In the mid-80's, John Harer with Don Zagier found a striking formula for the orbifold euler characteristic of the moduli space of curves, which connects it with the values of the Riemann zeta function at negative integers. Here it is:
$$\chi(\mathcal{M_g}) = \frac{\zeta(1-2g)}{2-2g}$$
In my talk, I will try to explain where it comes from. It will involve lots of polygon gluings, generating functions, some combinatorics and a little bit of integration over the space of hermitian matrices.