Contents -- Preface -- 1 The Dynamo Effect -- 1 Introduction: from industrial dynamos to the magnetic field of stars and planets -- 1.1 Generation of an electric current by a rotating conductor -- 1.2 Magnetic fields of astrophysical objects -- 1.3 Fluid dynamos -- 2 The MHD approximation -- 2.1 The approximations and equations of magnetohydrodynamics -- 2.2 Relation with a two-fluid model -- 2.3 Boundary conditions -- 2.4 Relevant dimensionless numbers -- 2.5 Some limits of the MHD equations -- 3 Advection and diffusion of a passive vector -- 3.1 More or less useful analogies -- 3.2 Accelerated diffusion -- 3.3 Local amplification -- 3.4 Expulsion of a transverse magnetic field from a rotating eddy -- 3.5 Effect of differential rotation and the Herzenberg dynamo -- 4 Necessary conditions for dynamo action -- 4.1 Magnetic energy -- 4.2 The critical magnetic Reynolds number -- 4.3 "Antidynamo theorems" -- 5 Laminar dynamos -- 5.1 The Ponomarenko dynamo -- 5.2 Spatially periodic dynamos -- 6 Turbulent dynamos -- 6.1 Dynamos generated by small scale turbulence? -- 6.2 The dynamo threshold at large kinetic Reynolds number -- 6.3 Effect of turbulence on the dynamo threshold -- 6.4 Is it useful to have scale separation? -- 7 Saturation of the magnetic field above the dynamo threshold -- 7.1 Laminar dynamos -- 7.2 Saturation in the high Re or small Pm limit -- 8 Conclusion -- 2 An Introduction to Equivariant Bifurcations and Spontaneous Symmetry Breaking -- 1 Introduction -- 2 The classical bifurcation theory -- 2.1 Elementary bifurcations -- 2.2 Bifurcation with symmetry: a general approach -- 2.3 Existence theory -- 3 Equivariant dynamical systems -- 3.1 The Hopf bifurcation -- 3.2 Relative equilibria and relative periodic orbits -- 3 Bose-Einstein Condensation of Trapped Atomic Gases -- 1 Introduction.

2 Creating Bose-Einstein condensates -- 2.1 Bose-Einstein condensation in alkalis -- 2.2 Other systems -- 3 Measuring Bose-Einstein condensates -- 3.1 Density and velocity distributions -- 3.2 Imaging techniques -- 3.3 Measuring other quantities -- 4 Manipulating Bose-Einstein condensates -- 4.1 Dipole force -- 4.2 Coherent coupling -- 4.3 Resonant scattering -- 4.4 Manipulation of collisions -- 5 Experiments on Bose-Einstein condensates -- 5.1 Thermodynamics -- 5.2 Dynamics and superfluidity -- 5.3 Atom optics -- 5.4 Interactions with light -- 5.5 Collision resonances -- 6 Conclusion -- 4 An Introduction to Bose-Einstein Condensation Experiments -- 1 Introduction -- 2 Magnetic trap -- 2.1 Preliminary -- 2.2 The Ioffe trap -- 2.3 Noise-induced heating -- 2.4 Collisions -- 3 Adiabatic compression -- 3.1 Preliminaries -- 3.2 How to modify the phase space density by adiabatic compression? -- 3.3 Collision rate -- 4 Collective oscillations -- 4.1 Averaging method -- 4.2 Monopole mode -- 4.3 Dipole mode -- 4.4 Quadrupole mode -- 4.5 The role of the classical mean field -- 5 Evaporative cooling -- 5.1 Evaporation in the collisionless regime -- 5.2 Role of collisions forced evaporative cooling -- 6 Bose-Einstein condensation -- 6.1 The Gross-Pitaevskii equation -- 6.2 Hydrodynamic description -- 6.3 Superfluidity -- 7 Collective excitations: BEC and the classical gas in the hydrodynamic regime -- 7.1 Introduction -- 7.2 Equation of continuity -- 7.3 Equation for the velocity field -- 7.4 Collective modes -- 8 Rotational properties of a BEC -- 8.1 Moment of inertia -- 8.2 Introduction to Vortices -- 8.3 The rotating bucket experiment -- 8.4 Vortex nucleation -- 5 Highlights of the Theory of Bose-Einstein Condensates -- 1 Introduction and outline of the set of lectures.

2 Equilibrium thermodynamics of a dilute B-E gas at low temperature -- 2.1 Thermodynamics of the dilute Bose gas with an energy perturbed to first order -- 2.2 Thermodynamics of the dilute Bose gas -- 3 Kinetic theory of a dilute Bose gas -- 3.1 Boltzmann-Nordheim kinetic theory and B-E condensation -- 3.2 Dynamics before collapse -- 3.3 Post collapse dynamics -- 4 Kinetic theory and the Gross-Pitaevskii equation -- 4.1 The kinetic theory at the mean field order -- 4.2 Hydrodynamics of the coupled kinetic equations -- 5 The Gross-Pitaevskii equation and superfluid mechanics: an example -- 5.1 The general solution of the Euler-Tricomi equation near a body -- 6 Summary and conclusion -- 6 Mixing -- 1 Interaction leads to uniformity -- 2 The elementary interaction -- 2.1 Composition rule -- 3 Practical mixtures: limit cases -- 3.1 The confined mixture -- 3.2 The ever dispersing mixture -- 3.3 Kinetics -- 3.4 Spectral consequences -- 4 Changing dimensions -- 5 Kinetic equation -- 6 Conclusion -- 7 Flow of Complex Fluids -- 1 Introduction -- 1.1 Brief review of Newtonian fluid flow -- 1.2 Interaction structure/flow: criteria for non-Newtonian effects -- 1.3 Rheological measurements -- 1.4 Elongational viscosity -- 2 Examples of interaction flow/structure -- 3 Non-Newtonian shear viscosity -- 4 Elastic effects in non-Newtonian fluids -- 4.1 Viscoelastic fluids: first normal stress difference -- 4.2 Normal stress effects on free boundary problems -- 4.3 Elongational viscosity -- 5 Yield stress fluids -- 8 Elements of Similarity and Singularity -- 1 Introduction -- 1.1 Thales and the origin of similarity -- 1.2 The selection of relevant parameters -- 1.3 Similarity solution of the diffusion equation -- 2 Free surface problems in the lubrication approximation -- 2.1 Thin film equations.

2.2 Nonlinear diffusion and the spreading of a drop -- 2.3 The capillary regime of drop spreading -- 3 Second type similarity in a porous medium -- 3.1 The renormalization group -- 3.2 The Barenblatt equation -- 3.3 Perturbation expansion and renormalization group solution -- 4 Shocks Burgers equation and finite time singularities -- 9 Topics on Volume Preserving Maps and Permutations Related to Hydrodynamics and Electrostatics -- 1 A dynamical system involving permutations -- 2 Volume preserving maps -- 2.1 Definition -- 2.2 Volume and orientation preserving diffeomorphisms -- 2.3 Permutation maps -- 2.4 Dense subsets of volume preserving maps -- 2.5 Doubly stochastic measures -- 3 Geometric interpretation of the Euler equations -- 3.1 Usual description of the Euler equations -- 3.2 Geometrical interpretation of the Euler equation -- 3.3 The Shortest Path Problem -- 4 Approximate geodesies and nonlinear electrostatics -- 4.1 Definition of approximate geodesies -- 4.2 The time-discrete SPP -- 4.3 The closest point problem and the polar factorization of Maps -- 4.4 A nonlinear cutoff for electrostatics -- 4.5 The discrete approximate geodesic equation.