of modern-day research, with the huge roles played by algorithmic thinking and by statistics, both with their power, their paradoxes and their intricacies.

For further details see here: course page

C3.2 Geometric Group Theory

Department: Maths

Lecturer: Prof Panos Papazoglou

Course Weight: 1 unit/16 lectures

Assessment Method: written exam in TT

General Prerequisites: Some familiarity with Cayley graphs, fundamental group and covering spaces (as for example in the course B3.5 Topology & Groups) would be a helpful though not essential prerequisite.

Course Synopsis: The aim of this course is to introduce the fundamental methods and problems of geometric group theory and discuss their relationship to topology and geometry. The first part of the course begins with an introduction to presentations and the list of problems of M. Dehn. It continues with the theory of group actions on trees and the structural study of fundamental groups of graphs of groups.The second part of the course focuses on modern geometric techniques and it provides an introduction to the theory of Gromov hyperbolic groups. 

C3.5 Lie Groups

Department: Maths

Lecturer: Prof Pierrick Bousseau

Course Weight: 1 unit/16 lectures

Assessment Method: written exam in TT

General Prerequisites: ASO: Group Theory, A5: Topology and ASO: Multidimensional Analysis and Geometry are all useful but not essential. It would be desirable to have seen notions of derivative of maps from Rn to Rm, inverse and implicit function theorems, and submanifolds of Rn. Acquaintance with the notion of an abstract manifold would be helpful but not really necessary. 

Course Synopsis: The theory of Lie Groups is one of the most beautiful developments of pure mathematics in the twentieth century, with many applications to geometry, theoretical physics and mechanics. The subject is an interplay between geometry, analysis and algebra. Lie groups are groups which are simultaneously manifolds, that is geometric objects where the notion of differentiability makes sense, and the group multiplication and inversion are differentiable maps. The majority of examples of Lie groups are the familiar groups of matrices. The course does not require knowledge of differential geometry: the basic tools needed will be covered within the course.

C3.11 Riemannian Geometry

Department: Maths

Lecturer: Prof Andrew Dancer

Course Weight: 1 unit/16 lectures

Assessment Method: written exam in TT

General Prerequisites: Differentiable Manifolds is required. An understanding of covering spaces will be strongly recommended.
Course Synopsis: Riemannian Geometry is the study of curved spaces and provides an important tool with diverse applications from group theory to general relativity. The surprising power of Riemannian Geometry is that we can use local information to derive global results. This course will study the key notions in Riemannian Geometry: geodesics and curvature. Building on the theory of surfaces in R3 in the Geometry of Surfaces course, we will describe the notion of Riemannian submanifolds, and study Jacobi fields, which exhibit the interaction between geodesics and curvature. We will prove the Hopf--Rinow theorem, which shows that various notions of completeness are equivalent on Riemannian manifolds, and classify the spaces with constant curvature.The highlight of the course will be to see how curvature influences topology. We will see this by proving the Cartan--Hadamard theorem, Bonnet--Myers theorem and Synge's theorem.

C3.12 Low-dimensional Topology and Knot Theory

Department: Maths

Lecturer: Prof Andras Juhasz

Course Weight: 1 unit/ 16 lectures

Assessment Method: written exam in TT

General Prerequisites: B3.5 Topology and Groups (MT) and C3.1 Algebraic Topology (MT) are essential. We will assume working knowledge of the fundamental group, covering spaces, homotopy, homology, and cohomology. B3.2 Geometry of Surfaces (MT) and C3.3 Differentiable Manifolds (MT) are useful but not essential, though some prior knowledge of smooth manifolds and bundles should make the material more accessible.

Course synopsis: Low-dimensional topology is the study of 3- and 4-manifolds and knots. The classification of manifolds in higher dimensions can be reduced to algebraic topology. These methods fail in dimensions 3 and 4. Dimension 3 is geometric in nature, and techniques from group theory have also been very successful. In dimension 4, gauge-theoretic techniques dominate.This course provides an overview of the rich world of low-dimensional topology that draws on many areas of mathematics. We will explain why higher dimensions are in some sense easier to understand, and review some basic results in 3- and 4-manifold topology and knot theory.

C5.4 Networks

Department: Maths

Lecturer: Prof Peter Grindrod

Course Weight: 1 unit/16 lectures

Assessment Method: mini-project

General prerequisites: Basic notions of linear algebra, probability, dynamical systems, and some computational experience. The student may use the language of their choice for computational experiments. Relevant notions of graph theory will be reviewed and illustrated.

