abstract: Square-tiled surfaces (STSs) are branched covers of the standard square-torus $\mathbb{C}/\mathbb{Z}[i]$ branched exactly over a single point.
Geometrically, they can be thought of as surfaces built out of Euclidean unit squares, with sides identified in parallel pairs.
In this talk we will use a combinatorial model to explore the topological and geometrical statistics of square-tiled surfaces.
In particular, we will study the expected genus of an $n$-square-tiled surface and see the rareness and frequency in STSs of some
geometrical properties of the square-torus.