For a stratum of translation surfaces, we can consider different notions of the thin part: the subset of translation surfaces that have a short saddle connection or the subset of translation surfaces that have a cylinder with short circumference.

A result of Masur-Smillie shows that the volume of the thin part for saddle connections is quadratic in the length of the saddle connection. However, the dependence of the volume on the genus was not known.

We will review some methods from Eskin-Masur-Zorich and apply these in the case of the minimal stratum. With these, we bound the expected number of saddle connections up to a given length and the number of cylinders up to a given circumference. This gives us explicit, genus-dependent bounds on the volumes of the corresponding thin parts of minimal strata.

This talk is based on work with Howard Masur and Kasra Rafi (available at https://arxiv.org/abs/1809.10769).