I will discuss recent work calculating the top weight cohomology of the moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. The key idea is that this piece of cohomology is encoded combinatorially via the relationship between the boundary complex of a compactification of $\mathcal{A}_{g}$ and the moduli space of tropical abelian varieties.

This is joint work with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.