The unstable foliation, that changes the horizontal components of period coordinates, plays an important role in the study of translation surfaces, including their deformation theory, and in the understanding of horocycle invariant measures.

In this talk, we show that measures of large dimension equidistribute in affine invariant manifolds and give an effective rate. An analogous result in the setting of homogeneous dynamics is crucially used in the effective equidistribution results of Lindenstrauss-Mohammadi and Lindenstrauss--Mohammadi--Wang. Background knowledge on translation surfaces and homogenous dynamics will be explained.