The space of translation surfaces comes equipped with a natural Lebesgue measure, called the Masur-Veech measure, which is the starting point for the ergodic-theoretic study of dynamics on the space of translation surfaces. In fact, a wider class of (locally) affine measures arises naturally, and by work of Eskin-Mirzakhani-Mohammadi, many dynamical questions can be reduced to the study of these 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 in progress of a conjecture 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. The techniques are expected to be useful for other problems concerning affine measures.