Lecture courses for MT 2014: Lecture courses run for all 8 weeks of term, unless indicated otherwise. All lecture slots with a starting time only are for one hour. Two hour slots are indicated explicitly. Quantum Field Theory Quantum Field Theory [24 hours] area: PT, CM, Astro, foundational course syllabus (written by J. Cardy, F. Essler, A. Lukas, A. Starinets): Classical field theory, Noether's theorem, canonical quantization, path integral formulation of quantum mechanics, path integrals in field theory: generating functionals, finite temperature field theory, Feynman diagrams, Feynman rules, divergences and regularisation, renormalisation and renormalisation group, scattering and S-matrices, response functions, path integrals for fermions. sequel: Advanced Quantum Field Theory for Particle Physics (HT), Conformal Field Theory (TT), Quantum Field Theory in Curved Space-Time (TT) lecturer: Volker Braun department: Maths course website: link location, times: Fisher Room, Denys Wilkinson Building, Physics Department. Mondays at 10am, Wednesdays at 9am and Thursdays at 10am. Statistical Mechanics Statistical Mechanics [16 hours] area: CM, Astro remark: This course can be taken by students who have not studied this subject before (e.g., as Physics A1) but would like to be able to follow the more specialised courses offered in Hilary and Trinity that require familiarity with Statistical Mechanics. syllabus: Classical mechanics: Newton's second law, D'Alembert's principle, Lagrange's equations, Hamilton's equations. Probability: probability density functions, moment generating function, central limit theorem. Fluid mechanics: material derivative, Euler and Navier-Stokes equations, energy equation. Random walks, Brownian motion, diffusion equation. Loschmidt's paradox. Liouville equation, BBGKY hierarchy, Boltzmann equation. The collision integral for a hard sphere gas. Boltzmann H theorem. Maxwellian distribution. Definition of entropy and temperature. Gibbs and Helmholtz free energies. Thermodynamic relations. Classical statistical mechanics. Ergodic theorem, equiprobability. Microcanonical ensemble for the hard sphere gas, entropy. Canonical ensemble. Selected applications and extensions: for example, chemical potential, phase change, binary alloys, surface energy, radiative transfer, polymer solution theory, Arrhenius kinetics, nucleation theory, percolation theory, renormalisation. lecturer: Andrew Fowler department: Maths course website: Maths course C5.3 location, times: Mathematical Institute, Mondays and Tuesdays at 11am (L5). Kinetic Theory Kinetic Theory [24 hours] areas: PT, CM, Astro, foundational course syllabus (written by J. Binney, P. Dellar, R. Golestanian, J. Magorrian, A. Schekochihin): Part I: Basic Kinetic Theory of Gases. Liouville Theorem. BBGKY hierarchy and derivation of Boltzmann's equation. H-theorem, Maxwell's distribution. Derivation of fluid equations. Transport: viscosity and thermal diffusvity. Onsager symmetries. Part II: Plasma Kinetics (Charged Particles in Electromagnetic Fields). Kinetics in an external field. Plasma: charged particles and self-consistent electromagnetic fields. Debye screening. Landau collision integral. Outline of the derivation of two-fluid equations and magnetohydrodynamics. Collisionless plasma in electrostatic field. Dielectric permittivity, Landau damping, kinetic instabilities, waves. Outline of the quasilinear theory and nonlinear approximations. Part III: Kinetics of Gravitating Objects. Self-gravitating kinetics and the resultant fluid equations. Invariants of motion and the Jeans theorem. Non-Maxwellian (collisionless) equilibria. Anisotropic distributions. Part IV: Kinetics of Quasiparticles. Phonons. UV catastrophe. sequels: Advanced Fluid Dynamics (HT), Plasma Physics (HT), Galactic and Planetary Dynamics (HT) lecturer: Dellar, Schekochihin, Binney department: Physics/Maths course website: link location, times: Fisher Room, Denys Wilkinson Building, Physics Department. Mondays, 3pm-5pm and Tuesdays at 12pm. Viscous Flow Viscous Flow [16 hours] areas: CM, Astro remark: This course is particularly recommended to the students who have not studied basic Fluid Dynamics (e.g., as Physics B1) and would like to be able to follow the more specialised courses offered in Hilary and Trinity and requiring familiarity with this subject. syllabus: Euler's identity and Reynolds' transport theorem. The continuity equation and incompressibility condition. Cauchy's stress theorem and properties of the stress tensor. Cauchy's momentum equation. The incompressible Navier-Stokes equations. Vorticity. Energy. Exact solutions for unidirectional flows; Couette flow, Poiseuille flow, Rayleigh layer, Stokes layer. Dimensional analysis, Reynolds number. Derivation of equations for high and low Reynolds number flows. Thermal boundary layer on a semi-infinite flat plate. Derivation of Prandtl's boundary-layer equations and similarity solutions for flow past a semi-infinite flat plate. Discussion of separation and application to the theory of flight. Slow flow past a circular cylinder and a sphere. Non-uniformity of the two