  An increasingly important area of interest for mathematicians is the study of
Abelian differentials. This growing interest can be attributed to the
interdisciplinary role this subject plays in modern mathematics, as various
problems of algebraic geometry, dynamical systems, geometry and topology lead
to the study of such objects. It comes as a natural consequence that we can
employ in our study algebraic, analytic, combinatorial and dynamical
perspectives. These lecture notes aim to provide an expository introduction to
this subject that will emphasize the aforementioned links between different
areas of mathematics. We will associate to an Abelian differential a flat
surface with conical singularities such that the underlying Riemann surface is
obtained from a polygon by identifying edges with one another via translation.
We will focus on studying these objects in families and describe some
properties of the orbit as we vary the polygon by the action of
$GL_2^{+}(\mathbb{R})$ on the plane.
