Course subject: Applied Mathematics (AMATH)

The Hilbert space of states, observables, and time evolution. Feynman path integral and Greens functions. Approximation methods. Co-ordinate transformations, angular momentum, and spin. The relation between symmetries and conservation laws. Density matrix, Ehrenfest theorem, and decoherence. Multiparticle quantum mechanics. Bell inequality and basics of quantum computing.

Tensor analysis. Curved space-time and the Einstein field equations. The Schwarzschild solution and applications. The Friedmann-Robertson-Walker cosmological models.

Elements of asymptotic analysis. Techniques of perturbation theory such as Poincare-Lindstedt, matched asymptotic expansions and multiple scales. Applications to various areas form an essential aspect of the course. Previous courses in real analysis and differential equations at the undergraduate level are strongly recommended.

Discretization methods for partial differential equations, including finite difference, finite volume and finite element methods. Application to elliptic, hyperbolic and parabolic equations. Convergence and stability issues, properties of discrete equations, and treatment of non-linearities. Stiffness matrix assembly and use of sparse matric software. Students should have completed a course in numerical computation at the undergraduate level.

The main themes are well-posedness of problems, Hilbert space methods, variational principles and integral equation methods. Topics include: first-order nonlinear partial differential equations, quasilinear hyperbolic systems, potential theory, eigenfunctions and eigenvalues, semi-groups, and power series solutions. Applications from various areas will be used to motivate and illustrate the theory. A previous course in partial differential equations at the undergraduate level is strongly recommended.

Concepts of stability and boundedness, basic stability criteria, comparison methods, large scale systems, method of decomposition and aggregation, method of several Lyapunov functions, method of vector Lyapunov functions, method of higher derivatives. Stability problems in ecology, mechanics, neural networks and control systems. Students should have completed AM751 or equivalent.

Mathematical methods, stability of parellel flows for unstratified and stratified fluids, Rayleigh-Taylor instability, centrifugal instability, barotropic and baroclinic instabilities, the effects of viscosity and the Orr-Sommerfeld equation, transition to turbulence, averaged equations, closure problem, homogeneous isotropic turbulence, turbulent boundary layers, effects of stratification. Students should have completed an introductory undergraduate course in fluid mechanics.

Review of basics of quantum information and computational complexity; Simple quantum algorithms; Quantum Fourier transform and Shor factoring algorithm: Amplitude amplification, Grover search algorithm and its optimality; Completely positive trace-preserving maps and Kraus representation; Non-locality and communication complexity; Physical realizations of quantum computation: requirements and examples; Quantum error-correction, including CSS codes, and elements of fault-tolerant computation; Quantum cryptography; Security proofs of quantum key distribution protocols; Quantum proof systems. Familiarity with theoretical computer science or quantum mechanics will also be an asset, though most students will not be familiar with both.

Review of relativistic quantum mechanics and classical field theory. Quantization of free quantum fields (the particle interpretation of field quanta). Canonical quantization of interacting fields (Feynman rules). Application of the formalism of interacting quantum fields to lowest¿order quantum electrodynamic processes. Radiative corrections and renormalization.

Introduction to the differential geometry of Lorentzian manifolds. The priniciples of general relativity. Causal structure and cosmological singularities. Cosmological space-times with Killing vector fields. Friedmann-Lemaitre cosmologies, scalar, vector and tensor perburbations in the linear and nonlinear regimes. De Sitter space-times and inflationary models.

Review mathematical formulation of operational quantum theory; theory of measurements and decoherence; quantum-classical contrast; review of historical perspectives on interpretation, including EPR paradox; Bell's theorem, non-locality and contextuality; PBR theorem; selected topics including overviews of current interpretations of quantum mechanics and critical experiments in quantum foundations.

Dynamic mathematical modelling of biological process at the cellular level. Intracellular networks: metabolism, signal transduction, and genetic regulatory networks. Neural networks: from biophysical modelling of single neurons to the analysis of network behaviour. Modelling will be carried out primarily through ordinary differential equations; analysis will involve application of dynamical systems tools and simulations. Other relevant modelling frameworks (PDEs, delay equations, stochastic methods) will be touched on as time allows.