Lecture courses for HT 2015: Advanced Fluid Dynamics Advanced Fluid Dynamics [16 hours] area: CM, Astro prequels: Kinetic Theory (MT), Viscous Flow (MT) prerequisites: basic familiarity with fluid equations and stress tensors as provided, e.g., by Kinetic Theory (MT). syllabus (written by P. Dellar, A. Schekochihin, J. Yeomans): Introduction to the dynamics of fluids with stress tensors more complex than the viscous and Euler momentum fluxes. Part I: Magnetohydrodynamics. MHD equations: Maxwell stress, magnetic pressure and tension, flux freezing, magnetic diffusion, magnetic reconnection, Zeldovich rope dynamo. Conservation laws. Helicity, Taylor relaxation, force-free solutions. Simple MHD equilibria. MHD waves, Elsasser variables and Elsasser solutions. Lagrangian MHD, Cauchy solution, action principle. Energy principle, instabilities: sausage, kink, interchange (overview). Braginskii stress tensor. Part II: Non-Newtonian fluids. Stokes flow, reciprocity and minimal dissipation, forces and torques on rigid bodies. Stokeslets, the Oseen tensor, multipole expansions. Microscopic bead-spring models of polymers, derivation of upper convected Maxwell model. Properties of viscoelastic fluids: normal stress differences, rheological flows, die swell, rod climbing, elastic waves, elastic instabilities, analogies with MHD. Liquid crystals and active suspensions. sequel: Astrophysical Fluid Dynamics (TT), Advanced Plasma Physics (TT), Topics in Soft and Active Matter Physics (TT), Turbulence (TT) lecturer: Alex Schekochihin, Paul Dellar department: Maths and Physics course website: TBA location, times: Fisher Room, Denys Wilkinson Building, Physics. Mondays at 3pm. Geophysical Fluid Dynamics Geophysical Fluid Dynamics [16 hours] area: Astro prequel: Waves and Compressible Flow (HT) pre-requisite: basic familiarity with fluid equations as provided, e.g., by Kinetic Theory (MT), Maths B6a or an equivalent undergraduate course (e.g., Physics B1). syllabus (written by D. Marshall): Rotating frames of reference, Rossby number, geostrophic and hydrostatic balance, thermal wind relation, pressure coordinates. Shallow water and reduced gravity models, f and beta-planes, conservation laws for energy and potential vorticity (relation to particle relabelling symmetry?), inertia-gravity waves, equations for nearly geostrophic motion, Rossby waves, Kelvin waves. Linearised equations for a stratified, incompressible fluid, internal gravity waves, vertical modes. Quasigeostrophic approximation: quasigeostrophic potential vorticity equation and Rossby waves solutions, vertical propagation and trapping. Barotropic and baroclinic instability, necessary conditions for instability of zonal flow, Eady model of baroclinic instability, qualitative discussion of frontogenesis. Wave-mean flow interaction, transformed Eulerian mean, Eliassen-Palm flux, non-acceleration theorem. Angular momentum and Held-Hou model of Hadley circulations. Applications to Mars and slowly-rotating planets. Giant planets: Multiple jets, stable eddies and free modes. Ekman layers, spin-down and upwelling. Sverdrup balance and ocean gyres, western intensification, simple models for the vertical structure of ocean circulation. Energetics and simples models of the meridional overturning circulation. lecturer: Andrew Wells department: Physics course website: TBA location, times: Dobson Room, AOPP Building, Physics. Mondays 12pm and Tuesdays 11am. Monday week 3 is in the Robert Hooke Room, AOPP Building. Nonlinear Systems Nonlinear Systems [16 hours] area: CM, Astro prequels/pre-requisites: Nonequilibrium Statistical Physics (MT) syllabus: Part I: Bifurcations. Bifurcation theory: standard codimension one examples (saddle-node, pitchfork, trans-critical, Hopf); normal forms; Conservative and non-conservative systems. Part II: Nonlinear Oscillations. Van der Pol and Duffing's equations. Poincaré-Lindstedt method; Method of multiple scales; Krylov-Bogoliubov method of Averaging; Relaxation and forced oscillations; Synchronization of coupled oscillators. Part III: Maps. Poincaré sections and first-return maps. Stability and periodic orbits; bifurcations of one-dimensional maps. Two-dimensional maps: Hénon map, Chirikov (standard) map. Part IV: Chaos in Maps and Differential Equations. Maps: Logistic map, Bernoulli shift map, symbolic dynamics, Skinny