Counting square-tiled surfaces with prescribed real and imaginary foliations

Let X be a closed, connected, hyperbolic surface of genus 2. Is it more likely for a simple closed geodesic on X to be separating or non-separating? How much more likely? In her thesis, Mirzakhani gave very precise answers to these questions. One can ask analogous questions for square-tiled surfaces of genus 2 with one horizontal cylinder. Is it more likely for a such a square-tiled surface to have separating or non-separating horizontal core curve? How much more likely? Recently, Delecroix, Goujard, Zograf, and Zorich gave very precise answers to these questions. Surprisingly enough, their answers were exactly the same as the ones in Mirzakhani’s work. In this talk we explore the connections between these counting problems, showing they are related by more than just an accidental coincidence.