Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.
In this lecture series, we will describe new joint work that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by π symmetry.
We define "sufficiently large" in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.
The five lectures will introduce novel and broadly applicable techniques, organized as follows:
An introduction to orbit closures, their rank, their boundary in the WYSIWYG partial compactification, and cylinder deformations.
Reconstructing orbit closures from their boundaries (this talk will explicate a preprint of the same name).
Recognizing loci of covers using cylinders (this talk will follow a preprint titled “Generalizations of the Eierlegende-Wollmilchsau”).
An overview of the proof of the main theorem; marked points (following the preprint “Marked Points on Translation Surfaces”); and a dichotomy for cylinder degenerations.
Completion of the proof of the main theorem.
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