Tropical curves, graph complexes, and the cohomology of M_g

Joint with Søren Galatius and Sam Payne. The cohomology ring H^*(M_g,Q)
of the moduli space of curves of genus g is not fully understood, even
for g small. For example, in the 1980s Harer-Zagier showed that the
Euler characteristic (up to sign) grows super-exponentially with g
---yet most of this cohomology is not explicitly known. I will explain
how we obtained new results on the rational cohomology of moduli
spaces of curves of genus g, via Kontsevich's graph complexes and the
moduli space of tropical curves.