Being currently on a research fellowship my teaching is on a voluntary basis. I offer projects for 4th and 5th year (masters) students, summer projects and postgraduate research projects.


Below you can find some examples of past projects. They are intended as an indication what a possible project might involve. I usually modify projects to suit individual interests and preliminary knowledge.
 

What does an atom look like?


This is a popular 4h project which exposes you to some mathematical ideas behind quantum mechanics, such as Hilbert spaces and operators. You will solve the time independent Schroedinger equation for the hydrogen atom and discuss its solutions by displaying them graphically using the software package Mathematica.

There is no physics background required. A purely mathematical version of this project would be a study of the spherical harmonic functions (eigenfunctions of the spherical part of the Laplace operator) and their relation to the Lie algebra su(2).


 

Computation of Weyl group orbits and Cayley graphs


Weyl groups are finite reflection groups acting on a vector space. The computation of their orbits can be a formidable challenge even with a computer. In August 2010 I posed this as a summer project for 3rd year student Justin Paston-Cooper who rewrote algorithms of an article by Denis Snow for Mathematica. You can find the resulting Mathematica Demonstration here.


 

Quantum spin-chains and the su(2) Lie algebra


Picture a chain of atoms with magnetic spin which can point up or down. In quantum mechanics the spin degree of freedom is described through Pauli matrices (2 by 2 complex matrices whose trace is zero), they form the simplest representation of the Lie algebra su(2). The dynamics of the quantum spins is governed by an operator (here a matrix) called the Hamiltonian. In 2010 Paul Stephen Milne studied under my supervision the eigenvalue problem of the Hamiltonian using a technique known as Bethe ansatz: the eigenvalues and eigenvectors are described in terms of special polynomials whose roots have the physical interpretation of momenta of quasiparticles known as "spin waves". Computation of the eigenvectors for the Hamiltonian corresponds in the model we studied to finding the irreducible representations of the su(2) Lie algebra.






 

4H Project briefs


Mathematical Physics
Algebra

5H Project briefs

PhD projects

Projects for doctoral students can be found on my research page.

 

Summer projects

I might offer 6-8 week summer projects for interested students. Contact me.

  • R Fornear (2008): Symmetric Functions
  • J Paston-Cooper (2010): Weyl orbits
  • P S Milne (2010): the Heisenberg spin-chain

Course archive