Baker's map, Smale's Horseshoe Map. Differential equations: Lorenz equations, Rossler equations. lecturer: Irene Moroz department: Maths course website: Maths course B5.6 location, times: Mathematical Institute, Mondays at 11am (L4 in weeks 1–4, 6–8; L1 in week 5) and Tuesdays at 4pm (L5 in weeks 1–6; L3 in weeks 7 and 8). Advanced Quantum Field Theory Advanced Quantum Field Theory for Particle Physics [24 hours] area: PT syllabus (written by X. de la Ossa, G. Ross): Quantum Electrodynamics: Introduction, photon propagator, scalar electrodynamics (Feynman rules, radiative corrections), canonical quantization, fermions (fermions propagator, path integral and Feynman rules), spinor electrodynamics, sample calculations (scattering in spinor electrodynamics), beta function in QED. Non-Abelian Quantum Field Theory: SU(N) local gauge theory, path integral, gauge fixing, BRST, spontaneous symmetry breaking, anomalies, introduction to the standard model. sequel: The Standard Model (TT), Beyond the Standard Model (TT), Non-perturbative Methods in Quantum Field Theory (TT) lecturer: Guido Bell department: Physics course website: TBA location, times: Fisher Room, Denys Wilkinson Building, Physics. Week 2 - 8 Tuesdays 13:00-14:00, Thursdays 9:00-10:00. Week 2,3,5 & 7 , DWB Seminar Room 15:00-16:00 Thursdays. Quantum CMP II Quantum Condensed Matter Physics II [24 hours] area: CM Prequel/pre-requisite: Introduction to Quantum Condensed Matter Physics (MT) syllabus (written by J. Chalker and F. Essler): Phase transitions: transfer matrix methods, spontaneous symmetry breaking in the Ising model, Landau theory of phase transitions.Fermi liquid theory. BCS theory of superconductivity. Strong interactions: Mott insulators. Ferromagnetism and antiferromagnetism, Holstein-Primakoff transformation. Quantum Hall effect: integer and fractional QHE; fractional statistics. Disorderd systems: random potential and localization. sequel: Advanced Quantum Condensed Matter Physics (TT), Topics in Quantum Condensed Matter Physics (TT) lecturer: TBA department: Physics course website: TBA location, times: Not running in 2015. Waves and Compressible Flow Waves and Compressible Flow [16 hours] area: Astro prequel: Viscous Flow (MT), Kinetic Theory (MT) syllabus: Equations of inviscid compressible flow including flow relative to rotating axes. Models for linear wave propagation including Stokes waves, internal gravity waves, inertial waves in a rotating fluid, and simple solutions. Theories for Linear Waves: Fourier Series, Fourier integrals, method of stationary phase, dispersion and group velocity. Flow past thin wings. Nonlinear Waves: method of characteristics, simple wave flows applied to one-dimensional unsteady gas flow and shallow water theory. Shock Waves: weak solutions, Rankine–Hugoniot relations, oblique shocks, bores and hydraulic jumps. sequel: Geophysical Fluid Dynamics (TT), Astrophysical Fluid Dynamics (TT), Turbulence (TT) lecturer: Ian Hewitt department: Maths course website: Maths course B5.4 location, times: Mathematical Institute, Wednesdays at 9am (L3) and Thursdays at 9am (L3 in weeks 1–3, 5, 7–8; L4 in weeks 4 and 6). String Theory I String Theory I [16 hours] area: PT pre-requisite: Quantum Field Theory (MT) syllabus (written by P. Candelas): String actions, equations of motion and constraints, open and closed strings --- boundary conditions, Virasoro algebra, ghosts and BRS, physical spectrum, elementary consideration of D branes, Veneziano amplitude. sequel: String Theory II (TT), Introduction to Gauge-String Duality (TT) lecturer: Philip Candelas department: Maths course website: TBA location, times: Mathematical Institute, Mondays at 9am (L6), Wednesdays at 9am (L5 in weeks 1-6, 8; L4 in week 7). Networks Networks [16 hours] area: CM pre-requisites: Maths C5.3 or another undergraduate course in Statistical Mechanics syllabus: 1. Introduction and Basic Concepts: nodes, edges, adjacencies, weighted networks, unweighted networks, degree and strength, degree distribution, other types of networks. 2. Small Worlds: clustering coefficients, paths and geodesic paths, Watts-Strogatz networks [focus is on modelling and heuristic calculations]. 3. Toy Models of Network Formation: preferential attachment, generalizations of preferential attachment, network optimization. 