dimensional approximation; Oseen's equation. Lubrication theory: bearings, squeeze films, thin films; Hele–Shaw cell and the Saffman-Taylor instability. sequels: Advanced Fluid Dynamics (HT), Waves and Compressible Flow (HT) lecturer: Sarah Waters department: Maths course website: Maths course B5.3 location, times: Mathematical Institute, Mondays at 2pm (L3) and Wednesdays at 2pm (L3 in weeks 1-5, 7-8; L2 in week 6). General Relativity I General Relativity I [16 hours] areas: PT, CM, Astro, foundational course remark: Some students may have studied this subject before (for example, as Physics B5). syllabus: Review of Newtonian gravitation theory and problems of constructing a relativistic generalisation. Review of Special Relativity. The equivalence principle. Tensor formulation of special relativity (including general particle motion, tensor form of Maxwell's equations and the energy momentum-tensor of dust). Curved space time. Local inertial coordinates. General coordinate transformations, elements of Riemannian geometry (including connections, curvature and geodesic deviation). Mathematical formulation of General Relativity, Einstein's equations (properties of the energy-momentum tensor will be needed in the case of dust only). The Schwarzschild solution; planetary motion, the bending of light, and black holes. sequels: General Relativity II (HT), Cosmology (HT), Quantum Field Theory in Curved Space-Time (TT) lecturer: Ricardo Monteiro department: Maths course website: Maths course C7.5 location, times: Mathematical Institute, Mondays 5pm (L5) and Tuesdays 5pm (L4). Quantum Condensed Matter Physics Introduction to Quantum Condensed Matter Physics [16 hours] areas: PT, CM, Astro, foundational course syllabus: (written by J.~Chalker and F.~Essler): Second quantisation. Ideal Fermi and Bose gases in second quantization. Weakly interacting Bose gas: Bogoliubov theory; superfluidity. Weakly interacting fermions: mean-field theory; Hartree-Fock approximation. Linear response theory. sequel: Quantum Condensed Matter Physics II (HT) lecturer: C6 course lecturers department: Physics course website: Physics course C6 location, times: Dennis Sciama Lecture Theatre, Physics. Mondays at 9am, Wednesdays at 10am and Thursdays at 9am. This is part of the Physics C6 Course. Nonequilibrium Statistical Physics Nonequilibrium Statistical Physics [8 hours] areas: PT, CM, Astro, foundational course syllabus: (written by R.~Golestanian): Stochastic Processes. Brownian motion; Langevin and Fokker-Planck equations. Normal and anomalous diffusion. Brownian ratchets. Molecular motors. sequel: Soft Matter Physics (HT) lecturer: C6 course lecturers department: Physics course website: Physics course C6 location, times: Dennis Sciama Lecture Theatre, Physics. Mondays at 9am, Wednesdays at 10am and Thursdays at 9am. This is part of the Physics C6 Course. Perturbation Methods Perturbation Methods [16 hours] areas: PT, CM, Astro, foundational course syllabus: Asymptotic expansions. Asymptotic evaluation of integrals (including Laplace's method, method of stationary phase, method of steepest descent). Regular and singular perturbation theory. Multiple-scale perturbation theory. WKB theory and semiclassics. Boundary layers and related topics. Applications to nonlinear oscillators. Applications to partial differential equations and nonlinear waves. sequel: Applied Complex Variables (HT) lecturer: Jim Oliver department: Maths course website: Maths course C5.5 location, times: Mathematical Institute, Thursdays 11am-1pm (L3). Scientific Computing I Scientific Computing I [12 hours] areas: PT, CM, Astro syllabus: See Maths Graduate Handbook. sequel: Scientific Computing II (HT) lecturer: Nick Trefethen department: Maths course website: location, times: Mathematical Institute, Tuesdays and Fridays at 9am (L3) [weeks 1-6]. Numerical Solutions to Differential Equations Numerical Solutions to Differential Equations I [16 hours] areas: PT, CM, Astro syllabus: Development and analysis of numerical methods for initial value problems. We begin by considering classical techniques for the numerical solution of ordinary differential equations. The problem of stiffness is discussed in tandem with the associated questions of step-size control and adaptivity: Initial value problems for ordinary differential equations: Euler, multistep and Runge–Kutta; stability; stiffness; error control: symplectic and adaptive algorithms. The remaining lectures focus on the numerical solution of initial value problems for partial differential equations, including parabolic and hyperbolic problems: Initial value problems for partial differential equations: parabolic equations, hyperbolic equations; explicit and implicit methods; accuracy, stability and convergence, Fourier analysis, CFL condition. sequel: Numerical Solutions to Differential Equations II (HT) lecturer: Ian Sobey department: Maths course website: Maths course B6.1 location, times: Mathematical Institute, Tuesdays at 10am (L3) and Fridays at 10am (L2). Numerical Linear Algebra Numerical Linear Algebra [16 hours] areas: PT, CM, Astro syllabus: Common problems in