Research
My
interests are differential geometry, twistor theory, quaternionic geometry and related
topics. In
my thesis, I considered a hyperkahler 8-manifold with a 2-torus acting
on it by isometries. I did not require that the action be by
triholomorphic isometries, for this case has already been considered by
Hitchin, Karlhede, Lindstrom, Rocek (see also the related paper by
Pedersen and Poon). I was able to express the metric in terms of a
function on R^6 satisfying some systems of non-linear second order
partial differential equations. These are 8-dimensional analogues of
the Boyer-Finley equation, essentially.
The figure on the right is actually rather irrelevant to my thesis, but it represents a beautiful geometric
example of a conformally compactifiable manifold: it represents hyperbolic n-space and its n-1 dimensional sphere at infinity. I am representing hyperbolic n-space as one sheet of Q(v,v) = -1 with respect to a n+1 dimensional Minkowski space of signature n pluses (for spacelike directions) and 1 minus (for timelike directions). Then the manifold of generators of the null cone Q(v,v) = 0 is an n-1 dimensional sphere, which lies at infinity with respect to hyperbolic n-space.
The figure on the right is actually rather irrelevant to my thesis, but it represents a beautiful geometric
example of a conformally compactifiable manifold: it represents hyperbolic n-space and its n-1 dimensional sphere at infinity. I am representing hyperbolic n-space as one sheet of Q(v,v) = -1 with respect to a n+1 dimensional Minkowski space of signature n pluses (for spacelike directions) and 1 minus (for timelike directions). Then the manifold of generators of the null cone Q(v,v) = 0 is an n-1 dimensional sphere, which lies at infinity with respect to hyperbolic n-space.
