Postgraduate Courses

The aim is to train the PhD students associated with the Condensed Matter Theory Group in mathematical and computational methods. There are two compulsory components in the postgraduate course: Mathematical and Computational Methods and Directed Reading.

The Mathematical and Computational Methods course is associated with the Doctoral Training Centre (DTC) in Theory and Simulations of Materials. Students are required to follow all the lectures and the problem solving classes. Students should also take the exam at the end of the course. For the numerical methods classes, the students will require a laptop. (A laptop is provided by the DTC for DTC students only.)

For the Directed Reading, we urge that it includes problem solving and/or project based exercises and that some form of assessment is undertaken.

The courses will take place at the Whiteley Suite, Chemistry Building. [ timetable ]

Organiser: Kim Christensen.

Mathematical Methods

• Vector Spaces & Tensors (3 lectures, Dr. Arash A. Mostofi).
Definition of a vector space; dimensionality, orthogonality, linear dependence. Schwarz inequality. Eigenvectors, eigenvalues and normal modes. Tensor algebra, suffix notation, rank. Transformation law. Tensor fields. Representation of grad, div and curl. Principal axes and diagonalisation. Neumann's principle. Introduction to non-Euclidean tensors: contravariant and covariant tensors; metric tensor.
• Green's Functions (2 lectures, Dr. Arash A. Mostofi).
Variation of parameters method for inhomogeneous ODEs. Green's functions for initial value and boundary value problems.
• Hilbert Spaces (4 lectures, Dr. Arash A. Mostofi).
Definition of a Hilbert space; dimensionality, orthogonality, linear dependence, Wronskian. Sturm-Liouville Theory; self-adjoint operators, eigenfunctions, eigenvalues, weight function. Eigenfunction expansions, completeness. Examples of orthogonal functions to include spherical harmonics and Legendre polynomials.
• Integral Transforms (3 lectures, Dr. Arash A. Mostofi).
Continuous and discrete Fourier transforms. Laplace transforms.
• Complex Analysis and Contour Integration (4 lectures, Prof. Adrian P. Sutton).
Cauchy-Riemann relations, analytic functions, Laurent's theorem. Residue theorem. Cauchy's theorem. Principal values. Jordan's lemma. Inverse integral transforms.
• Partial Differential Equations (4 lectures, Prof. Adrian P. Sutton).
Derivation from conservation laws. Laplace, Poisson, diffusion, and wave equations. Separation of variables. Solution by Green's function, integral transform, and eigenfunction expansion methods. Series solution of ODEs.
• Response Theory (2 lectures, Dr. Paul Tangney).
Response functions, connection to Green's functions, Dyson equation, Kramers-Kronig relations (Hilbert transform). Linear response, fluctuation-dissipation theorem.
• Calculus of Variations (4 lectures, Dr. Paul Tangney).
Functionals and their differentiation. Euler-Lagrange equations. Symmetry and Noether's theorem. Variational principles. Lagrange multipliers. Rayleigh-Ritz methods; connection to Sturm-Liouville problems and constrained minimisation.
• Additional Contacts - 10 Rapid Feedback (problem solving) Classes.

Computational Methods

• Introduction to *nix and C++ (8 lectures, Dr. James Spencer)
1. Introduction to command line. Files and directories. Help. Text editors. History. Environment variables.
2. Handling data. Input, output and redirection.
3. Version control.
4. Simple scripts. Executing programs.
5. Basic structure of a C++ program. Compilation and execution. Variable types, declaration and assignment. Return codes. Simple arithmetic expressions. Order of precedence.
6. Loops and conditional expressions. Functions. Basic input and output. Elementary maths functions.
7. Arrays and pointers. Libraries.
8. Debugging.


• Numerical methods (17 lectures, Dr. Peter D. Haynes)
1. Representing numbers on a computer. Underflow and overflow. Loss of accuracy. Quadratic equations.
2. Root finding. Bracketing. Bisection. Secant method. Newton-Raphson method.
3. Linear interpolation. Cubic splines. Chebyshev polynomials and Padé approximants.
4. Quadrature. Trapezium rule. Romberg integration. Gaussian quadrature and orthogonal polynomials.
5. Recurrence relations. Stability of iteration. Continued fractions. Steed's method for Bessel functions.
6. Discrete Fourier transforms. Danielson-Lanczos lemma and fast Fourier transforms. Fourier interpolation.
7. Ordinary differential equations. Finite differences. Euler method. Runge-Kutta methods.
8. Partial differential equations. Crank-Nicholson method for the diffusion equation.
9. Representing vectors and tensors on a computer: arrays. Basic linear operations. Scaling of operations. Memory hierarchy. Matrix-matrix multiplication. Blocking and introduction to numerical libraries: BLAS/LAPACK.
10. Matrix analysis. Range, null-space and rank. Vector and matrix norms. Singular value decomposition.
11. Solving general systems of linear equations. Matrix representation. Formal solution in terms of inverse. Gaussian elimination with full or partial pivoting. LU decomposition and back substitution. Evaluating determinants.
12. Special linear systems. Tridiagonal and banded matrices. Positive definite matrices and the Cholesky decomposition. Sparse systems.
13. Symmetric eigenproblems and matrix diagonalisation. Householder and Givens reductions. Symmetric QR.
14. Iterative methods. Power method. Formulation as an optimisation problem. Steepest descents. Conjugate gradients.
15. Random numbers. Monte Carlo methods.

Directed Reading