linear algebra. Matrix structure, singular value decomposition. QR factorization, the QR algorithm for eigenvalues. Direct solution methods for linear systems, Gaussian elimination and its variants. Iterative solution methods for linear systems. Chebyshev polynomials and Chebyshev semi-iterative methods, conjugate gradients, convergence analysis, preconditioning. lecturer: Jared Tanner department: Maths course website: Maths course C6.1 location, times: Mathematical Institute, Mondays at 3pm (L3) and Fridays at 3pm (L3 in weeks 1–3 and 5–8; L2 in week 4). Groups and Representations Groups and Representations [24 hours] areas: PT, CM, Astro syllabus: (written by A. Lukas): Basics on groups, representations, Schur's Lemma, representations of finite groups, Lie groups, Lie algebras, Lorentz and Poincare groups, SU(n), SO(n), spinor representations, roots, classification of simple Lie algebras, weights, representations and Dynkin formalism. lecturer: Andre Lukas department: Physics course website: link location, times: Fisher Room, Denys Wilkinson Building, Physics Department. Thurdays, 2pm and Fridays 11am-1pm. Algebraic Topology Algebraic Topology [16 hours] areas: PT syllabus: Chain complexes of free Abelian groups and their homology. Short exact sequences. Delta (and simplicial) complexes and their homology. Euler characteristic. Singular homology of topological spaces. Relative homology and the Five Lemma. Homotopy invariance and excision (details of proofs not examinable). Mayer-Vietoris Sequence. Equivalence of simplicial and singular homology. Degree of a self-map of a sphere. Cell complexes and cellular homology. Application: the hairy ball theorem. Cohomology of spaces and the Universal Coefficient Theorem (proof not examinable). Cup products. Künneth Theorem (without proof). Topological manifolds and orientability. The fundamental class of an orientable, closed manifold and the degree of a map between manifolds of the same dimension. Poincaré Duality (without proof). lecturer: Christopher Douglas department: Maths course website: Maths course C3.1 location, times: Mathematical Institute, Tuesdays at 2pm (L3 in weeks 1-7), Tuesday at 1.30pm (L3 week 8 only) and Wednesdays at 3pm (L5 in weeks 1-4, 6-8, L2 in week 5). Algebraic Geometry Algebraic Geometry [16 hours] areas: PT syllabus: Affine algebraic varieties, the Zariski topology, morphisms of affine varieties. Irreducible varieties. Projective space. Projective varieties, affine cones over projective varieties. The Zariski topology on projective varieties. The projective closure of affine variety. Morphisms of projective varieties. Projective equivalence. Veronese morphism: definition, examples. Veronese morphisms are isomorphisms onto their image; statement, and proof in simple cases. Subvarieties of Veronese varieties. Segre maps and products of varieties, Categorical products: the image of Segre map gives the categorical product. Coordinate rings. Hilbert's Nullstellensatz. Correspondence between affine varieties (and morphisms between them) and finitely generate reduced k-algebras (and morphisms between them). Graded rings and homogeneous ideals. Homogeneous coordinate rings. Categorical quotients of affine varieties by certain group actions. The maximal spectrum. Primary decomposition of ideals. Discrete invariants projective varieties: degree dimension, Hilbert function. Statement of theorem defining Hilbert polynomial. Quasi-projective varieties, and morphisms of them. The Zariski topology has a basis of affine open subsets. Rings of regular functions on open subsets and points of quasi-projective varieties. The ring of regular functions on an affine variety in the coordinate ring. Localisation and relationship with rings of regular functions. Tangent space and smooth points. The singular locus is a closed subvariety. Algebraic re-formulation of the tangent space. Differentiable maps between tangent spaces. Function fields of irreducible quasi-projective varieties. Rational maps between irreducible varieties, and composition of rational maps. Birational equivalence. Correspondence between dominant rational maps and homomorphisms of function fields. Blow-ups: of affine space at appoint, of subvarieties of affine space, and general quasi-projective varieties along general subvarieties. Statement of Hironaka's Desingularisation Theorem. Every irreducible variety is birational to hypersurface. Re-formulation of dimension. Smooth points are a dense open subset. lecturer: Greg Berczi department: Maths course website: Maths course C3.4 location, times: Mathematical Institute, Mondays and Wednesdays at 12noon (C1) Geometric Group Theory [16 hours] areas: PT syllabus: Free groups. Group presentations. Dehn's problems. Residually finite groups. Group actions on trees. Amalgams, HNN-extensions, graphs of groups, subgroup theorems for groups acting on trees. Quasi-isometries. Hyperbolic groups. Solution of the word and conjugacy problem for hyperbolic groups. If time allows: Small Cancellation Groups, Stallings Theorem, Boundaries. lecturer: Panos Papazoglou department: Maths course website: Maths course C3.2