G_2 structures, rational curves, and ODEs

Consider the space M of parabolas y=ax^2+bx+c, with (a, b, c) as coordinates on M. Two parabolas generically intersect at two (possibly complex) points, and we can define a conformal structure on M by declaring two points to be null separated iff the corresponding parabolas are tangent. A simple calculation of discriminant shows that this conformal structure is flat.

In this talk (based on joint works with Godlinski and Sokolov) I shall show how replacing parabolas by rational plane curves of higher degree allows constructing curved conformal structures in any odd dimension. In dimension seven one can use this "twistor" construction to find G_2 structures in a conformal class.