Course Synopsis: Network Science provides generic tools to model and analyse systems in a broad range of disciplines, including biology, computer science and sociology. This course aims at providing an introduction to this interdisciplinary field of research, by integrating tools from graph theory, statistics and dynamical systems. Most of the topics to be considered are active modern research areas. This year the course has been altered to incorporate some new material on dynamically evolving networks and the analysis of scaling properties of growing networks.

C5.6 Applied Complex Variables

Department: Maths

Lecturer: Prof James Oliver

Course Weight: 1 unit/16 lectures

Assessment Method: written exam in TT

General Prerequisites: The course requires second year core analysis (A2 complex analysis). It continues the study of complex variables in the directions suggested by contour integration and conformal mapping. A knowledge of the basic properties of Fourier Transforms is assumed. Part A Waves and Fluids and Part C Perturbation Methods are helpful but not essential.

Course synopsis: The course begins where core second-year complex analysis leaves off, and is devoted to extensions and applications of that material. The solution of Laplace's equation using conformal mapping techniques is extended to general polygonal domains and to free boundary problems. The properties of Cauchy integrals are analysed and applied to mixed boundary value problems and singular integral equations. The Fourier transform is generalised to complex values of the transform variable, and used to solve mixed boundary value problems and integral equations via the Wiener-Hopf method.

C7.4 Intro to Quantum Information

Department: Maths

Lecturer: Prof Artur Ekert

Course Weight: 1 unit/16 lectures 

Assessment Method: written exam in TT

General Prerequisites: Quantum Theory. The course material should be of interest to physicists, mathematicians, computer scientists, and engineers. The following will be assumed as prerequisites for this course:
- elementary probability, complex numbers, vectors and matrices; - Dirac braket notation; - a basic knowledge of quantum mechanics especially in the simple context of finite dimensional state spaces (state vectors, composite systems, unitary matrices, Born rule for quantum measurements); - basic ideas of classical theoretical computer science (complexity theory) would be helpful but are not essential. Prerequisite notes will be provided giving an account of the necessary material. It would be desirable for you to look through these notes slightly before the start of the course.

Course Synopsis: The classical theory of computation usually does not refer to physics. Pioneers such as Turing, Church, Post and Goedel managed to capture the correct classical theory by intuition alone and, as a result, it is often falsely assumed that its foundations are self-evident and purely abstract. They are not! Computers are physical objects and computation is a physical process. Hence when we improve our knowledge about physical reality, we may also gain new means of improving our knowledge of computation. From this perspective it should not be very surprising that the discovery of quantum mechanics has changed our understanding of the nature of computation. In this series of lectures you will learn how inherently quantum phenomena, such as quantum interference and quantum entanglement, can make information processing more efficient and more secure, even in the presence of noise. 

C7.6 General Relativity II

Department: Maths

Lecturer: Dr Christopher Couzens

Course Weight: 1 unit/16 lectures

Assessment Method: written exam in TT

Prerequisites: C7.5 General Relativity I

Course synopsis: In this, the second course in General Relativity, we have two principal aims. We first aim to increase our mathematical understanding of the theory of relativity and our technical ability to solve problems in it. We apply the theory to a wider class of physical situations, including gravitational waves and black hole solutions. Orbits in the Schwarzschild solution are given a unified treatment which allows a simple account of the three classical tests of Einstein's theory. This leads to a greater understanding of the Schwarzschild solution and an introduction to its rotating counterpart, the Kerr solution. We analyse the extensions of the Schwarzschild solution show how the theory of black holes emerges and exposes the radical consequences of Einstein's theory for space-time structure.

C7.7 Random Matrix Theory

Department: Maths

Lecturer: Prof Louis-Pierre Arguin

Course Weight: 1 unit/16 lectures

Assessment Method: written exam in TT

General Prerequisites: There are no formal prerequisites, but familiarity with basic concepts and results from linear algebra and probability will be assumed, at the level of A0 (Linear Algebra) and A8 (Probability).

Course synopsis: Random Matrix Theory provides generic tools to analyse random linear systems. It plays a central role in a broad range of disciplines and application areas, including complex networks, data science, finance, machine learning, number theory, population dynamics, and quantum physics. Within Mathematics, it connects with asymptotic analysis, combinatorics, integrable systems, numerical analysis, probability, and stochastic analysis. This course aims to provide an introduction to this highly active, interdisciplinary field of research, covering the foundational concepts, methods, questions, and results.

