In the moduli space of Abelian differentials, big invariant subvarieties come from topology!- Paul Apisa (Yale University)
Abstract: It is a beautiful fact that any holomorphic one-form on a genus g Riemann surface can be presented as a collection of polygons in the plane with sides identified by translation. Since GL(2, R) acts on the plane (and polygons in it), it follows that there is an action of GL(2, R) on the collection of holomorphic one-forms on Riemann surfaces. This GL(2, R) action can also be described as the group action generated by scalar multiplication and Teichmuller geodesic flow. By work of McMullen in genus two, and Eskin, Mirzakhani, and Mohammadi in general, given any holomorphic one-form, the closure of its GL(2, R) orbit is an algebraic variety. While McMullen classified these orbit closures in genus two, little is known in higher genus. In the first part of the talk, I will describe the Mirzakhani-Wright boundary of an invariant subvariety (using mostly pictures) and a new result about reconstructing an orbit closure from its boundary. In the second part of the talk, I will define the rank of an invariant subvariety - a measure of size related to dimension - and explain why invariant subvarieties of rank greater than g/2 are loci of branched covers of lower genus Riemann surfaces. This will address a question of Mirzakhani. No background on Teichmuller theory or dynamics will be assumed. This material is work in progress with Alex Wright.
Translation surfaces with multiple short saddle connections- Ben Dozier (Stony Brook University)
Abstract: The SL_2(R) action on strata of translation surfaces allows us to answer many questions about the straight-line flow on individual translation surfaces (and this flow is in turn closely connected to billiards on rational polygons). By the pioneering work of Eskin-Mirzakhani, to understand dynamics on strata one is led to study "affine" measures. It is natural to ask about the interaction between volumes of certain subsets of surfaces and the geometric properties of the surfaces. I will discuss a proof of a bound on the volume, with respect to any affine measure, of the locus of surfaces that have multiple independent short saddle connections. This is a strengthening of the regularity result proved by Avila-Matheus-Yoccoz. A key tool is the new smooth compactification of strata due to Bainbridge-Chen-Gendron-Grushevsky-Moller, which gives a good picture of how a translation surface can degenerate.
What does your average translation surface look like?- Anja Randecker (Heidelberg University)
Abstract: Almost every talk on translation surfaces starts with a double pentagon or with an octagon. But are these n-gons generic examples of translation surfaces? We will look at this question from the point of view of the geometry of the translation surfaces. Specifically, we prove an upper bound for the average diameter (resp. covering radius) in a stratum of translation surfaces of large genus. The first part of my talk assumes only basic knowledge about translation surfaces, the second part assumes more background to give a sketch of the proof. Both are based on joint work with Howard Masur and Kasra Rafi (available at https://arxiv.org/abs/1809.10769).