In her thesis, Mirzakhani proved that the number of simple closed geodesics (that is, closed geodesics without self-intersection) of length at most $L$ in a fixed hyperbolic surface $X$ is asymptotic to a constant times $L^{6g - 6}$, as $L \to \infty$ (here, $g \geq 2$ is the genus of $X$). More recently, Eskin-Mirzakhani-Mohammadi presented an alternative proof of Mirzakhani's theorem by relating the count in question to lattice point counts in Teichmuller balls of large radius, a count that was previously studied by Athreya-Bufetov-Eskin-Mirzakhani. Their approach enabled them to obtain a more precise result, namely, to obtain a power-saving error term for the count in the question.
In this short course, we aim to highlight the connection between lattice point counts in Teichmuller space and simple closed geodesic counts on a fixed hyperbolic surface. Our exposition follows in part Arana-Herrera's 2021 paper which obtains power-saving error terms for a closely related count (that is, mapping class group orbits of a fixed filling curve), and follows in part a work in progress of mine.

More information: https://drive.google.com/drive/folders/1_0qR8OtN737UFGdYwt3jkhl86Nz7zmUE?usp=sharing