Collisionless Plasma Physics

Department: Physics

Lecturer: Dr Daniel Kennedy and Dr Plamen Ivanov

Course Weight: 1 unit/18 lectures

Assessment Method: take-home exam or homework completion

General Prerequisites: Kinetic Theory (MT), an undergraduate course on Electricity and Magnetism
Course Synopsis: 
Part I. Plasma waves:
Cold plasma waves in a magnetised plasma. WKB theory of cold plasma wave propagation in an inhomogeneous plasma, cut-offs and resonances. Hot plasma waves in a magnetised plasma. Cyclotron resonance.
Part II. Kinetics of strongly magnetised plasmas:
Kinetic description of a collisionless, magnetised plasma; kinetic MHD. Barnes damping, firehose and mirror instabilities. Particle motion. Drift kinetics. Drift waves and the ion-temperature-gradient instability. Electron drift kinetics (time permitting): kinetic Alfvén waves, electron-temperature-gradient instabilities.

Cosmology

Department: Physics

Lecturer: Dr David Alonso 

Course Weight: 1 unit/16 lectures

Assessment Method: written exam in TT

Prerequisites: General Relativity I (MT) or equivalent. 

Einstein field equations and the Friedman equations, universe models, statistics of expanding background, relativistic cosmological perturbations, observations, from the Hubble flow to the CMB.

Galactic and Planetary Dynamics

Department: Physics

Lecturer: Prof John Magorrian 

Course Weight: 1 unit/16 lectures

Assessment Method: TBC

Prerequisites: Kinetic Theory (MT)

Course Synopsis: Review of Hamiltonian mechanics. Orbit integration. Classification of orbits and integrability. Construction of angle-action variables. Hamiltonian perturbation theory. Simple examples of its application to the evolution of planetary and stellar orbits. Methods for constructing equilibrium galaxy models. Applications. Fundamentals of N-body simulation. Dynamical evolution of isolated galaxies. Interactions with companions.

Geophysical Fluid Dynamics

Department: Physics

Lecturer: Prof Tim Woollings 

Course Weight: 1 unit/16 lectures

Assessment Method: written exam in TT

Course Synopsis: Rotating frames of reference. Geostrophic and hydrostatic balance. Pressure coordinates. Shallow water and reduced gravity models, f and β–planes, potential vorticity. Inertia-gravity waves, dispersion relation, phase and group velocity. Rossby number, equations for nearly geostrophic motion, Rossby waves, Kelvin waves. Linearised equations for a stratified, incompressible fluid, internal gravity waves, vertical modes. Quasigeostrophic approximation: potential vorticity equation, Rossby waves, vertical propagation and trapping. Eady model of baroclinic instability. Overview of large-scale structure and circulation of atmospheres and oceans, poleward heat transport. Angular momentum and Held-Hou model of Hadley circulations. Applications to Mars and slowly-rotating planets. Tide-locked exoplanets. Giant planets: Multiple jets, stable eddies and free modes.

High Energy Density Plasma Physics

Department: Physics

Lecturer: Prof Peter Norreys and Dr Ramy Aboushelbaya

Course Weight: 1 unit/16 lectures

Assessment Method: homework completion only 

Course Synopsis: In this course, the topics will be introduced for first principles. The student will be taken through the fundamental physics of laser energy absorption in matter up to and including the new laser QED plasma regime at extreme intensities. The student will be introduced to hydrodynamic motion via first principles derivation of the Navier-Stokes equations as well as compression and rarefaction waves. Then a thorough grounding in hydrodynamic instabilities will be provided, including the Rayleigh-Taylor instability and the applications of linear theory. This will be followed by the extension to the convective instability; mode coupling; the Kelvin-Helmholtz; shock stability and the Richtmyer-Meshkov instability. The behaviour of shock waves in one dimension will then be discussed, including the derivation of the Rankine-Hugoniot equations; the effects of boundaries and interfaces; blast waves and shocks in solids. Following that, the physics of convergent shocks will be described. These include homogeneous expansion/contraction self-similar flows as well as shock dynamics. The hydrodynamic behaviour is governed by the equations of state including thermodynamic properties, so the student will be introduced to equations of state for gases, plasmas, solids and liquids. For thermal energy transport, the thermal energy transport equation is derived, as are the effects of the conductivity coefficients, inhibited thermal transport, electron-ion energy exchange, before electron degeneracy effects are introduced. The physics of radiation energy transport will be described, including radiation as a fluid and the Planck distribution function; radiation flux definition; solutions to the radiation energy transfer equations; material opacities; non-LTE radiation transport; radiation dominated hydrodynamics. Finally, dimensionless scaling laws for hydrodynamics will be outlined, ones that provide the student with a link between the fascinating detailed microphysics of laboratory plasma phenomena and exquisite astrophysical observations.

