Normal functions and the geometry of moduli spaces of curves

In this talk, I will begin by recalling the classification of normal functions over $\mathcal M_{g,n}$, the moduli space of $n$-pointed smooth projective curves of genus $g$. I'll then explain how they can be used to resolve a question of Eliashberg, how they generate the tautological ring of $\mathcal M_{g,n}$, and how they can be used to strengthen slope inequalities of the type proved by Moriwaki.