p-torsion of curves in characteristic p

The Torelli locus---the image of the moduli space of curves (Mg) in
the moduli space of abelian varieties (Ag)---is much-studied but still
mysterious.  In characteristic p, Ag has a beautiful stratification by
the isomorphism type of A[p], and examples show that Mg is far from
transverse to this stratification.  In an ongoing project, we develop
tools to understand (and perhaps make principled conjectures about)
which strata of Ag meet Mg.  In this talk, we explain some of the
structures involved (including a crash course on finite group schemes)
and then give some new results about them.  Parts of this are joint
work with Bryden Cais and Rachel Pries.