4. Additional Summary Statistics and Other Useful Concepts: modularity and assortativity, degree-degree correlations, centrality measures, communicability, reciprocity and structural balance. 5. Random Graphs: Erdos-Rényi graphs, configuration model, random graphs with clustering, other models of random graphs or hypergraphs; application of generating-function methods [focus is on modelling and heuristic calculations; material in this section forms an important basis for sections 6 and 7]. 6. Community Structure and Mesoscopic Structure: linkage clustering, optimization of modularity and other quality functions, overlapping communities, other methods and generalizations. 7. Dynamics on (and of) Networks: general ideas, models of biological and social contagions, percolation, voter and opinion models, temporal networks, other topics. 8. Additional Topics: games on networks, exponential random graphs, network inference, other topics of special interest to students [depending on how much room there is and interest of current students] sequel: Complex Systems (TT) lecturer: Mason Porter department: Maths course website: Maths course C5.4 location, times: Mathematical Institute, Fridays 2–4pm (L3 in weeks 1–4, 6, 8; L1 in weeks 5 and 7). Plasma Physics Plasma Physics [16 hours] area: Astro prequel: Kinetic Theory (MT) pre-requisites: Kinetic Theory (MT) or a basic introductory course in Plasma Physics. syllabus (written by F. Parra): Part I: Magnetised plasmas. Particle motion in magnetic field, adiabatic invariants. Drift kinetics, drift waves and instabilities. Kinetic MHD and CGL (double-adiabatic) equations. Two-fluid (Braginskii) equations, MHD. Part II: Plasma waves. Cold plasma dispersion relation. Hot-plasma dispersion relation for electrostatic waves. Hot-plasma dispersion relation for electromagnetic waves. Landau, Barnes (transit-time) and cyclotron damping. Quasilinear theory. sequel: Advanced Plasma Physics (TT) lecturer: F Parra department: Physics course website: TBA location, times: Fisher Room, Denys Wilkinson Building, Physics. Mondays 2pm, Tuesdays 12pm. Supersymmetry and Supergravity Supersymmetry and Supergravity [24 hours] area: PT pre-requisite: Quantum Field Theory (MT) syllabus (written by J. Conlon): Motivations for supersymmetry, spinor algebras and representations, supersymmetry algebra and representations, extended supersymmetry and BPS states, superfields, SUSY field theories, non-renormalisation theorems, SUSY breaking, the MSSM and its phenomenology, rescaling anomalies, NSVZ beta function, basic properties of supergravity. lecturer: Joe Conlon department: Physics course website: TBA location, times: Fisher Room, Denys Wilkinson Building, Physics Tuesdays 11-12. Then the Denys Wilkinson Seminar Room on Tuesdays 12-1 and on Thurs 11:30-12:30. Stellar Astrophysics Stellar Astrophysics [8 hours] area: Astro syllabus (written by P. Podsiadlowski): Part I: Modern Topics in Stellar Astrophysics. Late stages of stellar evolution; massive stars; supernovae, millisecond pulsars, hypernovae, gamma-ray bursts; compact binaries; the origin of elements, chemical evolution of the Universe. Part II: Accretion discs, Theory and Applications. Accretion disc theory, thin and thick discs; disc instabilities (thermal instability, gravitational instabilities [Toomre criterion]); optically thin advection-dominated flows, super-Eddington accretion. lecturer: P. Podsiadlowski department: Physics course website: As part of course C1 in physics location, times: Dennis Sciama Lecture Theatre, Denys Wilkinson Building, Physics. Week 1: Tues 9-10. Week 2: Mon 10-11, Tues 9-10, Weds 12-13. Week 3: Tues 9-10, Thurs 10-11. Week 4: Tues 9-10, Weds 12-13. Week 5: Tues 9-10, Weds 12-13, Thurs 10-11. Week 6-8: Tues 9-10, Weds 12-13. (Physics C1 timetable). High Energy Astrophysics High Energy Astrophysics [8 hours] area: Astro prequel: Stellar and Atomic Astrophysics (HT) syllabus (written by G. Cotter): Physics of interactions between high-energy particles and radiation (synchrotron, inverse-Compton, thermal Bremsstrahlung). Relativistic jets. lecturer: G. Cotter department: Physics course website: As part of course C1 in physics location, times: Dennis Sciama Lecture Theatre, Denys Wilkinson Building, Physics. Week 1: Tues 9-10. Week 2: Mon 10-11, Tues 9-10, Weds 12-13. Week 3: Tues 9-10, Thurs 10-11. Week 4: Tues 9-10, Weds 12-13. Week 