Abstract: Questions about billiards on rational polygons can be converted into questions about the straight-line flow on translation surfaces. These in turn can be converted (via renormalization) into questions about the dynamics of the SL_2(R) action on strata of translation surfaces. 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 measures 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 compactification of strata due to Bainbridge-Chen-Gendron-Grushevsky-Moller, which gives a good picture of how a translation surface can degenerate.