The moduli space of Riemann surfaces X equipped with a holomorphic one-form w can be stratified by prescribing the orders of the zeors of w. These strata have attracted considerable interest as the natural domain for Teichmuller dynamics, but are also very interesting and mysterious from the point of view of algebraic geometry, geometric topology, etc.

In this talk we discuss work in progress with Chen, Gendron, Grushevsky, and Moeller on constructing a "good" compactification of these strata which could be viewed as an analog of the Deligne-Mumford compactification of the moduli space of curves.

This talk will complement (but not depend on) Grushevsky's recent talk on this work at the Fields Institute.