Geometry and Dynamics in the Teichmüller and Moduli Spaces of Flat Surfaces

This work consists of two projects studying geometry and dynamics in the Teichmüller and moduli spaces of quadratic and Abelian differentials. In the first half of this thesis, we prove that the Masur-Veech volume of an Avila-Gouëzel-Yoccoz ball of a sufficiently small fixed radius is uniformly bounded on the principal stratum of quadratic differentials. We also give some estimates on the Avila-Gouëzel-Yoccoz norms of the vectors of a certain relative cohomology basis that captures the geometry of the flat metric on the underlying surface. In the second half, we show that for almost every choice of the slit the slit double cover of a fixed translation surface with uniquely ergodic vertical flow also has uniquely ergodic vertical flow.