Nonequilibrium Statistical Physics

Department: Physics

Lecturer: Prof Ramin Golestanian 

Course Weight: 1 unit/16 lectures

Assessment Method: TBC

Course Synopsis:  Stochastic Langevin dynamics. Brownian motion. Nonequilibrium kinetics. Master equation. Fokker-Planck equation. Kramers rate theory and mean first-passage time. Brownian ratchets. Multiplicative noise. Path integral formulation and Martin-Siggia-Rose method. Fluctuation theorems.

 
Quantum Matter 2: Quantum Fluids

Department: Physics

Lecturer: Prof Sid Parameswaran

Course Weight: 1 unit/16 lectures 

Assessment Method: written exam in week 0 TT or homework completion

Course Synopsis: “Quantum fluids” are systems of many interacting particles where the role of quantum statistics is significant, and can lead to macroscopic quantum effects. This courses focuses on the simplest examples, ones where Galilean invariance is a good approximation, i.e.  the role of crystal lattices or imperfections is ignored. We will first discuss phenomenological and microscopic models of superfluids of bosons (such as Helium 4), before discussing the case of charged superfluids. This will lead us naturally into discussions of the Meissner effect and the Anderson-Higgs mechanism. To describe electronic superconductors (or paired sermonic superfluids such as Helium 3) microscopically, we will first need to take a detour through the theory of the interacting electrons gas and Landau’s theory of Fermi liquids, before discussing the Bardeen-Cooper-Schrieffer theory. Time permitting, the course will close with a discussion of arguably the most exotic quantum fluids discovered to date: the two-dimensional  quantum Hall liquids that form out of electron gases placed in high magnetic fields, which give rise to fractional charge.

Quantum Matter 3: Quantum Dynamics and Information in Many-particle Systems

Department: Physics

Lecturer: Prof Fabian Essler

Course Weight: 1 unit/16 lectures (continues in Trinity term)

Assessment Method: written exam in week 0 TT or homework completion

Course Synopsis: 

1 Elements of Quantum Statistical Mechanics
1.1 Pure and Mixed States, (Reduced) Density Matrices 
1.2 Entropy, Ensembles and Typicality
2 Eigenstates of Local Many-Particle Hamiltonians
2.1 Tight-Binding Model of Spinless Fermions
2.2 Entanglement Measures
2.3 Entanglement Entropy of Energy Eigenstates
2.4 The Spin-1 Aklt Chain
2.5 Matrix Product State Methods
2.6 Symmetry Protected Topological Order
3 Quantum Many-Particle Dynamics
3.1 Quantum Quenches
3.2 (Generalized) Thermalization
3.3 Eigenstate Thermalization Hypothesis
3.4 Bbgky Hierarchy
3.5 Self-Consistent Time-Dependent Mean-Field Approximation
3.6 Quantum Boltzmann Equation
4 Open and Driven Quantum Systems
4.1 Quantum Master Equations
4.2 Periodically Driven Systems And Quantum Circuits

String Theory I

Department: Maths

Lecturer: Prof Xenia de la Ossa

Course Weight: 1 unit/16 lectures

Assessment Method: TBC

Prerequisites: Quantum Field Theory (MT)

Course Synopsis: Historical background, Dolen-Horn-Schmid duality, the Veneziano and Virasoro-Shapiro amplitudes. Nambu-Goto and Polyakov world-sheet actions, equations of motion and constraints, open and closed strings and their corresponding boundary conditions. Old covariant quantization: the Virasoro algebra, physical state conditions, ghosts, critical spacetime dimension, and spacetime particle spectrum. Basic considerations of light-cone gauge quantization. Vertex operators and string scattering amplitudes. Strings in background fields, spacetime effective action. Circle compactification, elementary consideration of D-branes, T-duality.

Supersymmetry and Supergravity

Department: Maths

Lecturer: Dr Michele Levi

Course Weight: 1 unit/16 lectures 

Assessment Method: written exam in week 0 TT

Course Synopsis: 
1. Context and Motivation.
2. Spinors Preliminary.
3. Supersymmetry Algebra.
4. Superspace and Superfields.
5. Chiral Superfields and Supersymmetric Actions.
6. Supersymmetric Gauge Theories.
7. Spontaneous Symmetry Breaking.