The twistor approach to four-dimensional geometry has been
very fruitful for physicists working in Minkowski space-time as well as
for mathematicians working in Riemannian (positive definite signature)
geometry.  It also provides a framework for relating these two subjects
via analytic continuation.

The way symmetries work in a quantum field theory changes significantly
when one formulates such a theory in positive definite signature.  I'll
argue that in this context the symmetries of twistor theory match well
the symmetries of the Standard Model as well as a chiral formulation of
general relativity, providing an intriguing possibility for a unified
theory.