5: Tues 9-10, Weds 12-13, Thurs 10-11. Week 6-8: Tues 9-10, Weds 12-13. (Physics C1 timetable). General Relativity II General Relativity II [16 hours] area: PT/Astro prequel/pre-requisite: General Relativity I (MT) syllabus: The Lie derivative and isometries. Linearised General Relativity and the metric of an isolated body. The Schwarzschild solution and its extensions; Eddington-Finkelstein coordinates and the Kruskal extension. Stationary, axisymmetric metrics and orthogonal transitivity; the Kerr solution and its properties; interpretation as rotating black hole. The Einstein field equations with matter; the energy-momentum tensor for a perfect fluid; equations of motion form the conservation law. Cosmological principles, homogeneity and isotropy; cosmological models; the Friedman--Robertson--Walker solutions; observational consequences. lecturer: X. de la Ossa department: Maths course website: Maths course C7.6 location, times: Mathematical Institute, Tuesdays and Thursdays at 10am (C1), Monday at 5pm (C4) in week 5 only. Applied Complex Variables Applied Complex Variables [16 hours] area: PT/CM/Astro prequel: Perturbation Methods (MT) syllabus: Review of tensors, conservation laws, Navier equations. Antiplane strain, torsion, plane strain. Elastic wave propagation, Rayleigh waves. Ad hoc approximations for thin materials; simple bifurcation theory and buckling. Simple mixed boundary value problems, brittle fracture and smooth contact. Perfect plasticity theories for granular materials and metals. lecturer: Peter Howell department: Maths course website: Maths course C5.6 location, times: Mathematical Institute, Mondays at 3pm (L3 in weeks 1–6, 8; L2 in week 7) and Thursdays at 3pm (L3 in weeks 1–3, 5, 7, 8; L1 in weeks 4 and 6). Scientific Computing II Scientific Computing II [12 hours] area: PT/CM/Astro prequel/pre-requisite: Scientific Computing I (MT) syllabus: see Maths graduate handbook lecturer: Nick Trefethen department: Maths course website: Scientific Computing for DPhil Students location, times: Mathematical Institute, Tuesdays at 9am (L3) and Fridays at 9am (L3 in weeks 2–4, 7; L2 in weeks 5 and 6). This course will run in weeks 2–7. Numerical Solutions to Differential Equations II Numerical Solutions to Differential Equations II [16 hours] area: PT/CM/Astro prequel: Numerical Solutions to Differential Equations I (MT) syllabus: Numerical methods for boundary value problems. We begin by developing numerical techniques for the approximation of boundary value problems for second-order ordinary differential equations. Boundary value problems for ordinary differential equations: shooting and finite difference methods. Then we consider finite difference schemes for elliptic boundary value problems. This is followed by an introduction to the theory of direct and iterative algorithms for the solution of large systems of linear algebraic equations which arise from the discretisation of elliptic boundary value problems. Boundary value problems for PDEs: finite difference discretisation; Poisson equation. Associated methods of sparse numerical algebra: sparse Gaussian elimination, iterative methods. lecturer: Jared Tanner department: Maths course website: Maths course B6.2 location, times: Mathematical Institute, Mondays at 10am (L3) and Fridays at 10am (L3 in weeks 1–4, 7–8; L5 in week 5; L4 in week 6). Differential Geometry (Manifolds) Differential Geometry (Manifolds) [16 hours] area: PT/Astro syllabus: A manifold is a space such that small pieces of it look like small pieces of Euclidean space. Thus a smooth surface is an example of a (2-dimensional) manifold. Manifolds are the natural setting for parts of classical applied mathematics such as mechanics, as well as general relativity. They are also central to areas of pure mathematics such as topology and certain aspects of analysis. In this course we introduce the tools needed to do analysis on manifolds. We prove a very general form of Stokes' Theorem which includes as special cases the classical theorems of Gauss, Green and Stokes. We also introduce the theory of de Rham cohomology, which is central to many arguments in topology. lecturer: Nigel Hitchin department: Maths course website: Maths course C3.3 location, times: Mathematical Institute, Thursdays at 12noon (C3 in week 1; C4 in weeks 2–8) and Fridays at 12noon (C4).