"Symmetry, Pure
and (not so) Simple": An essay attempting to describe some
differential geometry with a minimum of technical jargon.
For the last few years I have been trying to understand various kinds of geometric structures (Riemannian, conformal, projective, symplectic, Poisson, etc.) from the "symmetry" point of view, i.e., through the lens of Lie theory.
Most recently, I recast Cartan's method of equivalence in the completely invariant language of Lie algebroids. Using this method one can construct, in a completely invariant fashion, the curvature invariants of geometric structures. These are the invariants measuring the local obstruction to the existence of infinitesimal isometries. Previous implementations of Cartan's method have depended on local coordinate calculations or on the a priori choice of a model (e.g., Euclidean space, in the case of Riemannian structures). I am currently applying my reformulation of Cartan's method to uniformization problems in differential geometry.
Past research has included Hamiltonian perturbation theory, geometric mechanics, and fluid mechanics.
I carry out my mathematical research from home - or, more often, during tramps through the beautiful New Zealand bush - while homeschooling two daughters and one son. I am married to Thyra Blaom, a 777 pilot for Air New Zealand.