Sponsored by the Summer Math Foundation,
the Stony Brook Department of Mathematics,
and Undergraduate Research and Creative Activities (URECA)

We present a twistor correspondence for half-flat almost-Grassmannian structures on real manifolds. An almost-Grassmannian structure is (essentially) a factorization of the tangent bundle, which determines two preferred families of tangent subspaces, and this structure is said to be half-flat if one of these families is integrable. We provide global results when the underlying manifold is a Grassmannian of 2-planes, and show there exist nontrivial deformations of the standard almost-Grassmannian structure. Whereas twistor constructions typically involve moduli of closed curves in a complex manifold, we utilize and expand upon the more flexible approach pioneered by LeBrun and Mason using moduli of curves-with-boundary.

We describe the boundary of linear subvarieties in the moduli space of multi-scale differentials. Linear subvarieties are algebraic subvarieties of strata of (possibly) meromorphic differentials that in local period coordinates are given by linear equations. The main examples of such are affine invariant submanifolds, that is, closures of SL(2,R)-orbits. We prove that the boundary of any linear subvariety is again given by linear equations in generalized period coordinates of the boundary. Our main technical tool is an asymptotic analysis of periods near the boundary of the moduli space of multi-scale differentials which yields further techniques and results of independent interest.