Abstract: The Teichmüller flow acts on the moduli space of the space of quadratic differentials on Riemann surfaces with prescribed zeros and poles, known as “strata".
I will speak on the volumes of these strata, which can be evaluated in terms of multiple zeta values. This field has seen work by Masur, Veech, Konsevitch, McMullen, Eskin, Mirzakhani, Okunkov, Athreya, Goujard, …
But the underlying idea is straightforward. Suppose you have a pile of $N$ unit squares of lined paper, and you wish to assemble them edge to edge so that the lines continue each other, as to form a surface of genus g (say 2, or 3). How many ways are there of doing it, and more specifically, how does the number grow with $N$.
If time permits, I will then sketch why the Teichmüller flow is ergodic on the strata.