My main fields of interest are differential geometry and twistor theory, in particular, I have been studying lately quaternionic-kahler 8-manifolds that have a conformal compactification, and the relationship between the 8-manifold and its geometry at infinity, by making use of the associated twistor space.

What is differential geometry? I try to answer this question in a very subjective way here.

Consider hyperbolic space Hⁿ, with its usual model given by an open ball in ⁿ, and its usual metric. Note that there is a natural sphere S^n-1 at infinity. What kind of structure should we give to that sphere at infinity? After looking at the particular expression of the hyperbolic metric, it seems that it might be a good idea to endow that S^n-1 with the conformal structure represented by the round metric.

Thus we have, on one hand an n-dimensional Riemannian manifold Hⁿ, and on the other we have a hypersurface at infinity S^n-1 with its natural conformal structure. How are these 2 related? Hyperbolic space Hⁿ has 1 dimension greater than S^n-1 but its structure is a Riemannian metric. On the other hand, S^n-1 might have 1 dimension less, but its structure is a conformal structure, which is less restrictive locally. It turns out however that their global automorphism groups are the same. Thus from the point of view of global structure (ie the global automorphism group), hyperbolic space is equivalent (from that point of view) to its geometry at infinity.

Thus it seems that the geometry at infinity could hold a lot of information about the original space. Claude LeBrun constructed out of a real analytic Lorentzian (3, 1) manifold with (real analytic) conformal structure, a quaternionic-kahler 8-manifold whose geometry at infinity was essentially the original 4-manifold. It makes use of the twistor space of the quaternionic-kahler 8-manifold, which is constructed (the twistor space that is) from the space of null geodesics of a complexification of the original 4-manifold. My thesis investigates further this construction, and thus involves a mixture of real and complex geometry, twistor theory and some Lie groups.