The twistor program of Roger Penrose
has always had an appeal, for it places light particles as the
fundamental objects from which space-time should be derived. This
would seem to be a reasonable premise. In early work in this
domain, I studied the Riemannian analogue of spinor equations which
lead to integral formulae for massless fields, showing their
connections to harmonic morphisms and semi-conformal maps.
With my doctoral student Mohammad Wehbe, we generalize the notion of
shear-free congruence on Minkowski space to other space-times by
solving a Cauchy-type problem, which simultaneously produces a metric
and a ray congruence (Journal Math. Phys. 2012). In further work
with M. Wehbe, we show how the basic objects of twistor theory can
naturally be defined on a combinatorial graph. This once more
shows how a graph contains a lot of information that may be realized in
the right context (topic one above). The value zero must belong
to the geometric spectrum in order that it admit the appropriate
structure. The twistor correspondence now translates to the
correspondence between a graph and its line graph (Comm. Math. Phys.
2011).