Compactifications of strata of holomorphic one-forms

Abstract: The moduli space of compact Riemann surfaces X equipped with
a holomorphic one-form w has a natural stratification according to the
orders of the zeoes of w.   While in recent years tremendous advances
have been made in understanding important dynamical systems on these
strata, from the point of view of global geometry and topology, strata
of holomorphic one-forms remain largely mysterious.  In this talk, we
discuss recent work on constructing a smooth compactification of these
strata which is analogous to the Deligne-Mumford compactification of
the moduli space of curves.  This is joint work with Chen, Gendron,
Grushevsky, and Moeller.