Research
I am a graduate student at the University of Edinburgh specialising in Differential Geometry, specifically the geometry of spinors. Currently my focus is on the use of conformal submersions in solving the twistor equation and generalisations thereof.
My current supervisor is Professor José Figueroa-O'Farrill, who has kindly taken over this year from Professor Dmitri Alekseevsky.
Some Notes
Here are some notes I've made and talks I've given whilst doing my PhD. If you find any mistakes, or maybe just want a clearer explanation of something then send me an email and I'll be happy to help.
Spin Spaces - These are some detailed notes on Clifford and spin representations I built up, including lists of the various real, complex and quaternionic structures they admit.
Clifford Representations Adding a Vector - An explanation of what happens to a complex spin representation when you add an extra dimension to your underlying inner product space.
The Fefferman-Graham Construction - A talk I gave on the method of Fefferman and Graham of constructing invariants of a conformal structure by building a bigger 'ambient' pseudo-Riemannian manifold and stealing its invariants.
G-Structures - These are the notes from a talk I gave a while back introducing G-structures, a very important framework in which we can view metrics, symplectic forms, complex structures and many other things merely as different reductions of the frame bundle of a manifold.
Complex and Quaternionic Structures as Matrices - Complex and quaternionic structures written as matrices, so you can understand these ideas easily.
The Maurer-Cartan Equation - A very brief derivation of the equation satisfied by the canonical form on a Lie group.
Associated Vector Bundles - I have not come across a proper set of notes on associated vector bundles so I wrote some myself. A word of motivation precedes the basic definitions and plenty of examples, and this is followed by some useful more technical points.
Teaching
During my time as a postgraduate student I have taught a variety of topics to first and second year undergraduates.